In this paper, we construct a nonstandard finite difference (NSFD) scheme for an SIR epidemic model of childhood disease with constant strategy. The dynamics of the obtained discrete model is investigated. First we show that the discrete model has equilibria which are exactly the same as those of the continuous model. Furthermore, we prove that the conditions for those equilibria to be globally asymptotically stable are consistent with the continuous model for any size of numerical time-step. The analytical results are confirmed by some numerical simulations.
Keywords:mathematical model; transmission dynamics; basic reproduction number; sensitivity analysis; control strategies
Childhood diseases are the most common form of infectious diseases. These are the diseases such as measles, mumps, chicken pox, rubella, poliomyelitis, etc. to which children are born susceptible and usually contract within five years. Because young children are in frequent contact with each other at school or other place, such a disease can be spread very quickly. Meanwhile, the development of vaccines against infectious children diseases has been booming and protecting children from the diseases. Hence vaccination is considered to be the most effective strategy against childhood diseases, it is essential for us to predict the optimal vaccine coverage level to prevent the spread of theses diseases. A universal effort to extend vaccination coverage to all children began in 1974, when the World Health Organization (WHO) founded the Expanded Program on Immunization (EPI). Mathematical models (see [1-7]) of deterministic type have often been used to provide deeper insights into the transmission dynamics of a childhood disease and to evaluate control strategies.
In this paper, the total population that is involved in the spread of infection is split into three epidemiological classes: a susceptible class (S), an infected class (I) and a removed class (R) denoting vaccinated as well as recovered people with permanent immunity. We assume that the efficacy of vaccine is 100%, and the natural death rates μ in the classes remain unequal to births, so that the total population N is realistically not constant. Citizens are born into the population at a constant birth rate A with extremely low childhood disease mortality rate. We denote the fraction of citizens vaccinated at birth each year as p (with ) and assume the rest are susceptible. A susceptible individual will move into the infected group through contact with an infected individual, approximated by an average contact rate β. An infected individual recovers at rate γ, and enters the removed class. The removed class also contains people who are vaccinated. The differential equations for the SIR (see [1,4,7]) epidemic model of childhood diseases with constant vaccination strategy are as follows:
The biological background requires that all parameters be nonnegative. Makinde  employed the Adomian decomposition method to compute an approximate non-perturbative solutions of model (1.1). Yildirim and Cherruault  by qualitative analysis revealed the vaccination reproductive number for disease control and eradication.
However, for practical purposes, it is often necessary to discretize the continuous model. The discrete dynamical system obtained from the discretization should contain as many qualitative properties of the continuous problem as possible. It is shown that many standard methods such as Euler method, Runge-Kutta method and some other standard finite schemes implemented in a dynamical system can lead to negative solutions for spurious dynamical behaviors such as converging to wrong equilibrium point or wrong periodic cycle or numerical instabilities [8-10]. In this paper, we propose a numerical scheme to solve model (1.1) by implementing a nonstandard finite difference (NSFD) scheme. This method was originally developed by Mickens [11-16]. The nonstandard scheme relied on the following important rules: the standard denominator h in standard discrete derivative is replaced by a denominator function , where ; the nonlinear terms are approximated in a nonlocal way using more than one mesh point. Here, h is the time-step size of numerical integration. Moreover, the fundamental principle for constructing NSFD scheme for differential equations is dynamic consistency, that is, the discretized model maintain essential dynamical properties such as positivity of solutions, boundedness of solutions, monotonicity of solutions, correct number and stability of fixed-points and other special solutions of the continuous model. This method has been applied to various problems in which the resulting discrete systems preserve dynamical properties of the related continuous models [5,17-19]. In , the NSFD scheme has been implemented in a special class of SIR epidemic models. Mickens  considered a SIR epidemic model with square-root dynamics.
This paper is organized as follows. In the next section, we present several important properties of solutions to the continuous model. A particular discretization is constructed in Section 3. We illustrate the global asymptotic stability of disease-free equilibrium and endemic equilibrium in Sections 4 and 5, and provided numerical examples to verify our results in Section 6. Finally, we provide a summary of the obtained results and present a possible extension of this work.
2 Dynamical properties of the continuous model
In this section, we review some dynamical properties of the SIR epidemic model with constant vaccination strategy (1.1). It should be noted that the system of continuous equations, if added together, satisfies the conservation law 
The exact solution of equation (2.1) is
All valid epidemic models must have this feature since negative population numbers cannot exist as a physical reality. Moreover, it is easy to verify that the domain
is a compact, positively invariant set for model (1.1).
Define the basic reproduction number as follows:
Then the following results can be summarized in Li et al..
Theorem 2For model (1.1), the following results hold.
