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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

A vaccination strategy based on linearization control techniques for fighting against epidemic diseases propagation

Santiago Alonso-Quesada1*, Manuel De la Sen1, Asier Ibeas2 and Raúl Nistal1

Author Affiliations

1 Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country, UPV/EHU, Campus de Leioa, Leioa, Bizkaia, 48940, Spain

2 Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, 08193, Spain

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Advances in Difference Equations 2013, 2013:364  doi:10.1186/1687-1847-2013-364

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/364


Received:7 February 2013
Accepted:15 November 2013
Published:12 December 2013

© 2013 Alonso-Quesada et al.; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper presents a vaccination strategy for fighting against the propagation of epidemic diseases. The disease propagation is described by a SIRS (susceptible plus infected plus recovered populations) epidemic model. The model takes into account that the recovered individuals lose the disease immunity after a finite time period. A control technique based on a model linearization approach is used to design the vaccination strategy in order to eradicate the infection from the population. Moreover, the controlled system is guaranteed to be positive and stable under such a vaccination control strategy. A simulation example illustrates the theoretical results relative to the stability and positivity of the controlled system while guaranteeing the eradication of the epidemics.

MSC: 93C10, 93C15, 93C95, 34D20, 34H15.

Keywords:
SIRS epidemic models; vaccination; nonlinear control; stability; positivity

1 Introduction

The study of propagation of epidemic diseases within a host population has been broadly dealt with for the last decades [1-3]. In this context, several epidemic models have been proposed for analyzing the behavior of different infectious diseases. Among them, the so-called SIRS epidemic model has been used to describe the propagation of epidemic diseases as, for instance, influenza, tetanus, diphtheria, hepatitis A and so on. Such diseases are characterized, among other features, by the fact that the immunization induced by vaccination is only temporary, i.e., such an immunization is not permanent. This feature motivates the use of a SIRS model to explain the disease propagation within the host population [3,4]. In such a compartmental model, the host population is divided into three categories, each one relative to an epidemic status. Namely, there is a susceptible (S) population composed of either individuals who have not been previously exposed to the pathogen or recovered (R) individuals who have lost the immunity after having successfully cleared the infection. Finally, there is an infected (I) population composed of individuals who are currently colonized by the pathogen. Moreover, there are three transitions between such population categories. The first one is the transition from the susceptible population to the infected one. Such a transition occurs when there is a contact between a susceptible individual and an infected one. In such situations there is a probability of disease transmission from the infected individual to the susceptible one who becomes infected after a latent period. This latent period is not taken into account in a SIRS model. However, the model can be extended to include such a time period giving place to a SEIRS model [2]. The transition from the infected category to the recovered one occurs when an infected individual has fought off the infection. Such a transition occurs after an infection period. Finally, the transition from the recovered category to the susceptible one occurs when a recovered individual loses the temporary immunity [1].

One of the purposes of analyzing epidemic models concerns the design of control strategies in order to eradicate the persistence of the infection in the host population. In this context, constant, regular and/or impulsive vaccination strategies have been proposed in several research papers [5-10]. Concretely, a regular vaccination strategy based on a feedback control law for exact input-output linearization of a SEIR epidemic model is designed in [8]. Moreover, such a control strategy is combined with an observer in [9]. The observer is designed to estimate on-line the susceptible and the infected (or exposed) populations since such measures are not available in a real situation where only the infectious population is measurable. The estimates provided by the observer are used to synthesize the control law instead of the true susceptible and infected populations. Such strategies achieve the eradication of the disease from the host population while guaranteeing the positivity and stability of the controlled system. Other potential situation is that the parameters of the epidemic model are not fully known, which may be circumvented by using adaptive control strategies [11].

Linearization and impulsive control techniques have been satisfactorily used for stabilizing highly nonlinear processes in applications of diverse nature. On the one hand, an active magnetic bearing system composed of a spindle rotating at different speeds is stabilized by means of a control strategy based on a linearization of the process in [12]. On the other hand, a fuzzy impulsive strategy is designed to achieve the stabilization in chaos control in [13]. Also, sliding mode control techniques have been applied in models of diseases propagation [14]. This use is motivated by the satisfactory results obtained in the application of such techniques in other kind of systems [15].

