On the date of the epidemic peak

Ali Moussaoui; Mohammed Meziane; Ali Moussaoui; Mohammed Meziane

doi:10.3934/mbe.2024126

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 2: 2835-2855. doi: 10.3934/mbe.2024126

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Research article Special Issues

On the date of the epidemic peak

Ali Moussaoui ^,,
Mohammed Meziane

Laboratory of Nonlinear Analysis and Applied Mathematics, Department of Mathematics, Faculty of Sciences, University of Tlemcen, Algeria

Academic Editor: Libin Rong

Received: 10 December 2023 Revised: 04 January 2024 Accepted: 16 January 2024 Published: 25 January 2024

Epidemiologists have used the timing of the peak of an epidemic to guide public health interventions. By determining the expected peak time, they can allocate resources effectively and implement measures such as quarantine, vaccination, and treatment at the right time to mitigate the spread of the disease. The peak time also provides valuable information for those modeling the spread of the epidemic and making predictions about its future trajectory. In this study, we analyze the time needed for an epidemic to reach its peak by presenting a straightforward analytical expression. Utilizing two epidemiological models, the first is a generalized $ SEIR $ model with two classes of latent individuals, while the second incorporates a continuous age structure for latent infections. We confirm the conjecture that the peak occurs at approximately $ T\sim(\ln N)/\lambda $, where $ N $ is the population size and $ \lambda $ is the largest eigenvalue of the linearized system in the first model or the unique positive root of the characteristic equation in the second model. Our analytical results are compared to numerical solutions and shown to be in good agreement.
- SEIR model,
- disease-age structure,
- final size
Citation: Ali Moussaoui, Mohammed Meziane. On the date of the epidemic peak[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 2835-2855. doi: 10.3934/mbe.2024126

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Abstract

Epidemiologists have used the timing of the peak of an epidemic to guide public health interventions. By determining the expected peak time, they can allocate resources effectively and implement measures such as quarantine, vaccination, and treatment at the right time to mitigate the spread of the disease. The peak time also provides valuable information for those modeling the spread of the epidemic and making predictions about its future trajectory. In this study, we analyze the time needed for an epidemic to reach its peak by presenting a straightforward analytical expression. Utilizing two epidemiological models, the first is a generalized $ SEIR $ model with two classes of latent individuals, while the second incorporates a continuous age structure for latent infections. We confirm the conjecture that the peak occurs at approximately $ T\sim(\ln N)/\lambda $, where $ N $ is the population size and $ \lambda $ is the largest eigenvalue of the linearized system in the first model or the unique positive root of the characteristic equation in the second model. Our analytical results are compared to numerical solutions and shown to be in good agreement.

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