Periodic Behaviour of an Epidemic in a Seasonal Environment with Vaccination

Miled El Hajji (), Dalal M. Alshaikh and Nada A. Almuallem
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Miled El Hajji: Department of Mathematics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia
Dalal M. Alshaikh: Department of Mathematics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia
Nada A. Almuallem: Department of Mathematics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia

Mathematics, 2023, vol. 11, issue 10, 1-20

Abstract: Infectious diseases include all diseases caused by the transmission of a pathogenic agent such as bacteria, viruses, parasites, prions, and fungi. They, therefore, cover a wide spectrum of benign pathologies such as colds or angina but also very serious ones such as AIDS, hepatitis, malaria, or tuberculosis. Many epidemic diseases exhibit seasonal peak periods. Studying the population behaviours due to seasonal environment becomes a necessity for predicting the risk of disease transmission and trying to control it. In this work, we considered a five-dimensional system for a fatal disease in a seasonal environment. We studied, in the first step, the autonomous system by investigating the global stability of the steady states. In a second step, we established the existence, uniqueness, positivity, and boundedness of a periodic orbit. We showed that the global dynamics are determined using the basic reproduction number denoted by R 0 and calculated using the spectral radius of an integral operator. The global stability of the disease-free periodic solution was satisfied if R 0 < 1 , and we show also the persistence of the disease once R 0 > 1 . Finally, we displayed some numerical investigations supporting the theoretical findings, where the trajectories converge to a limit cycle if R 0 > 1 .

Keywords: SVEIR epidemic model; seasonal environment; periodic solution; Lyapunov stability; uniform persistence; extinction; basic reproduction number (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (4)

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