The proposed SEIR model in [8,9] assumes that the whole host population is maintained constant for all time, i.e., such a model is valid when the mortality by causes relative to the disease is negligible, and the birth and naturally mortality rates are similar so that the whole host population may be considered time-invariant. In the same line, the current paper proposes a vaccination strategy based on a linearization control technique of the mapping from the vaccination control to the infected population in order to eradicate an epidemic disease propagation of which can be described by a SIRS epidemic model. The control law based on an input-output linearization technique is useful in order to design a vaccination strategy since the epidemic models in general, and then the SIRS model considered in this paper, are nonlinear systems. Concretely, they include a bilinear term which describes the potential disease transmission when contacts between susceptible and infected individuals happen. The main difference of the current research with respect to [8,9] is that the mortality from causes relative to the infection is appreciable. Also, the birth and mortality from nature causes may be different so that the whole host population may be time-varying. A normalized SIRS epidemic model is used to circumvent the complexity from the fact that the whole population is time-varying. Such a normalized model is obtained from the original SIRS one via a suitable variables change. Moreover, such a normalized model is used to synthesize the vaccination control law which ensures the eradication of the infection from the host population and the positivity of the normalized SIRS model as well as the original SIRS epidemic model. The mathematical proofs of the epidemics eradication and the positivity of both models constitute the main contribution of this paper. Also, a simulation example based on a rabbit hemorrhagic disease illustrates the theoretical results. Such a disease has been chosen in such a numerical study because of its high mortality [16].

Finally, the described control strategy may be extended to more complex epidemical models which consider more population categories than the presented SIRS model. In this sense, vaccinated, quarantine, susceptible population with different risk of catching infection, asymptomatic or symptomatic infected populations have been considered in other compartmental models for describing the propagation of infectious diseases within a host population [17-19].

Notation<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M1">View MathML</a> is the set of strictly positive real numbers and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M2">View MathML</a>. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M3">View MathML</a> is the set of strictly negative real numbers and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M4">View MathML</a>. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M5">View MathML</a> is the first open real quadrant and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M6">View MathML</a> is the first closed real quadrant. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M7">View MathML</a> is the third open real quadrant and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M8">View MathML</a> is the third closed real quadrant.

2 SIRS epidemic model

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M9">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M10">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M11">View MathML</a> be, respectively, the susceptible, infected and recovered or removed-by-immunity populations at time t. Consider a time-invariant SIRS epidemic model given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M12">View MathML</a>

(1)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M13">View MathML</a>

(2)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M14">View MathML</a>

(3)

subject to the initial conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M15">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M16">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M17">View MathML</a> under a vaccination function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M18">View MathML</a>. In the above SIRS model, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M19">View MathML</a> is the total population at any time instant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M20">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M21">View MathML</a> is the death rate from natural causes unrelated to the infection, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M22">View MathML</a> is the birth rate, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M23">View MathML</a> is the rate of losing immunity, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M24">View MathML</a> is the transmission constant (with the total number of infections per unity of time at time t being <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M25">View MathML</a>), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M26">View MathML</a> is the recovery rate (or <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M27">View MathML</a> the average duration of the infective period) and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M28">View MathML</a> is the probability of death from infection causes. The total population dynamics is obtained by summing up (1)-(3) yielding

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M29">View MathML</a>

(4)

so that the total population is time-varying. A normalized SIRS model is used to analyze and design a vaccination control strategy.

2.1 Normalized SIRS model

The SIRS model (1)-(3) is normalized with respect to the whole population by using the following variables change:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M30">View MathML</a>

(5)

By introducing this variables change in (1)-(3), one obtains the normalized SIRS model given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M31">View MathML</a>

(6)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M32">View MathML</a>

(7)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M33">View MathML</a>

(8)

By summing up (6)-(8), it follows that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M34">View MathML</a> so that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M35">View MathML</a> for all time irrespective of the vaccination function.

3 Vaccination strategy

The main control objective is that the infected population asymptotically tends to zero as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36">View MathML</a>, so the infection is eradicated from the population, while guaranteeing the positivity of the controlled system. A vaccination control law based on a static-state feedback linearization strategy is developed for achieving such a control objective. This technique requires a nonlinear coordinate transformation, based on the Lie derivatives theory, in the system representation [20].

