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Stability analysis of a logistic growth epidemic model with two explicit time-delays, the nonlinear incidence and treatment rates

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Abstract

In the present study, a time-delayed SIR epidemic model with a logistic growth of susceptibles, Crowley–Martin type incidence, and Holling type III treatment rates is proposed and analyzed mathematically. We consider two explicit time-delays: one in the incidence rate of new infection to measuring the impact of the latent period, and another in the treatment rate of infectives to analyzing the effect of late treatment availability. The stability behavior of the model is analyzed for two equilibria: the disease-free equilibrium (DFE) and the endemic equilibrium (EE). We derive the threshold quantity, the basic reproduction number \(R_0\), which determines the eradication or persistence of infectious diseases in the host population. Using the basic reproduction number, we show that the DFE is locally asymptotically stable when \(R_0< 1\), linearly neutrally stable when \(R_0= 1\), and unstable when \(R_0> 1\) for the time-delayed system. We analyze the system without a latent period, revealing the forward bifurcation at \(R_0= 1\), which implies that keeping \(R_0\) below unity can diminish the disease. Further, the stability behavior for the EE is investigated, demonstrating the occurrence of oscillatory and periodic solutions through Hopf bifurcation concerning every possible grouping of two time-delays as the bifurcation parameter. To conclude, the numerical simulations in support of the theoretical findings are carried out.

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Acknowledgements

The authors are grateful to the Delhi Technological University, Delhi, India, for providing financial support to carry out this research work. They are also indebted to the handling editor and the anonymous reviewers for their constructive comments and suggestions, which led to an improvement in the original manuscript.

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Authors and Affiliations

  1. Department of Applied Mathematics, Delhi Technological University, Delhi, 110042, India

    Kanica Goel &  Nilam

  2. Department of Mathematics, School of Basic Sciences and Research, Sharda University, Greater Noida, 201310, India

    Abhishek Kumar

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  1. Kanica Goel
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  2. Abhishek Kumar
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Goel, K., Kumar, A. & Nilam Stability analysis of a logistic growth epidemic model with two explicit time-delays, the nonlinear incidence and treatment rates. J. Appl. Math. Comput. 68, 1901–1928 (2022). https://doi.org/10.1007/s12190-021-01601-1

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  • DOI: https://doi.org/10.1007/s12190-021-01601-1

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