AN INFLUENZA MODEL WITH MULTIPLE INFECTIOUS STAGES

The paper presents a rigorous mathematical analysis of adeterministic model, of the transmission dynamics of an infuenza dis-ease with n distinct infectious stages. A deterministic model of the formSEInRS will be used. The rst is that standard incidence will be used forthe disease transmission process. The second is that a class of exposed (la-tent) individuals (E) is incorporated (exposed individuals are individualswho are infected but have yet to show clinical symptoms of the disease).In this paper, it is assumed that recovered individuals eventually lose theirinfection-acquired immunity and become fully susceptible again. In addi-tion to the above extensions, rigorous qualitative analysis will be providedfor the resulting autonomous SEInRS model. The main objective hereis to determine whether adding multiple infectious stages to the classicalSEIRS model (with standard incidence) will alter the transmission dy-namics of the SEIRS model (particularly in regards to the persistenceor elimination of the disease from the population). The paper presentsa rigorous mathematical analysis of a deterministic model, which uses astandard incidence function, for the transmission dynamics of a communi-cable disease with an arbitrary number of distinct infectious stages. It isshown, using a linear Lyapunov function, that the model has a globally-asymptotically stable disease-free equilibrium whenever the associated re-production threshold is less than unity. Further, the model has a uniqueendemic equilibrium when the threshold exceeds unity. Finally, a non-linear Lyapunov function is used to show the global asymptotic stabilityof the endemic equilibrium (for the special case). A notable feature ofthe resulting infuenza model is that it uses gamma distribution assump-tions for the average waiting times in the infectious (n) stages. Numericalsimulation results are presented to illustrate some of the main theoreticalresults. From epidemiological point of view, this study shows that thedisease being considered can be eliminated from the population wheneverthe associated reproduction number is brought to (and maintained at) avalue less unity. The disease will persist in the community whenever thereproduction number exceeds unity.