The dynamics equations (6)-(8) of the normalized SIRS model can be equivalently written as the following nonlinear control affine system:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M37">View MathML</a>

(9)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M38">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M39">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M40">View MathML</a> are, respectively, considered as the output signal, the input signal and the state vector of the system <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M42">View MathML</a> has been used with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M43">View MathML</a>

(10)

The first step to apply a coordinate transformation based on the Lie derivation is to determine the relative degree of the system. For such a purpose, the following definitions are taken into account: (i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M44">View MathML</a> is the kth-order Lie derivative of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M45">View MathML</a> along <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M46">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M47">View MathML</a> and (ii) the relative degree r of the system is the number of times that the output must be differentiated to obtain the input explicitly, i.e., the number r so that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M48">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M49">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M50">View MathML</a>.

From (10), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M51">View MathML</a>, while <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M52">View MathML</a>, so the relative degree of the system is 2 in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M53">View MathML</a>, i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M54">View MathML</a> except in the singular surface <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M55">View MathML</a> of the state space where the relative degree is not well defined. Since the relative degree of the system is exactly equal to the dimension of the state space for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M56">View MathML</a>, the nonlinear coordinate change

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M57">View MathML</a>

(11)

allows to represent the model in the so-called normal form in a neighborhood of any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M58">View MathML</a>. Namely,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M59">View MathML</a>

(12)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M60">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M61">View MathML</a>

(13)

The following result being relative to the input-output linearization of the system is established.

Theorem 1The state feedback control law

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M62">View MathML</a>

(14)

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M63">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M64">View MathML</a>are the controller tuning parameters, induces the linear closed-loop dynamics

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M65">View MathML</a>

(15)

around any point<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M58">View MathML</a>.

Proof The state equation for the closed-loop system

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M67">View MathML</a>

(16)

is obtained by introducing the control law (14) in (12) and taking into account the fact that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M68">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M69">View MathML</a> and the coordinate transformation (11). Also, it follows that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M70">View MathML</a> by direct calculations. Thus, the state equation of the closed-loop system in the state space defined by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M71">View MathML</a> can be written as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M72">View MathML</a>

(17)

Furthermore, the output equation of the closed-loop system is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M73">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M74">View MathML</a> since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M75">View MathML</a>. From (17) and the closed-loop output equation, it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M76">View MathML</a>

(18)

 □

Remark 1 The roots of the characteristic polynomial <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M77">View MathML</a> corresponding to the closed-loop dynamics (15) are given by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M78">View MathML</a>. Then the stability of the closed-loop dynamics is guaranteed if the control parameters are chosen strictly positive so that such roots have real parts being strictly negative. Moreover, such a choice implies the exponential convergence to zero of the output variable <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M79">View MathML</a> as time tends to infinity and, as a consequence, the eradication of the infection from the host population. However, the control parameters choice also has to guarantee the positivity of the susceptible, infected and recovered populations for all time as the nature of the system requires. This constraint implies that the model variables <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M80">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M81">View MathML</a> have to be such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M82">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a> so that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M84">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a> in view of the constraint <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M86">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a>. Such a positivity property implies additional conditions to be satisfied by the controller parameters <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M63">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M64">View MathML</a>. This analysis is carried out in Section 3.1 below.

3.1 Control parameters choice

The application of the control law (14), obtained from the exact input-output linearization strategy, makes the closed-loop dynamics of the normalized infected population be given by (15). Such a dynamics depends on the control parameters <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M63">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M64">View MathML</a>. Such parameters have to be appropriately chosen in order to guarantee the following suitable properties: (i) the stability and positivity of the controlled SIRS model, (ii) the eradication of the infection, i.e., the asymptotic convergence of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M10">View MathML</a> to zero as time tends to infinity and (iii) the non-negativity of the vaccination function for all time. The following theorems related to the choice of the controller tuning parameters in order to meet such properties are proven.

Theorem 2Assume that the initial condition<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M93">View MathML</a>fulfils<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M94">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M95">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M96">View MathML</a>. Moreover, both roots<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98">View MathML</a>of the characteristic polynomial<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M99">View MathML</a>associated with the closed-loop dynamics (15) are of strictly negative real part via an appropriate choice of the free-design controller parameters<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M63">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M64">View MathML</a>. Then the control law (14) guarantees the exponential stability of the transformed controlled SIRS model (12)-(13). Moreover, the normalized SIRS model (6)-(8) has the following properties: <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M80">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M103">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M104">View MathML</a>are bounded for all time, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M105">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M106">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M107">View MathML</a>exponentially as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M109">View MathML</a>.

