A robust study of the transmission dynamics of syphilis infection through non-integer derivative

Rashid Jan; Adil Khurshaid; Hammad Alotaibi; Mustafa Inc; Rashid Jan; Adil Khurshaid; Hammad Alotaibi; Mustafa Inc

doi:10.3934/math.2023314

AIMS Mathematics

2023, Volume 8, Issue 3: 6206-6232. doi: 10.3934/math.2023314

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A robust study of the transmission dynamics of syphilis infection through non-integer derivative

1.
Department of Mathematics, University of Swabi, Swabi 23561, KPK Pakistan
2.
Department of Mathematics and Statistics, Faculty of Science, Taif University, P.O. Box 1109, Taif 21944, Saudi Arabia
3.
Department of Mathematics, Firat University, Elazig 23119, Turkiye
4.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan

Received: 01 November 2022 Revised: 07 December 2022 Accepted: 13 December 2022 Published: 03 January 2023
MSC : 4C05, 92D25

One of the most harmful and widespread sexually transmitted diseases is syphilis. This infection is caused by the Treponema Palladum bacterium that spreads through sexual intercourse and is projected to affect $ 12 $ million people annually worldwide. In order to thoroughly examine the complex and all-encompassing dynamics of syphilis infection. In this article, we constructed the dynamics of syphilis using the fractional derivative of the Atangana-Baleanu for more accurate outcomes. The basic theory of non-integer derivative is illustrated for the examination of the recommended model. We determined the steady-states of the system and calculated the $ \mathcal{R}_{0} $ for the intended fractional model with the help of the next-generation method. The infection-free steady-state of the system is locally stable if $ \mathcal{R}_{0} < 1 $ through jacobian matrix method. The existence and uniqueness of the fractional order system are investigate by applying the fixed-point theory. The iterative solution of our model with fractional order was then carried out by utilising a newly generated numerical approach. Finally, numerical results are computed for various values of the factor $ \Phi $ and other parameters of the system. The solution pathways and chaotic phenomena of the system are highlighted. Our findings show that fractional order derivatives provide more precise and realistic information regarding the dynamics of syphilis infection.
- syphilis dynamics,
- Atangana-Baleanu,
- stability analysis,
- fixed-point theorem,
- numerical method,
- dynamical behavior
Citation: Rashid Jan, Adil Khurshaid, Hammad Alotaibi, Mustafa Inc. A robust study of the transmission dynamics of syphilis infection through non-integer derivative[J]. AIMS Mathematics, 2023, 8(3): 6206-6232. doi: 10.3934/math.2023314

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Abstract

One of the most harmful and widespread sexually transmitted diseases is syphilis. This infection is caused by the Treponema Palladum bacterium that spreads through sexual intercourse and is projected to affect $ 12 $ million people annually worldwide. In order to thoroughly examine the complex and all-encompassing dynamics of syphilis infection. In this article, we constructed the dynamics of syphilis using the fractional derivative of the Atangana-Baleanu for more accurate outcomes. The basic theory of non-integer derivative is illustrated for the examination of the recommended model. We determined the steady-states of the system and calculated the $ \mathcal{R}_{0} $ for the intended fractional model with the help of the next-generation method. The infection-free steady-state of the system is locally stable if $ \mathcal{R}_{0} < 1 $ through jacobian matrix method. The existence and uniqueness of the fractional order system are investigate by applying the fixed-point theory. The iterative solution of our model with fractional order was then carried out by utilising a newly generated numerical approach. Finally, numerical results are computed for various values of the factor $ \Phi $ and other parameters of the system. The solution pathways and chaotic phenomena of the system are highlighted. Our findings show that fractional order derivatives provide more precise and realistic information regarding the dynamics of syphilis infection.

References

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