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Abstract

Epidemic dynamics is to study disease spread, predict the trends of the disease, find outthe key factors in the process of the spread, and prevent the disease by the best methods.In epidemiological modelling, it is significant to consider an epidemic with multiple stage ofinfections i.e., where infectives progress through consecutive stages of infection. The infectedindividuals of some diseases, such as Chagas’disease, hepatitis C virus (HCV), gonorrhea andother sexually transmitted diseases, may advance through several infective stages, and havedi?erent ability to transmit these infections in di?erent stages of infection; their infectivityusually depends on the parasite or viral loads in infected individuals or vectors .In chapter1, we give the significance of investigating epidemic models, development of this filed andthe main work in this thesis.In chapter 2, the SEIRS epidemic model with vertical transmission and vaccinationis studied. It is proved that the global dynamics are completely determined by the basicreproduction number R0.If R0 < 1,the disease-free equilibrium is global asymptotically stableand the disease always dies out.If R0 > 1,a unique endemic equilibrium exists and is globallystable in the interior of the feasible region ,and the disease persists at the endemic equilibriumstate if it initially exits.At the same time,using global stability analysis of the model ,it isshown that the disease can be eradicated from the population if the vaccination coveragelevel exceeds a certain threshold value.It is also shown that the disease will persist withinthe population if the coverage level is below this threshold.In chapter 3, we present a time-delayed SIRS model with stage structure, where theperiod of infection is partitioned into two stage: the early and later stages. Immunity gainedby experiencing the disease is temporary;whenever infected,the disease individuals will returnto the susceptible class after a fixed period of tim.First, the local and global stabilities ofthe infection-free equilibrium are analyzed, respectively. Second, the endemic equilibriumis formulated , and the su?cient condition for its locally asymptotic stability are found.Finally, by using a Lyapunov functional, a su?cient criterion for the global stability of the endemic equilibrium is established.In chapter 4, an epidemic model with infectious force in infected and immune periodand treatment rate of infectious individuals is proposed to understand the e?ect of thecapacity for treatment of infective on the disease spread. It is assumed that treatment rateis proportional to the number of infective below the capacity and is a constant when thenumber of infective is greater than the capacity.It is proved that the existence and stabilityof equilibria for the model is not only related to the basic reproduction number but alsothe capacity for treatment of infective. It is found that a backward bifurcation occurs if thecapacity is small. It is also found that there exist bistable endemic equilibria if the capacityis low.In chapter 5 , a stage-structured epidemic model with a nonlinear incidence is inves-tigated. By carrying out global qualitative and bifurcation analysis, it is shown that themodel exhibits the bistability and undergoes the Hopf bifurcation and the Bogdanov-Takensbifurcation. Moreover, several numerical simulations are given to support the theoreticalanalysis.

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations
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Categories epidemic model with stage structure

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