Dynamical analysis, optimal control and spatial pattern in an influenza model with adaptive immunity in two stratified population

Mamta Barik; Chetan Swarup; Teekam Singh; Sonali Habbi; Sudipa Chauhan; Mamta Barik; Chetan Swarup; Teekam Singh; Sonali Habbi; Sudipa Chauhan

doi:10.3934/math.2022273

AIMS Mathematics

2022, Volume 7, Issue 4: 4898-4935. doi: 10.3934/math.2022273

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Dynamical analysis, optimal control and spatial pattern in an influenza model with adaptive immunity in two stratified population

1.
Amity Institute of Applied Science, Amity University, Noida, Sector-125, U. P., India
2.
Department of Basic Science, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh-Male Campus, 11673, Riyadh, Saudi Arabia
3.
Department of Mathematics, Graphic Era Hill university, Dehradun, Uttarakhand, India

Received: 10 October 2021 Revised: 02 December 2021 Accepted: 12 December 2021 Published: 28 December 2021
MSC : 34D20, 34D23, 34H05

Consistently, influenza has become a major cause of illness and mortality worldwide and it has posed a serious threat to global public health particularly among the immuno-compromised people all around the world. The development of medication to control influenza has become a major challenge now. This work proposes and analyzes a structured model based on two geographical areas, in order to study the spread of influenza. The overall underlying population is separated into two sub populations: urban and rural. This geographical distinction is required as the immunity levels are significantly higher in rural areas as compared to urban areas. Hence, this paper is a novel attempt to proposes a linear and non-linear mathematical model with adaptive immunity and compare the host immune response to disease. For both the models, disease-free equilibrium points are obtained which are locally as well as globally stable if the reproduction number is less than 1 (R₀₁ < 1 & R₀₂ < 1) and the endemic point is stable if the reproduction number is greater then 1 (R₀₁ > 1 & R₀₂ > 1). Next, we have incorporated two treatments in the model that constitute the effectiveness of antidots and vaccination in restraining viral creation and slow down the production of new infections and analyzed an optimal control problem. Further, we have also proposed a spatial model involving diffusion and obtained the local stability for both the models. By the use of local stability, we have derived the Turing instability condition. Finally, all the theoretical results are verified with numerical simulation using MATLAB.
- immune response,
- local stability,
- global stability,
- optimal control,
- diffusion,
- spatial pattern
Citation: Mamta Barik, Chetan Swarup, Teekam Singh, Sonali Habbi, Sudipa Chauhan. Dynamical analysis, optimal control and spatial pattern in an influenza model with adaptive immunity in two stratified population[J]. AIMS Mathematics, 2022, 7(4): 4898-4935. doi: 10.3934/math.2022273

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Abstract

Consistently, influenza has become a major cause of illness and mortality worldwide and it has posed a serious threat to global public health particularly among the immuno-compromised people all around the world. The development of medication to control influenza has become a major challenge now. This work proposes and analyzes a structured model based on two geographical areas, in order to study the spread of influenza. The overall underlying population is separated into two sub populations: urban and rural. This geographical distinction is required as the immunity levels are significantly higher in rural areas as compared to urban areas. Hence, this paper is a novel attempt to proposes a linear and non-linear mathematical model with adaptive immunity and compare the host immune response to disease. For both the models, disease-free equilibrium points are obtained which are locally as well as globally stable if the reproduction number is less than 1 (R₀₁ < 1 & R₀₂ < 1) and the endemic point is stable if the reproduction number is greater then 1 (R₀₁ > 1 & R₀₂ > 1). Next, we have incorporated two treatments in the model that constitute the effectiveness of antidots and vaccination in restraining viral creation and slow down the production of new infections and analyzed an optimal control problem. Further, we have also proposed a spatial model involving diffusion and obtained the local stability for both the models. By the use of local stability, we have derived the Turing instability condition. Finally, all the theoretical results are verified with numerical simulation using MATLAB.

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