GLOBAL DYNAMICS OF A TIME-DELAYED ZIKA VIRUS MODEL WITH INCUBATION PERIOD

Yuxi Li; Fuxiang Li

doi:10.11948/20260018

2027 Volume 17 Issue 1

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Article navigation > Journal of Applied Analysis & Computation > 2027 > 17(1): 349-373

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Yuxi Li, Fuxiang Li. GLOBAL DYNAMICS OF A TIME-DELAYED ZIKA VIRUS MODEL WITH INCUBATION PERIOD[J]. Journal of Applied Analysis & Computation, 2027, 17(1): 349-373. doi: 10.11948/20260018

Citation:

Yuxi Li, Fuxiang Li. GLOBAL DYNAMICS OF A TIME-DELAYED ZIKA VIRUS MODEL WITH INCUBATION PERIOD[J]. Journal of Applied Analysis & Computation, 2027, 17(1): 349-373. doi: 10.11948/20260018

GLOBAL DYNAMICS OF A TIME-DELAYED ZIKA VIRUS MODEL WITH INCUBATION PERIOD

Yuxi Li^1,,
Fuxiang Li^1, ,

1.
School of Science, China University of Geosciences (Beijing), 100083 Beijing, China

Author Bio: Email: hlgpn51567@gmail.com(Y. Li)
Corresponding author: Email: fuxiang_li@cugb.edu.cn(F. Li)

Abstract

Abstract

Zika virus (ZIKV), primarily transmitted by Aedes mosquitoes, can cause severe neurological complications and has triggered multiple outbreaks worldwide in recent years. In this paper, we develop a time-delayed transmission model incorporating the human incubation period and an exposed host compartment. The basic reproduction number $ R_0 $ is derived, and the local asymptotic stability of both the disease-free and endemic equilibria is established. By constructing suitable Lyapunov functionals, the global asymptotic stability of both equilibria is further demonstrated. Numerical simulations further illustrate the theoretical results, showing that the disease dies out when $ R_0 < 1 $ and persists when $ R_0 > 1 $. Furthermore, an optimal control framework involving five intervention measures is formulated and analyzed. Numerical results show that the proposed control strategies effectively reduce disease transmission, with vaccination and treatment exhibiting the greatest impact among the considered interventions. In addition, sensitivity analysis reveals the relative influence of model parameters on $ R_0 $. Overall, this model enhances our understanding of ZIKV transmission dynamics and provides theoretical support for the design of effective control strategies.
- Zika virus /
- time delay /
- global stability /
- Lyapunov functional /
- optimal control
MSC: 34K20, 37N25, 92D30

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References

[1]	R. Buathong, L. Hermann, B. Thaisomboonsuk, W. Rutvisuttinunt, C. Klungthong, P. Chinnawirotpisan, W. Manasatienkij, A. Nisalak, S. Fernandez, I. K. Yoon, P. Akrasewi and T. Plipat, Detection of Zika virus infection in Thailand, 2012–2014, Am. J. Trop. Med. Hyg., 2015, 93, 380–383. doi: 10.4269/ajtmh.15-0022 CrossRef Google Scholar
[2]	D. Denu and H. Son, Analysis and optimal control of a deterministic Zika virus model, J. Nonlinear Sci. Appl., 2022, 15, 88. Google Scholar
[3]	D. Diallo, A. A. Sall, C. T. Diagne, O. Faye, O. Faye, Y. Ba, K. A. Hanley, M. Buenemann, S. C. Weaver and M. Diallo, Zika virus emergence in mosquitoes in Southeastern Senegal, 2011, PLoS ONE, 2014, 9, e109442. doi: 10.1371/journal.pone.0109442 CrossRef Google Scholar
[4]	G. W. A. Dick, S. F. Kitchen and A. J. Haddow, Zika virus (I). Isolations and serological specificity, Trans. R. Soc. Trop. Med. Hyg., 1952, 46, 509–520. doi: 10.1016/0035-9203(52)90042-4 CrossRef Google Scholar
[5]	P. van den Driessche, Reproduction numbers of infectious disease models, Infect. Dis. Model., 2017, 2, 288–303. Google Scholar
[6]	P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 2002, 180, 29–48. doi: 10.1016/S0025-5564(02)00108-6 CrossRef Google Scholar
[7]	M. R. Duffy, T. -H. Chen, W. T. Hancock, et al., Zika virus outbreak on Yap Island, Federated States of Micronesia, N. Engl. J. Med., 2009, 360, 2536–2543. doi: 10.1056/NEJMoa0805715 CrossRef Google Scholar
[8]	D. Gao, Y. Lou, D. He, et al., Prevention and control of Zika as a mosquito-borne and sexually transmitted disease: A mathematical modeling analysis, Sci. Rep., 2016, 6, 28070. doi: 10.1038/srep28070 CrossRef Google Scholar
[9]	L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints, Optim. Control Appl. Methods, 2009, 30, 341–365. doi: 10.1002/oca.843 CrossRef Google Scholar
[10]	S. Guo, M. He and F. Li, Threshold dynamics of a time-delayed dengue virus infection model incorporating vaccination failure and exposed mosquitoes, Appl. Math. Lett., 2025, 161, 109366. doi: 10.1016/j.aml.2024.109366 CrossRef Google Scholar
[11]	J. Gutkes, N. P. Krabbe, K. Ausderau and E. L. Mohr, Macaque models of prenatal and postnatal Zika virus exposure and developmental outcomes, J. Pediatric Infect. Dis. Soc., 2025, 14, piaf024. doi: 10.1093/jpids/piaf024 CrossRef Google Scholar
[12]	J. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993. Google Scholar
[13]	M. Jamal, S. Batool, I. Ahmed, et al., Mathematical modeling of Zika virus with vertical transmission in the presence of Wolbachia-infected mosquitoes, J. Appl. Math. Comput., 2025, 71, 605–625. doi: 10.1007/s12190-024-02236-8 CrossRef Google Scholar
[14]	Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, MA, 1993. Google Scholar
[15]	R. S. Lanciotti, O. L. Kosoy, J. J. Laven, J. O. Velez, A. J. Lambert, A. J. Johnson, S. M. Stanfield and M. R. