# An NSFD scheme for SIR epidemic models of childhood diseases with constant vaccination strategy

- Qianqian Cui
^{1, 2}, - Jiabo Xu
^{3}, - Qiang Zhang
^{2}and - Kai Wang
^{4}Email author

**2014**:172

https://doi.org/10.1186/1687-1847-2014-172

© Cui et al.; licensee Springer 2014

**Received: **10 December 2013

**Accepted: **4 June 2014

**Published: **24 June 2014

## Abstract

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

## 1 Introduction

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.

*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 $0<p<1$) 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 [4] employed the Adomian decomposition method to compute an approximate non-perturbative solutions of model (1.1). Yildirim and Cherruault [7] 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 $0<\varphi (h)<1$, where $\varphi (h)=h+o({h}^{2})$; 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 [19], the NSFD scheme has been implemented in a special class of SIR epidemic models. Mickens [5] 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.* [3].

**Theorem 1**

*Model*(1.1)

*always has a disease*-

*free equilibrium*${E}^{0}=((1-p)A/\mu ,0,pA/\mu )$

*and has a unique endemic equilibrium*${E}^{\ast}=({S}^{\ast},{I}^{\ast},{R}^{\ast})$

*when*${\mathcal{R}}_{0}>1$,

*where*

**Theorem 2**

*For model*(1.1),

*the following results hold*.

- (i)
*If*${\mathcal{R}}_{0}<1$,*the disease*-*free equilibrium*${E}^{0}$*is globally asymptotically stable in**D*.*On the other hand*,*if*${\mathcal{R}}_{0}>1$, ${E}^{0}$*is unstable*; - (ii)
*If*${\mathcal{R}}_{0}>1$,*then the unique endemic equilibrium*${E}^{\ast}$*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.

**Theorem 3**

*For model*(3.1)

*or equivalent*(3.2),

*there always exists a disease*-

*free equilibrium*${E}_{d}^{0}=({S}_{d}^{0},{I}_{d}^{0},{R}_{d}^{0})=((1-p)A/\mu ,0,pA/\mu )$

*and has a unique endemic equilibrium*${E}_{d}^{\ast}=({S}_{d}^{\ast},{I}_{d}^{\ast},{R}_{d}^{\ast})$

*when*${\mathcal{R}}_{0}>1$,

*where*

## 4 Global asymptotic stability of disease-free equilibrium ${E}_{d}^{0}$

Now, we give the following theorem about the local stability of the disease-free equilibrium ${E}_{d}^{0}$ for model (3.1).

**Theorem 4** *If* ${\mathcal{R}}_{0}<1$, *the disease*-*free equilibrium point* ${E}_{d}^{0}$ *of discrete model* (3.1) *or equivalent* (3.2) *is locally asymptotically stable in* ${D}_{h}$. *On the other hand*, *if* ${\mathcal{R}}_{0}>1$, ${E}_{d}^{0}$ *is unstable*.

*Proof*Substituting the disease-free equilibrium ${E}_{d}^{0}$ to Jacobian matrix (4.1) yields

Obviously $|{\lambda}_{1}|<1$ for all *h*. From the definition of basic reproductive number (2.4), we can easily conclude that ${\mathcal{R}}_{0}<1$ is equivalent to $(1-p)\beta A<\mu (\mu +\gamma )$. Thus, if ${\mathcal{R}}_{0}<1$, then the magnitude of eigenvalue ${\lambda}_{2}$ is also strictly less than unity irrespective of *h*. This completes the proof. □

*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.* [[22], proof of Lemma 3.3], we obtain the following basic lemma.

**Lemma 1**

*For any solution*$({S}_{n},{I}_{n})$

*of model*(4.2),

*with the initial conditions*${S}_{0}>0$, ${I}_{0}>0$,

*we have that*

then we easily get the following lemma.

**Lemma 2**$P(s)$

*is strictly monotone increasing function on*$[0,+\mathrm{\infty})$

*with*

*and*

*Moreover*,

*if*${\mathcal{R}}_{0}>1$,

*then there exists a unique solution*$S={S}_{d}^{\ast}$

*of*$P(s)=1$

*such that*

By applying techniques in Izzo *et al.* [22], we now prove the global stability of the disease-free equilibrium for ${\mathcal{R}}_{0}\le 1$.

**Theorem 5** *If* ${\mathcal{R}}_{0}\le 1$, *then the disease*-*free equilibrium* ${E}_{d}^{0}$ *of model* (3.1) *is globally asymptotically stable*.

