Analysis Of An Age-structured SIS And SEIS Epidemic Model With Vertical Transmission

The spreading of the epidemics bring heavy disaster to human,therefore, it is of greatimportance to establish a reasonable model of mathematics and detect the propagation ofinfective diseases to research its law. The main contents of this paper consist of two parts.In the first part, an age-structured SIS epidemic model with vertical transmission is dis-cussed. With the assumption that the total population is invariant in the model, the ba-sic reproductive number R₀which determines the extinction of epidemic disease is obtained.The disease-free equilibrium is locally asymptotically stable when the basic reproductive num-ber R₀<1, and the disease is extinct. The disease-free equilibrium is unstable when the basicreproduction number R₀>1, and a positive endemic solution appears which is also locallyasymptotically stable under a certain condition.In the second part, an age-structured SEIS epidemic model with vertical transmissionis discussed. With the assumption that the total population is invariant in the model, theexistence and uniqueness of solution are proved, and continuous dependence of solution forinitial value are obtained. And then, by using the theory and method of Diferential andIntegral Equation, the basic reproductive number R₀is obtained. The disease-free equilibriumis globally asymptotically stable when the basic reproductive number R₀<1. The disease-freeequilibrium is unstable when the basic reproduction number R₀>1, and endemic equilibriumis locally asymptotically stable.