Novel tick-borne encephalitis transmission dynamics: A model-based strategy for epidemic control

Zakirullah; Shakir Ullah; M. Motawi Khashan; Kamal Shah; Thabet Abdeljawad; Zakirullah; Shakir Ullah; M. Motawi Khashan; Kamal Shah; Thabet Abdeljawad

doi:10.3934/nhm.2026023

Networks and Heterogeneous Media

2026, Volume 21, Issue 2: 496-530. doi: 10.3934/nhm.2026023

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Novel tick-borne encephalitis transmission dynamics: A model-based strategy for epidemic control

1.
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China
2.
Department of Mathematics, University of Malakand, Chakdara Dir(L), Khyber Pakhtunkhwa 18000, Pakistan
3.
Department of Basic Sciences, Common First Year, King Saud University, Riyadh 11451, Saudi Arabia
4.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
5.
Department of Fundamental Sciences, Faculty of Engineering and Architecture, Istanbul Gelisim University, Avcilar, Istanbul 34310, Turkey
6.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
7.
Department of Mathematics and Applied Mathematics School of Science and Technology, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa

Received: 21 November 2025 Revised: 03 March 2026 Accepted: 17 March 2026 Published: 30 March 2026

This study, we present a fractional-order compartmental model to describe the transmission dynamics of tick-borne encephalitis between human and vector (tick) populations. The model is first formulated in the classical integer-order sense and then extended to a fractional-order framework using the Caputo fractional derivative to incorporate memory effects. We analyzed the model with respect to positivity, boundedness, and stability, and identified the invariant region under the Caputo fractional operator. Furthermore, the model was fitted to the reported case data, and $ \mathcal{R}_0 $ was computed using the next-generation matrix method, resulting in $ \mathcal{R}_0 = 0.23 $. PRCC sensitivity analysis confirmed the effectiveness of vaccination in reducing disease transmission, with local and global stability established through Jacobian and Lyapunov analyses, respectively. We examined the effect of the fractional-order parameter $ \zeta $ using an efficient Adams-Moulton numerical approach, and contour and 3D surface plots are used to visualize the outcomes. The simulations showed that variations in key parameters reduced disease transmission and infection levels in both populations.
- tick-borne encephalitis,
- fractional-order mathematical model,
- stability analysis,
- Adams–Moulton method,
- parameter estimation
Citation: Zakirullah, Shakir Ullah, M. Motawi Khashan, Kamal Shah, Thabet Abdeljawad. Novel tick-borne encephalitis transmission dynamics: A model-based strategy for epidemic control[J]. Networks and Heterogeneous Media, 2026, 21(2): 496-530. doi: 10.3934/nhm.2026023

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Abstract

This study, we present a fractional-order compartmental model to describe the transmission dynamics of tick-borne encephalitis between human and vector (tick) populations. The model is first formulated in the classical integer-order sense and then extended to a fractional-order framework using the Caputo fractional derivative to incorporate memory effects. We analyzed the model with respect to positivity, boundedness, and stability, and identified the invariant region under the Caputo fractional operator. Furthermore, the model was fitted to the reported case data, and $ \mathcal{R}_0 $ was computed using the next-generation matrix method, resulting in $ \mathcal{R}_0 = 0.23 $. PRCC sensitivity analysis confirmed the effectiveness of vaccination in reducing disease transmission, with local and global stability established through Jacobian and Lyapunov analyses, respectively. We examined the effect of the fractional-order parameter $ \zeta $ using an efficient Adams-Moulton numerical approach, and contour and 3D surface plots are used to visualize the outcomes. The simulations showed that variations in key parameters reduced disease transmission and infection levels in both populations.

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