*Proof*From (4.3) in Lemma 1, for any $\u03f5>0$, there exists an integer ${n}_{0}$ such that

*ϵ*is arbitrary, we conclude that if ${\mathcal{R}}_{0}\le 1$, then

We easily find that (4.7) implies ${lim}_{n\to +\mathrm{\infty}}{S}_{n}=\overline{S}=\frac{(1-p)A}{\mu}$. Meanwhile, combining (4.8) with (4.5), we have ${lim}_{k\to +\mathrm{\infty}}{V}_{{n}_{k}}=0$. Therefore, we have $\stackrel{\u02c6}{V}=0$, which yields ${lim}_{k\to +\mathrm{\infty}}{I}_{n}=0$. Since ${lim}_{k\to +\mathrm{\infty}}{N}_{n}=\frac{A}{\mu}$, we have ${lim}_{k\to +\mathrm{\infty}}{R}_{n}=\frac{pA}{\mu}$.

By applying (4.11), we conclude that (4.10) holds for any $n\ge {n}_{3}$. Thus, from (4.9)-(4.12), we conclude that ${E}_{0}^{d}$ is uniformly stable. Hence, if ${\mathcal{R}}_{0}\le 1$, ${E}_{0}^{d}$ is globally asymptotically stable. □

## 5 Global asymptotic stability of the endemic equilibrium ${E}_{d}^{\ast}$

In this section, we mainly discuss the global dynamics of the endemic equilibrium ${E}_{d}^{\ast}$ of model (3.1). Before we prove the stability of the endemic equilibrium, we first give the following lemma.

*The quadratic equation*$f(\lambda )={\lambda}^{2}-B\lambda +C=0$

*has two roots that satisfy*$|{\lambda}_{i}|<1$, $i=1,2$,

*if and only if the following conditions are satisfied*:

- (i)
$f(0)=C<1$,

- (ii)
$f(-1)=1+B+C>0$,

- (iii)
$f(1)=1-B+C>0$.

**Theorem 6** *If* ${\mathcal{R}}_{0}>1$, *then the endemic equilibrium point* ${E}_{d}^{\ast}$ *of discrete model* (3.1) *or equivalent* (3.2) *is locally asymptotically stable in* ${D}_{h}$.

*Proof*Assuming that ${\mathcal{R}}_{0}>1$ and substituting the endemic equilibrium point ${E}_{d}^{\ast}$ to Jacobian matrix (4.1) lead to

where $a=1+\varphi (\mu +\beta {I}_{d}^{\ast})>1$, $b=1+(\mu +\gamma )\varphi >1$, $c=\beta \varphi ({S}_{d}^{\ast}+(1-p)\varphi A)>0$.

Hence all the conditions in Lemma 3 are satisfied when ${\mathcal{R}}_{0}>1$. This proves that when ${\mathcal{R}}_{0}>1$, then the endemic equilibrium point ${E}_{d}^{\ast}=({S}_{d}^{\ast},{I}_{d}^{\ast},{R}_{d}^{\ast})$ is locally asymptotically stable for any *h*. □

Next, we will prove the permanence of model (3.1) for ${\mathcal{R}}_{0}>1$. Similar to the result of McCluskey in [25, 26], we first give the following lemma.

**Lemma 4**

*If*${I}_{n+1}<{I}_{n}$

*holds*,

*then*${S}_{n+1}<{S}_{d}^{\ast}$;

*inversely*,

*if*${S}_{n+1}\ge {S}_{d}^{\ast}$

*holds*,

*then*${I}_{n+1}\ge {I}_{n}$,

*where*

*Proof*By the second equation of model (3.1), we obtain

This completes the proof of Lemma 4. □

**Theorem 7**

*If*${\mathcal{R}}_{0}>1$,

*then for any solution of model*(3.1)

*with the initial conditions that*${S}_{0}>0$, ${I}_{0}>0$, ${R}_{0}>0$

*where the constant* ${l}_{0}$ *is sufficiently large such that* ${S}_{d}^{\ast}<{S}^{\mathrm{\Delta}}:=\frac{(1-p)A}{{k}^{\ast}}(1-{(\frac{1}{1+\varphi {k}^{\ast}})}^{{l}_{0}})$.

*Proof*For any positive constant

*ϵ*, there exists a sufficiently large positive integer ${n}_{0}$ such that

*ϵ*is arbitrary, we conclude that (5.1) holds. Now we prove that (5.2) holds. In fact, for any positive constant $q<1$, it is seen that

Thus, by applying Lemma 1, we have that ${I}_{n+1}\ge {I}_{n}$ for all $n\ge {n}_{1}+{l}_{0}+1$.

