Global Stability Of Numerical Schemes For Two Classes Of Epidemic Models

With the development of international communication, the deterioration ofenvironmental pollution, and the increase of the resistance of the media andpathogens, epidemic diseases become a serious problem that we have to be facedwith again. As is known to all, prevention is more significant than treatment to someextent. The analysis of the epidemic model is an important method for thetheoretical study of infectious diseases’ epidemic law, which is an important basisfor the prevention work.For the statistics of infectious diseases are not continuous but collected in acertain interval, and the discrete-time systems permit arbitrary time-step units, sothe discrete systems can use statistics more conveniently and effectively. On theother hand, epidemic models are usually constructed by suitable ordinarydifferential equations, and the analytic solutions of ordinary differential equation areusually difficult to obtain. So proper numerical methods are required to calculateanalytic approximations of the solutions perfectly. But the dynamic behaviors of thediscreted system is usually complex. Consequently, the construction of thenumerical schemes which can preserve the dynamic behaviors of the correspondingcontinuous systems is a significant topic.The paper constructs discrete-time schemes of a multigroup SIR epidemicmodel with nonlinear incidence rates and a multigroup SIR epidemic model withdistributed delay by applying nonstandard finite difference methods, respectively.Then the dynamic behaviors of the discrete-time schemes are studied. It is shownthat discrete-time schemes preserve not only the positivity and boundedness of thesolution unconditionally, but also the global asymptotic stability of the equilibria byapplying discrete-time analogue of Lyapunov functional techniques and a recentlydeveloped graph-theoretic approach. For any time-step, if the basic reproductionnumber is less than or equal to one, the disease-free equilibrium is the uniqueequilibrium and it is globally asymptotic stable; if it is greater than one, the uniqueendemic equilibrium is globally stable while the disease free equilibrium is unstable.