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

- 252 Downloads

## Abstract

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 \({\mathcal {R}}_0 > 1\). 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.

## Keywords

SIS epidemic model Vaccination Endemic equilibrium Global stability Geometric approach## Mathematics Subject Classification

92D25 34D23## 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, 10, 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

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 \(\varLambda\) individuals into population per unit time that may be either immigrants or newborns. The vaccination program is applied on the new members by a proportion \(\vartheta\). A proportion \(\sigma\) of susceptible individuals are also vaccinated, and the rate of losing immunity of vaccination is \(\xi\). The vaccine is supposed to be completely effective, and no vaccinated individual becomes infected. Susceptible individuals become infected at standard incidence rate \(\frac{\beta S I}{N},\) where \(\beta\) is transmission coefficient, whereas the rate of recovery is \(\eta\). The rate of natural death in population is \(\delta\) , and the mortality due to disease is also included in the model with a rate \(\alpha\). All values in the model are nonnegative except \(\varLambda\) and \(\delta\) that are assumed positive.

*N*is not fixed in general and it is expressed by the following equation:

*I*,

*S*, and

*V*are bounded. We can see that the feasible region

*the basic reproduction number*of the model. Furthermore, the total population sizes at two equilibria \(E^0\) and \(E^*\) are obtained, respectively, as follows:

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

- (I)\(-a_1+a_2 = -(\delta +\eta +\alpha )+\delta \beta \left( \frac{{\mathcal {R}}_0-1}{\delta \widetilde{{\mathcal {R}}}_0 +\alpha (\widetilde{{\mathcal {R}}}_0-{\mathcal {R}}_0)}\right);\) thus,$$\begin{aligned} b_1=\delta \beta \left( \frac{{\mathcal {R}}_0-1}{\delta \widetilde{{\mathcal {R}}}_0 +\alpha (\widetilde{{\mathcal {R}}}_0-{\mathcal {R}}_0)}\right) +(2\delta +\sigma +\xi )>0. \end{aligned}$$
- (II)We can see thatand therefore,$$\begin{aligned}&-a_1(2\delta +\sigma +\xi ) + a_3\sigma +(\delta +\eta +\alpha )(2\delta +\sigma +\xi )\\&\quad = (\delta +\eta +\alpha ) \left( \frac{\delta ({\mathcal {R}}_0-1)}{\delta \widetilde{{\mathcal {R}}}_0+\alpha (\widetilde{{\mathcal {R}}}_0-{\mathcal {R}}_0)}\right) (2\delta +\xi ), \end{aligned}$$$$\begin{aligned} b_2&= (\delta +\eta +\alpha ) \left( \frac{\delta ({\mathcal {R}}_0-1)}{\delta \widetilde{{\mathcal {R}}}_0+\alpha (\widetilde{{\mathcal {R}}}_0-{\mathcal {R}}_0)}\right) (2\delta +\xi )\\&+ a_2(2\delta +\alpha +\xi ) + \delta (\delta +\sigma +\xi )>0. \end{aligned}$$
- (III)We have$$\begin{aligned} b_3= & {} \delta (\delta +\sigma +\xi )(\delta +\eta +\alpha ) \left( \frac{\delta ({\mathcal {R}}_0-1)}{\delta \widetilde{{\mathcal {R}}}_0+\alpha (\widetilde{{\mathcal {R}}}_0-{\mathcal {R}}_0)}\right) \\&+(\beta (\delta +\xi )\\&-(\delta +\sigma +\xi )(\delta +\eta +\alpha ))\delta (\delta +\alpha ) \left( \frac{{\mathcal {R}}_0-1}{\delta \widetilde{{\mathcal {R}}}_0+\alpha (\widetilde{{\mathcal {R}}}_0-{\mathcal {R}}_0)}\right) \\&+\delta (\delta +\sigma +\xi )(\delta +\eta +\alpha ). \end{aligned}$$Besides, \(\widetilde{{\mathcal {R}}}_0>{\mathcal {R}}_0\) and \({\mathcal {R}}_0>1\) implies \(\beta (\delta +\xi )>(\delta +\sigma +\xi )(\delta +\eta +\alpha )\). Therefore,$$\begin{aligned} b_3>\delta (\delta +\sigma +\xi )(\delta +\eta +\alpha )\left( 1+\frac{\delta ({\mathcal {R}}_0-1)}{\delta \widetilde{{\mathcal {R}}}_0+\alpha (\widetilde{{\mathcal {R}}}_0-{\mathcal {R}}_0)}\right) >0. \end{aligned}$$
- (IV)We see that \(a_1 = (\delta +\eta +\alpha ) \left( 1-\frac{\delta ({\mathcal {R}}_0-1)}{\delta \widetilde{{\mathcal {R}}}_0+\alpha (\widetilde{{\mathcal {R}}}_0-{\mathcal {R}}_0)}\right) < (\delta +\eta +\alpha ),\) and as a result,$$\begin{aligned} b_1b_2-b_3&= (\delta +\xi )(b_2-a_2(\delta +\alpha )) \\&\quad+(\delta +\eta +\alpha )(b_2-\delta (\delta +\sigma +\xi ))\\&\quad+ (-a_1+a_2+\delta +\sigma )b_2\\&\quad+a_1\delta (\delta +\sigma +\xi )-a_3\sigma (\delta +\alpha )\\&= (\delta +\xi )(b_2-a_2(\delta +\alpha ))\\&\quad+((\delta +\eta +\alpha )-a_1)(b_2-\delta (\delta +\sigma +\xi ))\\&\quad+(a_2+\delta +\sigma )b_2-a_3\sigma (\delta +\alpha ). \end{aligned}$$$$\begin{aligned} b_1b_2-b_3> (\delta +\xi )(b_2-a_2(\delta +\alpha ))+(a_2+\delta +\sigma )b_2-a_3\sigma (\delta +\alpha ) >0. \end{aligned}$$

