On the global stability of the endemic state in an epidemic model with vaccination

This paper investigates an SIS epidemic model with variable population size including a vaccination program. Dynamics of the endemic equilibrium of the model are obtained, and it will be shown that this equilibrium exists and is locally asymptotically stable when R0>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_0 > 1$$\end{document}. In this case, the disease uniformly persists, and moreover, using a geometric approach we conclude that the model is globally asymptotically stable under some conditions. Also, a numerical discussion is given to verify the theoretical results.


Introduction
The susceptible-infected-susceptible (SIS) model is one of the most well-known type of epidemic models. These models are appropriate for some infections, for instance, common cold and influenza, or bacterial diseases such as meningitis and cholera, or sexually transmitted diseases, that do not cause permanent immunity after recovery. To immunize individuals from infection and control the infectious diseases, vaccination is usually preferred because of its efficiency compared with other drug and non-drug interventions. There are many epidemic models [1,3,8,13,14,16,19] and also some SIS epidemic models in which temporary or permanent vaccination has been included [6, 9-11, 17, 22]. These models may be deterministic [10,11] or stochastic [4,6,22], with constant [10,13] or variable [9] population size, and with general incidence [7,21] or a particular incidence such as standard, bilinear, and saturated [2,5,13]. In this paper, we introduce an SIS epidemic model with vaccination and standard incidence. In the next section, we first describe the model and then some basic properties of it will be given. Sections "Local asymptotical stability" and "Global asymptotical stability" are devoted to investigating the local and global stability of endemic equilibrium of the model, respectively. Finally, we summarize the conclusions in section "Conclusions", after a numerical consideration in section "Numerical examples".

Model description
Consider the following system: Here, the population has been divided into three subpopulations as susceptible, infected, and vaccinated individuals. The size of population in each class at time t is denoted by S, I, and V, respectively, whereas the number of all individuals in this time is given by N = S + I + V . The recruiting is done by entering a number of individuals into population per unit time that may be either immigrants or newborns. The

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vaccination program is applied on the new members by a proportion . A proportion of susceptible individuals are also vaccinated, and the rate of losing immunity of vaccination is . The vaccine is supposed to be completely effective, and no vaccinated individual becomes infected. Susceptible individuals become infected at standard incidence rate SI N , where is transmission coefficient, whereas the rate of recovery is . The rate of natural death in population is , and the mortality due to disease is also included in the model with a rate . All values in the model are nonnegative except and that are assumed positive.
From system (1), it can be seen that the total number of members N is not fixed in general and it is expressed by the following equation: We have dN dt ≤ − N , and thus, lim sup N(t) t→∞ ≤ . This also states that I, S, and V are bounded. We can see that the feasible region is a positively invariant set for system (1), and the system is well posed for mathematical and epidemiological considerations in .
The equilibria of the model (1) are obtained by solving the following equations: The system has two solutions: When Ī = 0 , the disease-free equilibrium (DFE) E 0 and when Ī > 0 , the endemic equilibrium E * = (I * , S * , V * ) , where , , .
Here, and thus, we have The quantity  0 is called the basic reproduction number of the model. Furthermore, the total population sizes at two equilibria E 0 and E * are obtained, respectively, as follows: and In the following sections, we investigate dynamics of the model (1) at the endemic equilibrium E * and its asymptotic stability will be obtained. The next section is devoted to consider the local asymptotic stability.

Local asymptotical stability
Letting F = SI N , the Jacobian matrix of the model (1) at E * has the following form: . .

We have
Obviously a 3 < 0 and a 1 = ( + + ) Moreover, notice that the equilibrium E * exists if  0 > 1 and this implies and thus, a 2 > 0.

Corollary 1 The endemic state E * is stable if parameter values and lie under the following line in theplane:
Corollary 1 gives minimum amount of vaccination proportions that is needed to the vaccine be effective. Indeed, those values of and that lie above the mentioned line in (9) are sufficient to the disease eradicated from population. When the proportion of vaccination in new members (9) ( ) + ( + + ) = ( − ( + + ))( + ). and in susceptibles is equal, i.e., = = , this optimum amount of vaccination will be as

Global asymptotical stability
The global asymptotical stability of the endemic state E * is discussed in this section. The analysis of the global stability of epidemic models is generally a difficult task, and the methods introduced for carrying this out are scant. To analyze the global asymptotic stability of the endemic state, we employ a geometric method developed by Li and Muldowney [12] and used by many authors [2,15,[18][19][20]23]. The following result has been proved in [12]. , respectively. When  0 > 1 , there exists an unique endemic state E * , and moreover, the disease-free equilibrium (DFE) E 0 ∈ is unstable. Now, using Acyclicity Theorem similar to the proof of Theorem 3.2 in [18] we find that when  0 > 1 , system (1) is uniformly persistent. On the other hand, we know that the (10) * = ( − ( + + ))( + ) ( + + ) + .

Theorem 2 Assume that f is a C 1 function on a simply connected open set
solutions are bounded because the total population size N is bounded. Considering this with the uniform persistence of the system, it can be concluded that there exists a compact absorbing set ⊂ . The Jacobian matrix of the system at E * was given by (6), and thus, its second additive compound matrix is where Matrix in Theorem 2 acts as a Lyapunov function. So, for the method to be applicable and efficient, this matrix must be chosen suitable. Now, we choose matrix as Therefore, The matrix can be written in the block form as = 11 12 21 22 , in which , n 1 = 2 + + + , n 2 = 2 + + + , Considering the vector norm |( 1 , 2 , 2 )| = max{| 1 |, | 2 | + | 3 |} for all ( 1 , 2 , 3 ) ∈ ℝ 3 as in [12], we have where Thus, we obtain and From the second equation in system (1) Therefore, we have and Hence, assuming yields to ( ) ≤ sup{ 1 , 2 }, 1 = ( 11 ) + | 12 | and 2 = ( 22 ) + | 21 |.   Figure 3 shows the number of infected individuals (sub-figure (a)) and the phase diagram of I(t) and S(t) (sub-figure (b)) when = 0.4 , and it can be seen that E 0 is stable. Figure 4 shows that I(t) does not vanish when = 0.05 , and in this case, E * is stable .

Conclusions
In this paper, a deterministic SIS epidemic model with temporary vaccination was studied. Vaccination includes both susceptible and new members, and disease transmission takes place at standard incidence rate. The number of individuals which are added to the population per unit time is constant and differs from the number of individuals that die and leave the population per unit time. Thus, the total population size is variable. Dynamics of the model at the endemic equilibrium were determined by the basic reproduction number  0 ; when  0 > 1 , the unique endemic equilibrium E * is locally asymptotically stable and uniformly persists. In addition, it was shown that in this case E * is globally asymptotically stable under some conditions by employing a geometric approach and second additive compound matrix method. A numerical discussion was also performed to support the theoretical results.
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