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Articles

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

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Pages 259-288 | Received 09 Aug 2021, Accepted 02 Feb 2022, Published online: 23 Feb 2022
 

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 R0. We establish the global asymptotic stability of the disease-free equilibrium (GAS) for R0<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 R0>1. For R0=1, the transcritical bifurcation appears. For R0>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 R0.

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Acknowledgments

All authors contributed equally and significantly in writing this article.

Disclosure statement

The authors declare that they have no competing interests.

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