Proof The dynamics of the normalized controlled SIRS model (15) can be equivalently written with the state equation (17) and the output equation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M110">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M74">View MathML</a>, by taking into account that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M112">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M113">View MathML</a>. The initial condition <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M114">View MathML</a> in such a realization is bounded since it is related to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M115">View MathML</a> via the coordinate transformation (11) and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M115">View MathML</a> is bounded. Such a controlled model is exponentially stable since the eigenvalues of the matrix A are the roots <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98">View MathML</a> of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M99">View MathML</a> which are assumed to be in the open left-half complex plane. Then the state vector <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M71">View MathML</a> exponentially converges to zero as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36">View MathML</a> while being bounded for all time. It implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M80">View MathML</a> is bounded for all time and converges exponentially to zero as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36">View MathML</a> from the boundedness and exponential convergence to zero of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M71">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36">View MathML</a> since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M79">View MathML</a>. Furthermore, the boundedness of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M104">View MathML</a> for all time and its exponential convergence to unity as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36">View MathML</a> are derived from the boundedness of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M80">View MathML</a>, the exponential convergence to zero of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M80">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36">View MathML</a> and the fact that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M132">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a>. Finally, from the second equation of (11), it follows that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M103">View MathML</a> is bounded and it exponentially converges to zero as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36">View MathML</a> from the boundedness and exponential convergence to zero of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M80">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M71">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36">View MathML</a>. The facts that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M105">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M107">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36">View MathML</a> imply directly that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M109">View MathML</a>. □

Theorem 3Assume that the parameters and the initial condition of a SIRS epidemic model describing the propagation of an epidemic disease satisfy that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M143">View MathML</a>

(19)

If a vaccination strategy based on the control law (14) with the tuning parameters<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M63">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M64">View MathML</a>is chosen such that the poles<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98">View MathML</a>of the controlled normalized system satisfy

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M148">View MathML</a>

(20)

for some real parameter<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M149">View MathML</a>where the upper-bound is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M150">View MathML</a>

(21)

with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M151">View MathML</a>

(22)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M152">View MathML</a>

then:

(i) the normalized populations are non-negative<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a>, i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M154">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M155">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M156">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a>,

(ii) the populations<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M10">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M9">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M11">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M161">View MathML</a>are non-negative<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a>,

(iii) the epidemics is eradicated from the population, i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M10">View MathML</a>tends asymptotically to zero as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M36">View MathML</a>, and

(iv) the vaccination control function is non-negative<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M165">View MathML</a>, i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M166">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a>.

Proof (i) The dynamics of the normalized controlled SIRS model (15) can be equivalently written with the state equation (17) and the output equation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M110">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M74">View MathML</a>, by taking into account that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M112">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M113">View MathML</a>. From such a realization and taking into account the first equation in (11) it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M172">View MathML</a>

(23)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a> for some constants <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M174">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M175">View MathML</a> being dependent on the initial conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M176">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M177">View MathML</a>, and where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98">View MathML</a> denote the eigenvalues of A, i.e., the roots of the characteristic polynomial <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M180">View MathML</a>, which may be fixed to desired values by appropriately adjusting the control parameters <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M63">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M64">View MathML</a>. The values <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M183">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M184">View MathML</a> are related to the initial conditions of the normalized SIRS model in its original realization, i.e., in the state space defined by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M185">View MathML</a> via (11). In this way, the constants <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M174">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M175">View MathML</a> can be obtained by solving the following set of linear equations:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M188">View MathML</a>

(24)

where (11) and (23) have been used. One obtains directly from (24) and by taking into account the assignment of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98">View MathML</a> in (20) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M190">View MathML</a>

(25)

Then it follows from (23) and (25) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M191">View MathML</a>

(26)

since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M192">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a> from the fact that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M194">View MathML</a> subject to the constraints (19)-(21). From (11), (23), (25) and the facts that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M195">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M196">View MathML</a>, which implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M197">View MathML</a> since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M198">View MathML</a>, it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M199">View MathML</a>

(27)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a>, since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M201">View MathML</a> from (20), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M202">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M203">View MathML</a>

(28)

by taking into account (20) and the fact that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M204">View MathML</a> from (21). The fact that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M205">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a> and (27) implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M207">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a> since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M24">View MathML</a>. From the fact that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M132">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a> and (11), it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M212">View MathML</a>

(29)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a>, and then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M214">View MathML</a>

