An epidemic model with time delays determined by the infectivity and disease durations

Masoud Saade; Samiran Ghosh; Malay Banerjee; Vitaly Volpert; Masoud Saade; Samiran Ghosh; Malay Banerjee; Vitaly Volpert

doi:10.3934/mbe.2023574

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 7: 12864-12888. doi: 10.3934/mbe.2023574

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An epidemic model with time delays determined by the infectivity and disease durations

1.
Peoples Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St, Moscow 117198, Russia
2.
Department of Mathematics and Statistics, IIT Kanpur, Kanpur 208016, India
3.
Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne 69622, France

Academic Editor: Jia Li

Received: 17 March 2023 Revised: 19 May 2023 Accepted: 24 May 2023 Published: 02 June 2023

We propose an epidemiological model with distributed recovery and death rates. It represents an integrodifferential system of equations for susceptible, exposed, infectious, recovered and dead compartments. This model can be reduced to the conventional ODE model under the assumption that recovery and death rates are uniformly distributed in time during disease duration. Another limiting case, where recovery and death rates are given by the delta-function, leads to a new point-wise delay model with two time delays corresponding to the infectivity period and disease duration. Existence and positiveness of solutions for the distributed delay model and point-wise delay model are proved. The basic reproduction number and the final size of the epidemic are determined. Both, the ODE model and the delay models are used to describe COVID-19 epidemic progression. The delay model gives a better approximation of the Omicron data than the conventional ODE model from the point of view of parameter estimation.
- epidemic model,
- distributed recovery and death rates,
- disease duration,
- time delay
Citation: Masoud Saade, Samiran Ghosh, Malay Banerjee, Vitaly Volpert. An epidemic model with time delays determined by the infectivity and disease durations[J]. Mathematical Biosciences and Engineering, 2023, 20(7): 12864-12888. doi: 10.3934/mbe.2023574

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Abstract

We propose an epidemiological model with distributed recovery and death rates. It represents an integrodifferential system of equations for susceptible, exposed, infectious, recovered and dead compartments. This model can be reduced to the conventional ODE model under the assumption that recovery and death rates are uniformly distributed in time during disease duration. Another limiting case, where recovery and death rates are given by the delta-function, leads to a new point-wise delay model with two time delays corresponding to the infectivity period and disease duration. Existence and positiveness of solutions for the distributed delay model and point-wise delay model are proved. The basic reproduction number and the final size of the epidemic are determined. Both, the ODE model and the delay models are used to describe COVID-19 epidemic progression. The delay model gives a better approximation of the Omicron data than the conventional ODE model from the point of view of parameter estimation.

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