Study On Dynamical Models Of Epidemic Diseases With Nonmonotone Incidence Rate

It is well known that infectious disease are the archenemy of mankind. Over the past decades, human made permanent struggle to all kinds of infectious and have obtained many brilliant achievements, especially after 20th century much more achievements have been obtained. However, it is long way to go for the infectious disease. The reports from the World Health Organization(WHO) show that infectious disease are still the essential factor for human's health and life. So, it is very important to study the infectious disease either from the experiment and from the theoretical analysis.For the mathematical workers, some mathematical models for the infectious disease have already made. By the theoretical analysis, quantitative analysis and computer simulations to these gotten mathematical models, we have obtained many results which are very close to the real disease. Since from the 1980s, a rapid progress has been made for the mathematical models of many kinds of disease, which makes the theoretical studies more perfect. Liapunov's stability approach, LaSalle invariance principle, limited equation theory, characteristic roots methods, bifurcation theory, K sequence monotone system theory and the center manifold theory are the main methods to study the mathematical models of disease.Inoculate vaccination for the epidemic individual has become one of effective methods for diseases control. In this paper, we focus on study a SIS epidemic model with vaccination and nonlinear incidence rate (?).We first obtained the equilibria of this model and then defined the basic reproduction number "R₀" determining whether the endemic diseases occur or not. When R₀ < 1,the model has only a global asymptotically stable disease-free equilibrium and the disease will naturally disappear; when R₀ > 1,the model has stable endemic equilibrium and the disease will always exist in the population.Let N(t) , S(t), I(t) and R(t) be the number of total population, susceptible individuals, infective individuals and vaccinated individuals at time t, respectively. First, we have the following assumption:1) b is the recruitment rate of the population andδ(0 <δ< 1) is the ratio of vaccination of b; 2) The infectious disease spread by the nonmonotone incidence rate (?),and for the vaccinated individuals, the incidence rate is (?).WhereβI measures the infectious force of the disease in the susceptible individuals,σβI measures the infectious force of the disease in the vaccinated individuals, and the parameterσdescribes the efficiency of the vaccine,the largerσis the worse efficiency of the vaccine,σ= 0 means the efficiency of the vaccine is 100% for the infectious disease control, andσ= 1 means the vaccination is ineffective;3) d is the natural death rate of the population;4) The constantμ≥0 is the ratio of vaccination of the susceptible individuals;5) The constantγ> 0 is the natural recovery rate of the infective individuals;6)ε> 0 is the rate at which vaccinated individuals lose immunity and return to the susceptible class;7)α≥0 is the parameter measures the psychological or inhibitory effect. Notice that whenα= 0,the nonmonotone incidence rate (?) becomes the bilinear incidence rateβS I;8) The recruitment individuals who are not vaccinated are susceptible individuals.Based on these assumption, we constructed the following mathematical model:We obtained some results as follow:Caseâ… ):whenσ= 1,we obtained the basic reproduction number, disease-free equilibrium,epidemic equilibrium are as follows:respectively, where andBy using the Liapunov function, LaSalle invariance principle and characteristic roots methods we have the following results for the stability of the equilibrium:Theorem 1 If R₀≤1,then there is no endemic equilibrium, and the disease-free equilibrium E₀ is globally asymptotically stable; if R₀ > 1,the disease-free equilibrium E₀ is unstable, the endemic equilibrium E⁺ is stable.Furthermore, we obtained the following schemes that can be used to control the spread of infectious disease:1) Decrease the effective contact rate small enough, do our most to and prevent the spread way from infectious disease;2) Reduce the recruitment rate b, decrease the mobility from one area to another;3) Improve the recovery rate y, speed up the study of drugs.Caseâ…¡): whenσ= 0,we obtained the basic reproduction number, disease-free equilibrium, epidemic equilibrium are as follows:respectively, whereThe stability of the disease-free equilibrium and endemic equilibrium is similar to the- orem 1,and here, we omitted the details. We only give some methods that can be used to control the spread of infectious disease:1) Methods used as same as the caseâ… ;2) Increase the probability of vaccination of the recruitment individuals big enough;3) Increase the vaccination rate of the susceptible individuals, especial in the poor area.Caseâ…¢):when 0 <σ< 1,we obtained the basic reproduction number and disease-freeequilibrium are as follows:respectively.Then we give a theorem for the global asymptotically stable of the disease-free equilibrium.Theorem 2 For the system (2.13), if R₀≤1,the disease-free equilibrium E_σ,is global asymptotically stable, and if R₀ > 1,the the disease-free equilibrium E_σis unstable.It is difficult to obtain the endemic equilibrium in this case. We just give a theorem to illustrate the stability of the endemic equilibrium.Theorem 3 If R₀ < 1,the system (2.13) nonexist positive equilibrium; if R₀ > 1. the system (2.13)probably exist positive equilibrium, and if the system (2.13) exist positive equilibrium E_σ⁺,then it was stable.