(30)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a> by taking into account (23), (25) and that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M216">View MathML</a>. By introducing the expression (20) for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98">View MathML</a> in (30), one obtains that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M218">View MathML</a>

(31)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a>, where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M220">View MathML</a>

(32)

Note that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M221">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a>, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M223">View MathML</a>, from (19)-(21), and that all of them are monotone decreasing functions which exponentially decrease to zero as time tends to infinity since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M194">View MathML</a>. Moreover, both <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M225">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M226">View MathML</a> decrease faster than <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M227">View MathML</a> so that the initial constraint <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M228">View MathML</a> implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M229">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M231">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a> is deduced from the fact that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M205">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M234">View MathML</a>, as it was previously proven.

Finally, the facts that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M205">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M207">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M231">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M132">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a> directly imply that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M154">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M155">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M156">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a>.

(ii) From (4) it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M244">View MathML</a>

(33)

so the total population <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M19">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a> since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M247">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M248">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M249">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M250">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a> from (5) by taking into account that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M154">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M155">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M254">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a>.

(iii) From (5), (23) and (33) it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M256">View MathML</a>

(34)

and then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M257">View MathML</a>

(35)

As a consequence, the infected population tends exponentially to zero as time tends to infinity, and then the infection is eradicated from the host population since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M194">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M259">View MathML</a> from (20).

(iv) The control law (14) can be equivalently written as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M260">View MathML</a>

(36)

in the state space defined by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M261">View MathML</a>. One can obtain by direct calculations and taking into account (13) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M262">View MathML</a>

(37)

The normalized infected population <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M263">View MathML</a> presents a unique maximum value at the time instant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M264">View MathML</a> when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M265">View MathML</a> as it can be deduced from (26). Furthermore, from (11) and (19), the fact that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M266">View MathML</a> is implied. As a consequence, one knows that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M267">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M268">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M269">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M270">View MathML</a>. In the following, the proof of the non-negativity of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M271">View MathML</a> is split into two parts. On the one hand, when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M267">View MathML</a>, i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M268">View MathML</a>, one can deduce that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M274">View MathML</a>

(38)

by taking into account that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M275">View MathML</a> from (29) and the facts that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M231">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M277">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M165">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M279">View MathML</a> from (19). From (38), if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M280">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M281">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M282">View MathML</a>. Otherwise, it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M283">View MathML</a>

(39)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M284">View MathML</a> and (20) have been used. From (39), if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M285">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M281">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M282">View MathML</a> since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M198">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M202">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M290">View MathML</a> from (19). Otherwise, from (39) it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M291">View MathML</a>

(40)

by using the upper-bound <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M292">View MathML</a> defined in (22). In summary, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M281">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M268">View MathML</a> irrespective of the value of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M295">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97">View MathML</a> whenever <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97">View MathML</a> and ε satisfy (20) and (21).

On the other hand, when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M269">View MathML</a>, i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M299">View MathML</a>, from (37) one obtains that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M300">View MathML</a>

(41)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M279">View MathML</a> from (19) has been used. By direct calculations, it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M302">View MathML</a>

(42)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M284">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M304">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M202">View MathML</a>, (26) and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M306">View MathML</a> have been used. Then from (41)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M307">View MathML</a>

(43)

where (22) has been used. If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M308">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M309">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M310">View MathML</a> from (43) by using the fact that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M311">View MathML</a> from (19). Otherwise, i.e., if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M312">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M313">View MathML</a> reaches its minimum value at the time instant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M264">View MathML</a> when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M315">View MathML</a> reaches its maximum value. Such a minimum value is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M316">View MathML</a>

(44)

where the fact that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M317">View MathML</a> since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M318">View MathML</a> has been used. From introducing the relation between <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98">View MathML</a> of (20) in (44), one obtains

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M321">View MathML</a>

(45)

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M322">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M323">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M310">View MathML</a> for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M202">View MathML</a>, where the facts that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M198">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M290">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M328">View MathML</a> from (19)-(20) have been used. Otherwise, i.e., if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M329">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M330">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M310">View MathML</a> from using the upper-bound <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M332">View MathML</a> defined in (22) for ε. In summary, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M281">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M299">View MathML</a> irrespective of the value of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M335">View MathML</a> whenever <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97">View MathML</a> and ε satisfy the constraints in (20) and (21). This fact completes the proof that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M281">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a> irrespective of the values for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M295">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M335">View MathML</a> if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97">View MathML</a> and ε satisfy the constraints in (20) and (21). □

Remark 2 The constraints in (19) relative to the initial conditions and the parameters of the SIRS model are fulfilled for the majority of the epidemic diseases. On the one hand, the disease transmission constant β is usually much higher than both the birth rate ν and the recovery rate γ so that the constraint about β in (19) can be considered. On the other hand, at the beginning of the infection propagation, the number of infected individuals is usually small enough and the almost population is susceptible so that the conditions in (19) relative to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M342">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M343">View MathML</a> are satisfied.

