Stability of general pathogen dynamic models with two types of infectious transmission with immune impairment

B. S. Alofi; S. A. Azoz; B. S. Alofi; S. A. Azoz

doi:10.3934/math.2021009

AIMS Mathematics

2021, Volume 6, Issue 1: 114-140. doi: 10.3934/math.2021009

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Research article

Stability of general pathogen dynamic models with two types of infectious transmission with immune impairment

B. S. Alofi ^{1
,
,},
S. A. Azoz ^{2
,
,}

1.
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt

Received: 09 June 2020 Accepted: 03 September 2020 Published: 30 September 2020
MSC : 34D20, 34D23, 37N25, 92B05

In this paper, we investigate the global properties of two general models of pathogen infection with immune deficiency. Both pathogen-to-cell and cell-to-cell transmissions are considered. Latently infected cells are included in the second model. We show that the solutions are nonnegative and bounded. Lyapunov functions are organized to prove the global asymptotic stability for uninfected and infected steady states of the models. Analytical expressions for the basic reproduction number $\mathcal{R}_{0}$ and the necessary condition under which the uninfected and infected steady states are globally asymptotically stable are established. We prove that if $\mathcal{R}_{0}$ < 1 then the uninfected steady state is globally asymptotically stable (GAS), and if $\mathcal{R}_{0}$ > 1 then the infected steady state is GAS. Numerical simulations are performed and used to support the analytical results.
- pathogen infection,
- cell-to-cell transmission,
- immune impairment,
- global stability
Citation: B. S. Alofi, S. A. Azoz. Stability of general pathogen dynamic models with two types of infectious transmission with immune impairment[J]. AIMS Mathematics, 2021, 6(1): 114-140. doi: 10.3934/math.2021009

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Abstract

In this paper, we investigate the global properties of two general models of pathogen infection with immune deficiency. Both pathogen-to-cell and cell-to-cell transmissions are considered. Latently infected cells are included in the second model. We show that the solutions are nonnegative and bounded. Lyapunov functions are organized to prove the global asymptotic stability for uninfected and infected steady states of the models. Analytical expressions for the basic reproduction number $\mathcal{R}_{0}$ and the necessary condition under which the uninfected and infected steady states are globally asymptotically stable are established. We prove that if $\mathcal{R}_{0}$ < 1 then the uninfected steady state is globally asymptotically stable (GAS), and if $\mathcal{R}_{0}$ > 1 then the infected steady state is GAS. Numerical simulations are performed and used to support the analytical results.

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