In preceding relations, we got \(b_1>0\), \(b_2>0\), \(b_3>0\) and \(b_1b_2-b_3>0\); thus, by using Routh–Hurwitz criterion the real part of all eigenvalues of the Jacobian matrix \(J^*\) must be negative. Therefore, the following theorem has been proven:

### Theorem 1

*For epidemic model* (1), *the endemic state* \(E^*\) *exists and is stable if* \({\mathcal {R}}_0>1\).

### Corollary 1

*The endemic state*\(E^*\)

*is stable if parameter values*\(\vartheta\)

*and*\(\sigma\)

*lie under the following line in the*\(\vartheta\)-\(\sigma\)

*plane:*

## 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].

### Theorem 2

*Assume that*

*f*

*is a*\(C^1\)

*function on a simply connected open set*\(D\subset \mathbb {R}^n\),

*the system*\(x'=f(x)\)

*has a unique equilibrium*\(\bar{x}\)

*in*

*D*,

*and there exists a compact absorbing set*\(\varTheta \subset D\).

*Then,*\(\bar{x}\)

*is globally stable in*

*D*

*if*

*in which*\(\varUpsilon =\varSigma _f\varSigma ^{-1}+\varSigma J^{[2]}\varSigma ^{-1}\).

*Here*, \(\varSigma\)

*is an*\(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \times \left( {\begin{array}{c}n\\ 2\end{array}}\right)\)

*matrix-valued function,*\(J^{[2]}\)

*is the second additive compound matrix of the Jacobian matrix*

*J*, \(\varSigma _f\)

*is obtained by*\((\sigma _{ij})_f=\left( \frac{\partial \sigma _{ij}}{\partial x}\right) ^\top . f(x)\),

*and*\(\varPsi (\varUpsilon )\)

*is the*

*Lozinski*\(\breve{\imath }\)

*measure of*\(\varUpsilon\)

*with respect*

*to a vector norm*|.|

*in*\(\mathbb {R}^m\),

*with*\(m=\left( {\begin{array}{c}n\\ 2\end{array}}\right)\),

*defined by*\(\varPsi (\varUpsilon )= \mathop {\lim }\limits _{h \rightarrow 0^+} \frac{|I+h\varUpsilon |-1}{h}\).