Remark 3 The process to select the desired poles <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98">View MathML</a>, via a suitable choice of ε, is the following. First, once one knows or has estimated the values for the parameters ν, β, ω, ρ and γ corresponding to the propagation of a specific epidemics, a value for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M347">View MathML</a> is chosen. Then the upper-bounds in (21)-(22) for the parameter ε can be computed in order to choose a value for ε satisfying <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M348">View MathML</a>.

4 Simulation results

An example based on the rabbit hemorrhagic disease in the United Kingdom is considered to illustrate the theoretical results presented in the paper. An initial population of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M349">View MathML</a> rabbits is used. Such an epidemics can be described by the SIRS model (1)-(4) with the parameter values: <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M350">View MathML</a> per day (p.d.), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M351">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M352">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M353">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M354">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M355">View MathML</a> Such values are commonly used in the literature [1,16]. The main characteristic of such an infection is its high mortality, note the value of the probability of dying from the infection (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M354">View MathML</a>) close to 1. The initial conditions for the individual populations are given by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M357">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M358">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M359">View MathML</a>.

Two sets of simulation results are presented to compare the time evolution of the populations within the SIRS mathematical model in two different situations, namely: (i) when no vaccination control actions are applied and (ii) if a vaccination based on the described feedback input-output linearization control technique is applied.

4.1 Epidemics evolution without vaccination

The time evolution of the system populations in the free-vaccination case, i.e., if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M360">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M41">View MathML</a> is displayed in Figure 1.

thumbnailFigure 1. Time evolution of the total and partial (susceptible, infected and recovered) populations without vaccination.

The population of rabbits disappears because of the high mortality of the infection as it can be seen in such a figure. As a consequence, a vaccination strategy has to be applied if the eradication of the epidemics is required while guaranteeing the persistence of the rabbits.

4.2 Epidemics evolution with a feedback control law

First, note that the considered initial condition and the parameters of the SIRS model for the propagation of the rabbit hemorrhagic disease satisfy the constraints in (19). Then the control law given by (14), or equivalently written as in (36), can be applied in order to eradicate the epidemics while guaranteeing the non-negativity of the populations and the vaccination control function. The free-design controller parameters <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M284">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M304">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98">View MathML</a> are the desired roots for the characteristic polynomial <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M99">View MathML</a> associated with the closed-loop dynamics, are prefixed in the following way. The desired dominant root <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97">View MathML</a> satisfying the constraint <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M368">View MathML</a> is chosen, namely, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M369">View MathML</a>. Then, the upper-bound in (21) for the value of ε is calculated, namely, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M370">View MathML</a>. The theoretical results developed in Section 3 prove that a choice of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M371">View MathML</a> is sufficient to guarantee the non-negativity of the populations and the vaccination control function in the controlled SIRS model as well as the eradication of the infectious disease. For such a purpose, the value <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M372">View MathML</a> is chosen. Such a choice for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M97">View MathML</a> and ε determines the value for the root <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98">View MathML</a> by the relation in (20), namely, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M375">View MathML</a>. Also, the values <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M376">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M377">View MathML</a> for the control law are derived from such a procedure. The time evolution of the respective populations is displayed in Figures 2 and 3 while the vaccination control function is shown in Figure 4.

thumbnailFigure 2. Evolution of the infected population with vaccination.

thumbnailFigure 3. Evolution of the susceptible, recovered and total populations with vaccination.

thumbnailFigure 4. Evolution of the vaccination control function in three different time ranges.

