An NSFD scheme for SIR epidemic models of childhood diseases with constant vaccination strategy
© Cui et al.; licensee Springer 2014
Received: 10 December 2013
Accepted: 4 June 2014
Published: 24 June 2014
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.
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.
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
is a compact, positively invariant set for model (1.1).
Then the following results can be summarized in Li et al. .
If , the disease-free equilibrium is globally asymptotically stable in D. On the other hand, if , is unstable;
If , then the unique endemic equilibrium is globally asymptotically stable in D.
3 The NSFD scheme
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.
4 Global asymptotic stability of disease-free equilibrium
Now, we give the following theorem about the local stability of the disease-free equilibrium for model (3.1).
Theorem 4 If , the disease-free equilibrium point of discrete model (3.1) or equivalent (3.2) is locally asymptotically stable in . On the other hand, if , is unstable.
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. □
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 .
Theorem 5 If , then the disease-free equilibrium of model (3.1) is globally asymptotically stable.
We easily find that (4.7) implies . Meanwhile, combining (4.8) with (4.5), we have . Therefore, we have , which yields . Since , we have .
By applying (4.11), we conclude that (4.10) holds for any . Thus, from (4.9)-(4.12), we conclude that is uniformly stable. Hence, if , is globally asymptotically stable. □
5 Global asymptotic stability of the endemic equilibrium
In this section, we mainly discuss the global dynamics of the endemic equilibrium of model (3.1). Before we prove the stability of the endemic equilibrium, we first give the following lemma.
Theorem 6 If , then the endemic equilibrium point of discrete model (3.1) or equivalent (3.2) is locally asymptotically stable in .
where , , .
Hence all the conditions in Lemma 3 are satisfied when . This proves that when , then the endemic equilibrium point is locally asymptotically stable for any h. □
This completes the proof of Lemma 4. □
where the constant is sufficiently large such that .
Thus, by applying Lemma 1, we have that for all .
Since , this leads to , which yields a contradiction. Hence the claim is proved.
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. □
Notice that , which implies that is uniformly stable. Finally, we hence obtain the theorem as follows.
Theorem 8 If , then the endemic equilibrium for model (3.1) or equivalent (3.2) is globally asymptotically stable.
6 Numerical simulations
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.
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.
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 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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