key: cord-018832-g96earfl
authors: Song, Xiuchao; Liu, Miaohua; Song, Hao; Ren, Jinshen
title: Dynamical Behavior of an SVIR Epidemiological Model with Two Stage Characteristics of Vaccine Effectiveness and Numerical Simulation
date: 2019-10-08
journal: Advances in Intelligent Systems and Interactive Applications
DOI: 10.1007/978-3-030-34387-3_29
sha: 
doc_id: 18832
cord_uid: g96earfl

An SVIR epidemiological model with two stage characteristics of vaccine effectiveness is formulated. By constructing the appropriate Lyapunov functionals, it is proved that the disease free equilibrium of the system is globally stable when the basic reproduction number is less than or equal to one, and that the unique endemic equilibrium of the system is globally stable when the basic reproduction number is greater than one.

In human history, infectious diseases have repeatedly brought great disaster to human survival. In recent years, the outbreak of some new infectious diseases (SARS, influenza A (H1N1), influenza A (H7N9), etc.) has caused a great impact on people's lives. Vaccines are biological agents made from bacteria, viruses, tumor cells and so on, which enable antibodies to produce specific immunity. Vaccination can provide immunity to those who are vaccinated, can eliminate the spread of some diseases (such as smallpox) [1] .

In recent years, more and more authors study the epidemiological models with vaccination [2] [3] [4] [5] . Some authors assume that vaccine recipients will not be infected [2, 3] ; some other authors assume that vaccine recipients may still be infected [4, 5] , but the probability of being infected is smaller than before vaccination. In fact, for some infectious diseases, the vaccinated individuals would not be infected for some time after vaccination. However, bacteria or viruses mutate as time goes by, and the efficacy of the vaccine is correspondingly affected, which makes it is possible for the vaccinated individuals to be infected. For example, the new H7N9 influenza virus mutates more quickly, and the effectiveness of the vaccine depends largely on the extent of the virus mutation [6] . Based on the above facts, we assume that vaccine effectiveness has two stage characteristics: in the first stage, the vaccinated individuals will not be infected; in the second stage, the vaccinated individuals will be infected, but the probability of infection will be smaller than before vaccination. Therefore, this paper studies the epidemiological model with two stage characteristics of vaccine effectiveness, On the basis of getting the basic reproductive number, by using appropriate functionals, the stability of the model is proved by the algebraic approach provided by the reference [8] .

In this work, we study the following epidemiological model:

The model (1) has the same dynamic behavior with the following system

2 Existence of Equilibria

Obviously, system (2) has a disease free equilibrium P 0 ðS 0 ; V 10 ; V 20 ; 0Þ, where

It can be found the unique endemic equilibrium P Ã ðS Ã ; V Ã 1 ; V Ã 2 ; I Ã Þ from the following equations,

and I Ã satisfies the following equation

Theorem. When R 0 1 the P 0 ðS 0 ; V 10 ; V 20 ; 0Þ is global stable. And P Ã is global stable when R 0 [ 1.

Proof. The global stability of P 0 is firstly proved. Consider the following Lyapunov functional

where

For simplicity, denote

Using the algebraic approach provided by the reference [8] , we will prove the function Hðx 1 ; x 2 ; x 3 Þ 0. Firstly, we can get five groups

x 2 x 3 g and the product of all functions within each group is one, then we have

As the nonnegativity of b i ði ¼ 1; 2. . .5Þ; b 4 must satisfy the following condition maxf0; peS 0 c 2 À kV 20 g b 4 minflð1 À qÞA; l V 20 ; peS 0 c 2 g;

It is easy to prove the existence of the positive number b 4 . So Hðx 1 ; x 2 ; x 3 Þ 0 and Hðx 1 ;

In summary, when R 0 \1 we have L 0 1 ; and when R 0 ¼ 1; we get L 0 1 0; and L 0 1 ¼ 0 if and only if S ¼ S 0 V 1 ¼ V 10 V 2 ¼ V 20 . The largest invariant set for (2) on the set fðS; V 1 ; V 2 ; IÞ 2 X : S ¼ S 0 ; V 1 ¼ V 10 ; V 2 ¼ V 20 g is fP 0 g. Using the literature [10] , we can prove the theorem.

The numerical simulations on system (2) were carried out. We can see that if R 0 1, then P 0 ðS 0 ; V 10 ; V 20 ; 0Þ is global stable (Fig. 1) and P Ã is globally stable when R 0 [ 1 (Fig. 2) . 

World Health Organization: Immunization against diseases of public health importance

Two different vaccination strategies in an SIR epidemic model with saturated infectious force

Stability analysis for SIS epidemic models with vaccination and constant population size

Global stability of an epidemic model with latent stage and vaccination

Global analysis for a general epidemiological model with vaccination and varying population

Characterization of H7N9 influenza A viruses isolated from humans

Long-term immunogenicity of hepatitis B vaccination in a cohort of Italian healthy adolescents

An algebraic approach to proving the global stability of a class of epidemic models

Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission

The stability of dynamical systems

Acknowledgements. This paper is supported by the Research Fund of Department of Basic Sciences at Air Force Engineering University (2019107).