Global asymptotic properties of an SEIRS model with multiple infectious stages

Abstract

The paper presents a rigorous mathematical analysis of a deterministic model, which uses a standard incidence function, for the transmission dynamics of a communicable disease with an arbitrary number of distinct infectious stages. It is shown, using a linear Lyapunov function, that the model has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction threshold is less than unity. Further, the model has a unique endemic equilibrium when the threshold exceeds unity. The equilibrium is shown to be locally-asymptotically stable, for a special case, using a Krasnoselskii sub-linearity trick. Finally, a non-linear Lyapunov function is used to show the global asymptotic stability of the endemic equilibrium (for the special case). Numerical simulation results, using parameter values relevant to the transmission dynamics of influenza, are presented to illustrate some of the main theoretical results.

Transmission dynamics of Trichomonas vaginalis: A mathematical approach

“…(19) It is worth mentioning that λ * m like λ * f 1 exists only when R N > 1. Solving for λ * f 2 in g(λ * f 2 ) = 0, the roots of g(λ * f 2 ) = 0 are explored using the Descartes rule of signs.…”

“…Quite recently some authors [34,19] did analyse some general SIR and SEIRS models with interesting dynamics. The later [19] did analyse SEIRS model incorporating multiple infectious stages.…”

“…Quite recently some authors [34,19] did analyse some general SIR and SEIRS models with interesting dynamics. The later [19] did analyse SEIRS model incorporating multiple infectious stages. Despite Trichomanas vaginalis being an old infection which is still affecting mankind no mathematical account of it has been carried out to the best knowledge of the authors.…”

Cross-immunity-induced backward bifurcation for a model of transmission dynamics of two strains of influenza

“…Further, let R 02 < 1 < R 01 , so that the boundary equilibrium E 1 exists (Theorem 4) and strain-2 dies out (Lemma 3). Consider the following non-linear Lyapunov function, of Goh-Volterra type (functions of this type have been used in the mathematical ecology/epidemiology literature, such as those in [7,22,26,28]):…”

Global Stability of an Epidemic Model with Two Infected Stages and Mass-Action Incidence

“…We can also cite more recent works. Particularly, the work of Melese and Gumel in [17], where for the proof of the endemic equilibrium stability, authors make a very strong assumption, which is very difficult to verify. We cite also and specially the work of M. Li, J. Graef, L. Wang and J. Karsai in [15], which deals with a similar system, but the authors used one contact rate.…”