Global behaviour of a class of discrete epidemiological SI models with constant recruitment of susceptibles

Abstract

Motivated by the recent paper [M.R.S. Kulenović, M. Nurkanović, and A.A. Yakubu, Asymptotic behaviour of a discrete-time density-dependent SI epidemic model with constant recruitment, J. Appl. Math. Comput. 67 (2021), pp. 733–753. DOI:10.1007/s12190-021-01503-2], in this paper, we consider the class of the SI epidemic models with recruitment where the Poisson function, a decreasing exponential function of the population of infectious individuals, is replaced by a general probability function that satisfies certain conditions. We compute the basic reproduction number ${\mathcal{R}}_0.$ We establish the global asymptotic stability of the disease-free equilibrium (GAS) for ${\mathcal{R}}_0<1.$ We use the Lyapunov function method developed in [P. van den Driessche and A.-A. Yakubu, Disease extinction versus persistence in discrete-time epidemic models, Bull. Math. Biol. 81 (2019), pp. 4412–4446], to demonstrate the GAS of the disease-free equilibrium and uniform persistence of the considered class of models. We show that the considered type of model is permanent for ${\mathcal{R}}_0>1$. For ${\mathcal{R}}_0=1,$ the transcritical bifurcation appears. For ${\mathcal{R}}_0>1,$ we prove the global attractivity result for endemic equilibrium and instability of the disease-free equilibrium. We apply theoretical results to specific escape functions of the susceptibles from infectious individuals. For each case, we compute the basic reproduction number ${\mathcal{R}}_0$.

Keywords:

• Basic reproductive number
• epidemic model
• persistence
• permanence
• stability

AMS CLASSIFICATIONS:

• 39A28
• 39A30
• 39A60
• 92D30

Acknowledgments

All authors contributed equally and significantly in writing this article.

Disclosure statement

The authors declare that they have no competing interests.