The vaccination control action achieves the control objectives as it is seen in Figures 2, 3 and 4. In this sense, the infection is eradicated from the population since the infected population exponentially converges to zero as Figure 2 shows. Also, all of the partial populations, the whole population and the vaccination control function are non-negative for all time. Such properties are coherent with the results proved in Theorem 3. A consequence of the vaccination control action is that the total population of the rabbits monotonically grows through time in a fast way, like it occurs in the absence of disease, as it can be seen in Figure 3. These simulation results point out the improvement of the use of a vaccination strategy in order to guarantee a suitable growth of the rabbit population against a high-mortality infectious disease.

Remark 4 The conditions in Theorem 3 are sufficient but non necessary to ensure the positivity of the controlled model. Concretely, the upper-bounds <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M292">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M332">View MathML</a> are sufficient to guarantee the non-negativity of the vaccination function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M380">View MathML</a> for all time. However, such upper-bounds can be relaxed in the current example by taking into account the results obtained from an exhaustive simulation work. In such a work, the non-negativity of the vaccination control function is maintained for all time although the value of the free-design parameter ε is not smaller than <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M370">View MathML</a>. In this context, the following section analyzes the influence of the parameter ε in the controlled system dynamics.

4.3 Influence of the control free-design parameter ε in the time evolution of the epidemics

Again, the rabbit hemorrhagic disease is considered for this study and the same dominant pole is chosen for the controlled system dynamics, namely, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M382">View MathML</a>. Four different values for the parameter ε are considered, namely, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M383">View MathML</a> (which corresponds to the non-dominant pole of the closed-loop dynamics located in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M375">View MathML</a>), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M385">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M386">View MathML</a>), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M387">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M388">View MathML</a>) and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M389">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M390">View MathML</a>) for analyzing the influence of such a parameter, and then the influence of the pole <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M98">View MathML</a> via the relation in (20), in the controlled system dynamics. Figure 5 displays the time evolution of the infected and total populations for the four different values of ε. The infected population increases until it reaches a maximum value, and then it exponentially decreases to zero as time tends to infinity. Moreover, such a maximum value is smaller and is reached earlier as smaller the parameter ε is. In this sense, a value for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M202">View MathML</a> and closed to unity is convenient for a fast eradication of the infection from the host population. On the other hand, the influence of ε in the time evolution of the total population is less appreciable. The total population exponentially increases in a fast way in all cases.

thumbnailFigure 5. Evolution of the infected and total populations with different values of the parameterεused to generate the vaccination function.

Figure 6 displays the time evolution of the vaccination control function for the different values of ε. One can see that the vaccination control function takes a large value at the initial time instant, and then it decreases until reaching a quasi-stationary regime where its value is maintained below a small threshold for any of the considered values of ε. The magnitude of the vaccination control function at such an initial time instant is larger as smaller the parameter ε is. Moreover, the vaccination control function shows an oscillatory behavior in the quasi-stationary regime if a value of ε closed to unity (concretely if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M393">View MathML</a>) is used while it does not oscillate if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/364/mathml/M394">View MathML</a>. As a consequence, a value for ε large enough so that the maximum of the vaccination control function does not exceed a prescribed threshold can be interesting in order to minimize the cost of the treatment of the infection by means of vaccines application. However, a large value of ε implies a non-appropriate time evolution of the infected population as it has been previously discussed in relation to Figure 5. As a consequence, a tradeoff between the treatment cost and the evolution of the epidemics has to be taken into account when choosing the value of the parameter ε used to generate the vaccination control function.

thumbnailFigure 6. Evolution of the vaccination function with different values of the parameterεused to generate the vaccination function.

5 Concluding remarks

A vaccination control strategy based on feedback input-output linearization techniques has been proposed to fight against the propagation of epidemic diseases within a host population. A SIRS epidemic model with known parameters is used to describe the propagation of the disease. The total population is time-varying and the model considers the mortality for causes related to the disease. The stability and the positivity properties of the closed-loop system have been proved in the case that true data of the susceptible and infected populations are available. Also, the eradication of the epidemic disease from the host population is guaranteed with such a control strategy. These theoretical results are complemented with some simulation results to illustrate the effectiveness of the proposed approach. Future research into the subject is going to deal with the combination of this control technique with the design of an observer to estimate the susceptible and infected populations when their true data are not available as it usually occurs in a real situation.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgements

The authors thank the Spanish Ministry of Economy and Competitiveness for its support of this work through Grants DPI2009-07197 and DPI2012-30651 as well as to the Basque Government for its support through Grants IT378-10, SAIOTEK SPE07UN04 and SAIOTEK SPE09UN12.

References

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