A robust study of the transmission dynamics of malaria through non-local and non-singular kernel

Rashid Jan; Sultan Alyobi; Mustafa Inc; Ali Saleh Alshomrani; Muhammad Farooq; Rashid Jan; Sultan Alyobi; Mustafa Inc; Ali Saleh Alshomrani; Muhammad Farooq

doi:10.3934/math.2023382

AIMS Mathematics

2023, Volume 8, Issue 4: 7618-7640. doi: 10.3934/math.2023382

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A robust study of the transmission dynamics of malaria through non-local and non-singular kernel

1.
Department of Mathematics, University of Swabi, Swabi 23561, KPK Pakistan
2.
King Abdulaziz University, College of Science & Arts, Department of Mathematics, Rabigh, Saudi Arabia
3.
Department of Medical Research, China Medical University, 40402 Taichung, Taiwan
4.
Department of Mathematics, Firat University 23119 Elazig, Turkey
5.
Mathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
6.
Department of Mathematics, Sheikh Taimur Academic Block-Ⅱ, University of Peshawar, 25120, Khyber Pakhtunkhwa, Pakistan

Received: 27 October 2022 Revised: 12 December 2022 Accepted: 04 January 2023 Published: 17 January 2023
MSC : 4C05, 92D25

It is valuable to measure the epidemiological significance of malaria, since there has been a growing interest in reducing malaria through improved local and national health care systems. We formulate the dynamics of malaria infection via a fractional framework to understand the intricate transmission route of malaria and to identify the role of memory for the control of malaria. The model is investigated for basic results, moreover, the basic reproduction number is determined symbolized by $ \mathcal{R}_0 $. We have shown the local stability of the disease-free steady-state of the system for for $ \mathcal{R}_0 < 1 $. The existence and uniqueness of the solution of the system are examined. The Adams Bashforth approach in fractional form is applied to analyse the numerical outcomes of the mathematical model. Furthermore, in order to realise more efficiently, the Atangana-Baleanu (ABC) fractional nonlocal operator, which was just invented, is used. The stability of the system is investigated through the fixed-point theorems of Krasnoselskii and Banach. The behaviour of the approximation solution is illustrated in terms of graphs across various fractional values and other factors of the systems. After all, a brief analysis of the simulation's findings is provided to explain how infection transmission dynamics occur in society.
- malaria,
- fractional derivatives,
- mathematical model,
- quantitative analysis,
- dynamical behaviour
Citation: Rashid Jan, Sultan Alyobi, Mustafa Inc, Ali Saleh Alshomrani, Muhammad Farooq. A robust study of the transmission dynamics of malaria through non-local and non-singular kernel[J]. AIMS Mathematics, 2023, 8(4): 7618-7640. doi: 10.3934/math.2023382

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Abstract

It is valuable to measure the epidemiological significance of malaria, since there has been a growing interest in reducing malaria through improved local and national health care systems. We formulate the dynamics of malaria infection via a fractional framework to understand the intricate transmission route of malaria and to identify the role of memory for the control of malaria. The model is investigated for basic results, moreover, the basic reproduction number is determined symbolized by $ \mathcal{R}_0 $. We have shown the local stability of the disease-free steady-state of the system for for $ \mathcal{R}_0 < 1 $. The existence and uniqueness of the solution of the system are examined. The Adams Bashforth approach in fractional form is applied to analyse the numerical outcomes of the mathematical model. Furthermore, in order to realise more efficiently, the Atangana-Baleanu (ABC) fractional nonlocal operator, which was just invented, is used. The stability of the system is investigated through the fixed-point theorems of Krasnoselskii and Banach. The behaviour of the approximation solution is illustrated in terms of graphs across various fractional values and other factors of the systems. After all, a brief analysis of the simulation's findings is provided to explain how infection transmission dynamics occur in society.

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