The global stability of the endemic state is expressed in the next theorem.

### Theorem 3

*When* \({\mathcal {R}}_0>1\) *and* \(\delta +\sigma >\xi\), *the endemic equilibrium* \(E^*\) *is globally asymptotic stable if* \(\delta +\xi > \max \{\sigma , \alpha \}+ \eta\).

### Proof

Considering the feasible region \(\varGamma\), we see that interior and boundary of \(\varGamma\) are \(\mathop {\varGamma }\limits ^{\circ } = \{(I, S, V)\in \varGamma : I>0 \}\) and \(\partial \varGamma = \varGamma \setminus \mathop {\varGamma }\limits ^{\circ } = \{(I, S, V)\in \varGamma : I=0 \}\), respectively. When \({\mathcal {R}}_0>1\), there exists an unique endemic state \(E^*\) , and moreover, the disease-free equilibrium (DFE) \(E^0 \in \partial \varGamma\) is unstable. Now, using *Acyclicity Theorem* similar to the proof of Theorem 3.2 in [18] we find that when \({\mathcal {R}}_0>1\), system (1) is uniformly persistent. On the other hand, we know that the 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 \(\varTheta \subset \varGamma\).

Matrix \(\varSigma\) 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 \(\varSigma\) as \(\varSigma =\frac{S}{I}\mathbb {I}_3\), where \(\mathbb {I}_3\) is the \(3\times 3\) identity matrix. Then, \(\varSigma ^{-1}=\frac{I}{S}\mathbb {I}_3\), \(\varSigma _f=\left( \frac{\frac{{\text{d}}S}{{\text{d}}t}}{I}-\frac{S\left( \frac{{\text{d}}I}{{\text{d}}t}\right) }{I^2}\right) \mathbb {I}_3\) and \(\varSigma _f\varSigma ^{-1}=\left( \frac{\frac{{\text{d}}S}{{\text{d}}t}}{S}-\frac{\frac{{\text{d}}I}{{\text{d}}t}}{I}\right) \mathbb {I}_3\).

Therefore, the conditions of Theorem 2 are hold and this concludes that \(E^*\) is globally asymptotically stable. \(\square\)

## Numerical examples

*I*(

*t*) and

*S*(

*t*) (sub-figure (b)) when \(\tau =0.4\), and it can be seen that \(E^0\) is stable. Figure 4 shows that

*I*(

*t*) does not vanish when \(\tau =0.05\), 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 \({\mathcal {R}}_0\); when \({\mathcal {R}}_0 >1\) , 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.