Since $\beta {S}^{\mathrm{\Delta}}-(\mu +\gamma )\varphi >0$, this leads to ${lim}_{n\to +\mathrm{\infty}}{V}_{n}=\mathrm{\infty}$, which yields a contradiction. Hence the claim is proved.

*n*sufficiently large. If this case holds, we get the conclusion of the proof. Second, we investigate the case that ${I}_{n}$ oscillates about $q{I}^{\ast}$ for all

*n*sufficiently large. Let ${N}_{1}<{N}_{2}$ be sufficiently large such that

If ${N}_{2}\ge {N}_{1}+{l}_{0}$, then by applying a similar discussion above, we obtain ${I}_{n+1}\ge \underline{I}$ for all ${N}_{1}+{l}_{0}\le n\le {N}_{2}-1$. We hence prove that ${I}_{n}\ge q\underline{I}$ for all ${N}_{1}\le n\le {N}_{2}$. Since the interval $[{N}_{1},{N}_{2}]$ is arbitrarily chosen, we conclude that ${I}_{n}\ge q\underline{I}$ for all *n* sufficiently large. Meanwhile, since *q* is also arbitrary, we conclude that ${lim}_{n\to +\mathrm{\infty}}{I}_{n}\ge \underline{I}$. This completes the proof. □

Notice that ${W}_{n}\le {W}_{0}$, which implies that ${E}_{d}^{\ast}$ is uniformly stable. Finally, we hence obtain the theorem as follows.

**Theorem 8** *If* ${\mathcal{R}}_{0}>1$, *then the endemic equilibrium* ${E}_{d}^{\ast}$ *for model* (3.1) *or equivalent* (3.2) *is globally asymptotically stable*.

## 6 Numerical simulations

- (i)
We choose $A=28$, $\mu =0.7$, $\gamma =15$, $p=0.5$, $\beta =0.7$ and $\varphi =0.2$. By calculation, we have that ${\mathcal{R}}_{0}=0.8917$ and the endemic equilibrium ${E}_{d}^{0}=(16,0,24)$. According to Theorem 5, the disease-free equilibrium ${E}_{d}^{0}$ of discrete model (3.1) or equivalent (3.2) is globally stable, which is shown in Figure 1.

- (ii)
Assuming the following parameter values: $A=28$, $\mu =0.7$, $\gamma =15$, $p=0.5$, $\beta =2$ and $\varphi =0.2$, by calculation, we have ${\mathcal{R}}_{0}=2.0382$ and the endemic equilibrium ${E}_{d}^{\ast}=(7.85,0.3634,24.5866)$. According to Theorem 8, the endemic equilibrium ${E}_{d}^{\ast}=({S}_{d}^{\ast},{I}_{d}^{\ast},{R}_{d}^{\ast})$ of discrete model (3.1) or equivalent (3.2) is globally stable, which is depicted in Figure 2.

## 7 Conclusion

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 ${\mathcal{R}}_{0}>1$, we offer Lemma 4. Applying the discrete Lyapunov functional technique (see [25, 26]) for both cases ${\mathcal{R}}_{0}\le 0$ and ${\mathcal{R}}_{0}>1$, 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 ${\mathcal{R}}_{0}$. 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.

## Declarations

### Acknowledgements

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).