3 The NSFD scheme
To derive the NSFD scheme, we define the following notation :
It is noted that discretized model (3.1) can be considered as using the standard Euler method for the first derivative and a nonlocal expression for nonlinear terms. It can be easily rearranged to get its explicit version
which is the exact finite scheme for the conservation law as expressed by equation (2.1) (see ). A straightforward calculation gives
Similar to the continuous-time case, discrete model (3.1) or equivalent (3.2) also has a compact, positivity invariant set
It is easy to verify that discrete model (3.1) or equivalent (3.2) has the same equilibrium as model (1.1) which is independent of h. It can be described as the following theorem.
In this section, we mainly discuss the global asymptotic stability of disease-free equilibrium . First, we consider the local stability of for model (3.1). In order to obtain the local stability of equilibria for discrete model (3.1) or equivalent (3.2), we notice that the first two equations in model (3.1) or (3.2) do not depend on the third equation, and therefore the third equation can be omitted without loss of generality properties. For the sake of simplicity, we define the following functions :
Obviously for all h. From the definition of basic reproductive number (2.4), we can easily conclude that is equivalent to . Thus, if , then the magnitude of eigenvalue is also strictly less than unity irrespective of h. This completes the proof. □
Next, let us consider the global asymptotic stability of disease-free equilibrium for model (3.1). Since the variable R does not appear in the first and the second equations, it is sufficient to consider the following 2-dimensional system:
Similar to the Izzo et al. [, proof of Lemma 3.3], we obtain the following basic lemma.
then we easily get the following lemma.
By applying techniques in Izzo et al., we now prove the global stability of the disease-free equilibrium for .
which yields that the sequence is monotone decreasing. Therefore, there exists a nonnegative constant such that . We will prove that for . In fact, if , then we have . Consequently, from (4.5), we have that . Meanwhile, by Lemma 1, we can conclude that . Suppose that , that is, . We first transform model (4.2) into the following form:
We claim that there exists a sequence such that if , the claim is evident. Now we consider the case . By applying the first inequality in (4.4), we obtain . Therefore, there exists a sequence such that . From the first equation of (4.6), we have
from which it is not difficult to obtain
Finally, we will prove that if , then is uniformly stable. First, we consider the case that there exists a nonnegative integer such that for any . From the first equation of (3.1), we have that for any ,
From the third equation of model (3.1), it is not difficult to obtain
Proof By the second equation of model (3.1), we obtain
This completes the proof of Lemma 4. □
We first claim that any solution of model (3.1) does not have the following property: there exists a nonnegative integer such that for all . Suppose on the contrary that there exists a solution of model (3.1) and a nonnegative such that for , it can be seen that for ,
We hence set
By the claim above, we are left to consider two possibilities. First, for all n sufficiently large. If this case holds, we get the conclusion of the proof. Second, we investigate the case that oscillates about for all n sufficiently large. Let be sufficiently large such that
If , then by applying a similar discussion above, we obtain for all . We hence prove that for all . Since the interval is arbitrarily chosen, we conclude that for all n sufficiently large. Meanwhile, since q is also arbitrary, we conclude that . This completes the proof. □
By Theorem 6, we easily obtain the permanence of model (3.1) for . Next, by constructing the Lyapunov function, we will prove that is globally asymptotically stable for . Consider the Lyapunov function as follows:
In the same way, we have
From (5.4) and (5.5), we have
From the first equation of models (3.1) and (5.7), we have
6 Numerical simulations
In this section, numerical simulations will be given to verify theoretical results obtained in the previous section. The simulation is performed using MATLAB software.
(i) We choose , , , , and . By calculation, we have that and the endemic equilibrium . According to Theorem 5, the disease-free equilibrium of discrete model (3.1) or equivalent (3.2) is globally stable, which is shown in Figure 1.
Figure 1. The solutionsof model (3.1) or equivalent (3.2) are globally asymptotically stable and converge to the disease-free equilibrium, when.
(ii) Assuming the following parameter values: , , , , and , by calculation, we have and the endemic equilibrium . According to Theorem 8, the endemic equilibrium of discrete model (3.1) or equivalent (3.2) is globally stable, which is depicted in Figure 2.
Figure 2. The solutionsof model (3.1) or equivalent (3.2) are globally asymptotically stable and converge to the disease-free equilibrium, when.
In this paper, we have proposed a discrete-time analogue of the continuous SIR epidemic model of childhood diseases with constant vaccination strategy which is derived by the NSFD scheme of Michens. In order to obtain the permanence of model (3.1) for , we offer Lemma 4. Applying the discrete Lyapunov functional technique (see [25,26]) for both cases and , it shown that the global dynamics of this discrete-time analogue of the continuous SIR epidemic model is fully determined only by the basic reproduction number . This shows dynamical consistency between the discrete SIR epidemic model and its corresponding continuous model. The NSFD scheme constructed in this paper is for the SIR epidemic model with constant vaccination strategy. For our future work, we will consider an epidemic model with varying vaccination strategy.
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
The work was supported by the National Natural Science Foundation of P.R. China (11201399, 11301451) and the Natural Science Foundation of Shihezi University (2013ZRKXYQ-YD05).
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