## Notes

## References

- 1.Allen, L.J.: Introduction to Mathematical Biology. Pearson/Prentice Hall, Upper Saddle River (2007)Google Scholar
- 2.Arino, J., McCluskey, C.C., van den Driessche, P.: Global results for an epidemic model with vaccination that exhibits backward bifurcation. SIAM J. Appl. Math.
**64**(1), 260–276 (2003)MathSciNetCrossRefGoogle Scholar - 3.Brauer, F., Castillo-Chavez, C.: Mathematical Models in Population Biology and Epidemiology, vol. 1. Springer, Berlin (2001)CrossRefGoogle Scholar
- 4.Chalub, F.A., Souza, M.O.: Discrete and continuous SIS epidemic models: a unifying approach. Ecol. Complex.
**18**, 83–95 (2014)CrossRefGoogle Scholar - 5.Farnoosh, R., Parsamanesh, M.: Disease extinction and persistence in a discrete-time SIS epidemic model with vaccination and varying population size. Filomat
**31**(15), 4735–4747 (2017)MathSciNetCrossRefGoogle Scholar - 6.Farnoosh, R., Parsamanesh, M.: Stochastic differential equation systems for an SIS epidemic model with vaccination and immigration. Commun. Stat. Theory Methods
**46**(17), 8723–8736 (2017)MathSciNetCrossRefGoogle Scholar - 7.Hu, Z., Teng, Z., Jiang, H.: Stability analysis in a class of discrete SIRS epidemic models. Nonlinear Anal. Real World Appl.
**13**(5), 2017–2033 (2012)MathSciNetCrossRefGoogle Scholar - 8.Jami, P., Khodabin, M., Hashemizadeh, E.: Numerical solution of stochastic SIR model via split-step forward Milstein method. J. Interpolat. Approx. Sci. Comput.
**2016**(1), 38–45 (2016)MathSciNetGoogle Scholar - 9.Jianquan, L., Zhien, M.: Global analysis of SIS epidemic models with variable total population size. Math. Comput. Model.
**39**(11), 1231–1242 (2004)MathSciNetCrossRefGoogle Scholar - 10.Kribs-Zaleta, C.M., Velasco-Hernandez, J.X.: A simple vaccination model with multiple endemic states. Math. Biosci.
**164**(2), 183–201 (2000)CrossRefGoogle Scholar - 11.Li, J., Ma, Z.: Qualitative analyses of SIS epidemic model with vaccination and varying total population size. Math. Comput. Model.
**35**(11), 1235–1243 (2002)MathSciNetCrossRefGoogle Scholar - 12.Li, M.Y., Muldowney, J.S.: A geometric approach to global-stability problems. SIAM J. Math. Anal.
**27**(4), 1070–1083 (1996)MathSciNetCrossRefGoogle Scholar - 13.Liu, X., Takeuchi, Y., Iwami, S.: SVIR epidemic models with vaccination strategies. J. Theoret. Biol.
**253**(1), 1–11 (2008)MathSciNetCrossRefGoogle Scholar - 14.Parsamanesh, M.: Global stability analysis of a VEISV model for network worm attack. Univ. Politehnica Bucharest Sci. Bull. Ser. A Appl. Math. Phys.
**79**(4), 179–188 (2017)MathSciNetGoogle Scholar - 15.Parsamanesh, M.: Global dynamics of an SIVS epidemic model with bilinear incidence rate. Ital. J. Pure Appl. Math.
**40**, 544–557 (2018)Google Scholar - 16.Rahmani, N., Khodabin, M., Hashemizadeh, E.: Numerical solution of stochastic SIR model by Bernstein polynomials. J. Interpol. Approx. Sci. Comput.
**2016**(1), 19–25 (2016)MathSciNetGoogle Scholar - 17.Safan, M., Rihan, F.A.: Mathematical analysis of an SIS model with imperfect vaccination and backward bifurcation. Math. Comput. Simul.
**96**, 195–206 (2014)MathSciNetCrossRefGoogle Scholar - 18.Sun, C., Hsieh, Y.H.: Global analysis of an SEIR model with varying population size and vaccination. Appl. Math. Model.
**34**(10), 2685–2697 (2010)MathSciNetCrossRefGoogle Scholar - 19.Yang, W., Sun, C., Arino, J.: Global analysis for a general epidemiological model with vaccination and varying population. J. Math. Anal. Appl.
**372**(1), 208–223 (2010)MathSciNetCrossRefGoogle Scholar - 20.Yang, Y.: Global stability of VEISV propagation modeling for network worm attack. Appl. Math. Model.
**39**(2), 776–780 (2015)MathSciNetCrossRefGoogle Scholar - 21.Zhang, X., Liu, X.: Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment. Nonlinear Anal. Real World Appl.
**10**(2), 565–575 (2009)MathSciNetCrossRefGoogle Scholar - 22.Zhao, Y., Jiang, D.: The threshold of a stochastic SIS epidemic model with vaccination. Appl. Math. Comput.
**243**, 718–727 (2014)MathSciNetzbMATHGoogle Scholar - 23.Zhou, X., Cui, J.: Modeling and stability analysis for a cholera model with vaccination. Math. Methods Appl. Sci.
**34**(14), 1711–1724 (2011)MathSciNetzbMATHGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.