## Authors’ Affiliations

## References

- Arafa AAM, Rida SZ, Khalil M: Solutions of fractional order model of childhood diseases with constant vaccination strategy.
*Math. Sci. Lett.*2013, 1: 17–23.View ArticleGoogle Scholar - Cui Q, Yang X, Zhang Q: An NSFD scheme for a class of SIR epidemic model with vaccination and treatment.
*J. Differ. Equ. Appl.*2014, 20: 416–422. 10.1080/10236198.2013.844802MathSciNetView ArticleGoogle Scholar - Li J, Zhang J, Ma Z: Global analysis of some epidemic models with general contact rate and constant immigration.
*Appl. Math. Mech.*2004, 4: 396–404.Google Scholar - Makinde OD: Adomian decomposition approach to a SIR epidemic model with constant vaccination strategy.
*Appl. Math. Comput.*2007, 184: 842–848. 10.1016/j.amc.2006.06.074MathSciNetView ArticleMATHGoogle Scholar - Mickens RE: A SIR-model with square-root dynamics: an NSFD scheme.
*J. Differ. Equ. Appl.*2010, 16: 209–216. 10.1080/10236190802495311MathSciNetView ArticleMATHGoogle Scholar - Wang L, Cui Q, Teng Z: Global dynamics in a class of discrete-time epidemic models with disease courses.
*Adv. Differ. Equ.*2013., 2013: Article ID 57. http://www.advancesindifferenceequations.com/content/2013/1/57Google Scholar - Yildirim A, Cherruault Y: Analytical approximate solution of a SIR epidemic model with constant vaccination strategy by homotopy perturbation method.
*Kybernetes*2009, 38: 1566–1575. 10.1108/03684920910991540View ArticleGoogle Scholar - Hu Z, Teng Z, Jiang H: Stability analysis in a class of discrete SIRS epidemic models.
*Nonlinear Anal., Real World Appl.*2012, 13: 2017–2033. 10.1016/j.nonrwa.2011.12.024MathSciNetView ArticleMATHGoogle Scholar - Suryanto A: A dynamically consistent nonstandard numerical scheme for epidemic model with saturated incidence rate.
*Int. J. Math. Comput.*2011, 13: 112–123.MathSciNetGoogle Scholar - Suryanto A: Stability and bifurcation of a discrete SIS epidemic model with delay.
*Proceedings of the 2nd International Conference on Basic Sciences*2012, 1–6. IndonesiaGoogle Scholar - Mickens RE:
*Nonstandard Finite Difference Model of Differential Equations*. World Scientific, Singapore; 1994.Google Scholar - Mickens RE:
*Application of Nonstandard Finite Difference Schemes*. World Scientific, Singapore; 2000.View ArticleGoogle Scholar - Mickens RE: Dynamic consistency: a fundamental principle for constructing nonstandard finite difference schemes for differential equations.
*J. Differ. Equ. Appl.*2005, 11: 645–653. 10.1080/10236190412331334527MathSciNetView ArticleMATHGoogle Scholar - Mickens RE: Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition.
*Numer. Methods Partial Differ. Equ.*2012, 3: 528–534.Google Scholar - Mickens RE: Nonstandard finite difference schemes for differential equations.
*J. Differ. Equ. Appl.*2002, 8: 823–847. 10.1080/1023619021000000807MathSciNetView ArticleMATHGoogle Scholar - Mickens RE: Numerical integration of population models satisfying conservation laws: NSFD methods.
*J. Biol. Dyn.*2007, 1: 427–436. 10.1080/17513750701605598MathSciNetView ArticleMATHGoogle Scholar - Ding X: A non-standard finite difference scheme for an epidemic model with vaccination.
*J. Differ. Equ. Appl.*2013, 19: 179–190. 10.1080/10236198.2011.614606View ArticleMATHGoogle Scholar - Mickens RE, Washington T: A note on an NSFD scheme for a mathematical model of respiratory virus transmission.
*J. Differ. Equ. Appl.*2012, 8: 525–529.MathSciNetView ArticleGoogle Scholar - Suryanto A, Kusumawinahyu WM, Darti I, Yanti I: Dynamically consistent discrete epidemic model with modified saturated incidence rate.
*Comput. Appl. Math.*2013, 32: 373–383. 10.1007/s40314-013-0026-6MathSciNetView ArticleMATHGoogle Scholar - Ross SL:
*Differential Equations*. Blaisdell, Waltham; 1964.MATHGoogle Scholar - Strogatz SH:
*Nonlinear Dynamics and Chaos*. Addison-Wesley, Reading; 1994.Google Scholar - Izzo G, Muroya Y, Vecchio A: A general discrete time model of population dynamics in the presence of an infection.
*Discrete Dyn. Nat. Soc.*2009., 2009: Article ID 143019 10.1155/2009/143019Google Scholar - Brauer F, Castillo-Chavez C:
*Mathematical Models in Population Biology and Epidemiology*. Springer, New York; 2001.View ArticleMATHGoogle Scholar - Elaydi S:
*An Introduction to Difference Equations*. 3rd edition. Springer, New York; 1992.Google Scholar - McCluskey CC: Global stability for an SEIR epidemiological model with varying infectivity and infinite delay.
*Math. Biosci. Eng.*2009, 6: 603–610.MathSciNetView ArticleMATHGoogle Scholar - McCluskey CC: Complete global stability for an SIR epidemic model with delay - distributed or discrete.
*Nonlinear Anal., Real World Appl.*2010, 10: 55–59.MathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.