Global dynamics of a two-strain flu model with a single vaccination and general incidence rate

Arturo J. Nic-May; Eric J. Avila-Vales; Arturo J. Nic-May; Eric J. Avila-Vales

doi:10.3934/mbe.2020400

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 7862-7891. doi: 10.3934/mbe.2020400

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Research article

Global dynamics of a two-strain flu model with a single vaccination and general incidence rate

Arturo J. Nic-May ^,,
Eric J. Avila-Vales ^,

Facultad de Matemáticas, Universidad Autónoma de Yucatán, Anillo Periférico Norte, Tablaje 13615, Mérida, Yucatán C.P. 97119, México

Received: 30 July 2020 Accepted: 01 November 2020 Published: 10 November 2020

Influenza remains one of the major infectious diseases that target humankind, therefore, understand transmission mechanisms and control strategies can help us obtain more accurate predictions. There are many control strategies, one of them is vaccination. In this paper, our purpose is to extend the incidence rate of a two-strain flu model with a single vaccination, which includes a wide range of incidence rates among them, some cases are not monotonic nor concave, which may be used to reflect media education or psychological effect. Our main aim is to mathematically analyze the effect of the vaccine for strain 1, the general incidence rate of strain 1 and the general incidence rate of strain 2 on the dynamics of the model. Four equilibrium points were obtained and the global dynamics of the model are completely determined via suitable Lyapunov functions. We illustrate our results by some numerical simulations. Our results showed that the vaccination is always beneficial for controlling strain 1, its impact on strain 2 depends on the force of infection of strain 2. Also, the psychological effect is always beneficial for controlling the disease.
- general nonlinear incidence rate,
- mathematical model,
- basic reproduction number,
- Lyapunov functional,
- globally asymptotically stable,
- vaccination,
- influenza
Citation: Arturo J. Nic-May, Eric J. Avila-Vales. Global dynamics of a two-strain flu model with a single vaccination and general incidence rate[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7862-7891. doi: 10.3934/mbe.2020400

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Abstract

Influenza remains one of the major infectious diseases that target humankind, therefore, understand transmission mechanisms and control strategies can help us obtain more accurate predictions. There are many control strategies, one of them is vaccination. In this paper, our purpose is to extend the incidence rate of a two-strain flu model with a single vaccination, which includes a wide range of incidence rates among them, some cases are not monotonic nor concave, which may be used to reflect media education or psychological effect. Our main aim is to mathematically analyze the effect of the vaccine for strain 1, the general incidence rate of strain 1 and the general incidence rate of strain 2 on the dynamics of the model. Four equilibrium points were obtained and the global dynamics of the model are completely determined via suitable Lyapunov functions. We illustrate our results by some numerical simulations. Our results showed that the vaccination is always beneficial for controlling strain 1, its impact on strain 2 depends on the force of infection of strain 2. Also, the psychological effect is always beneficial for controlling the disease.

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