Periodic solutions for chikungunya virus dynamics in a seasonal environment with a general incidence rate

Miled El Hajji; Miled El Hajji

doi:10.3934/math.20231269

AIMS Mathematics

2023, Volume 8, Issue 10: 24888-24913. doi: 10.3934/math.20231269

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Periodic solutions for chikungunya virus dynamics in a seasonal environment with a general incidence rate

Miled El Hajji ^,

Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia

Received: 09 July 2023 Revised: 16 August 2023 Accepted: 21 August 2023 Published: 25 August 2023
MSC : 34K13, 34K20, 34D23, 37B25, 49K40, 92D30

The chikungunya virus (CHIKV) infects macrophages and adherent cells and it can be transmitted via a direct contact with the virus or with an already infected cell. Thus, the CHIKV infection can have two routes. Furthermore, it can exhibit seasonal peak periods. Thus, in this paper, we consider a dynamical system model of the CHIKV dynamics under the conditions of a seasonal environment with a general incidence rate and two routes of infection. In the first step, we studied the autonomous system by investigating the global stability of the steady states with respect to the basic reproduction number. In the second step, we establish the existence, uniqueness, positivity and boundedness of a periodic orbit for the non-autonomous system. We show that the global dynamics are determined by using the basic reproduction number denoted by $ \mathcal{R}_0 $ and they are calculated using the spectral radius of an integral operator. We show the global stability of the disease-free periodic solution if $ \mathcal{R}_0 < 1 $ and we also show the persistence of the disease if $ \mathcal{R}_0 > 1 $ where the trajectories converge to a limit cycle. Finally, we display some numerical investigations supporting the theoretical findings.
- CHIKV epidemic model,
- seasonal environment,
- periodic solution,
- Lyapunov stability,
- uniform persistence,
- extinction,
- basic reproduction number
Citation: Miled El Hajji. Periodic solutions for chikungunya virus dynamics in a seasonal environment with a general incidence rate[J]. AIMS Mathematics, 2023, 8(10): 24888-24913. doi: 10.3934/math.20231269

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Abstract

The chikungunya virus (CHIKV) infects macrophages and adherent cells and it can be transmitted via a direct contact with the virus or with an already infected cell. Thus, the CHIKV infection can have two routes. Furthermore, it can exhibit seasonal peak periods. Thus, in this paper, we consider a dynamical system model of the CHIKV dynamics under the conditions of a seasonal environment with a general incidence rate and two routes of infection. In the first step, we studied the autonomous system by investigating the global stability of the steady states with respect to the basic reproduction number. In the second step, we establish the existence, uniqueness, positivity and boundedness of a periodic orbit for the non-autonomous system. We show that the global dynamics are determined by using the basic reproduction number denoted by $ \mathcal{R}_0 $ and they are calculated using the spectral radius of an integral operator. We show the global stability of the disease-free periodic solution if $ \mathcal{R}_0 < 1 $ and we also show the persistence of the disease if $ \mathcal{R}_0 > 1 $ where the trajectories converge to a limit cycle. Finally, we display some numerical investigations supporting the theoretical findings.

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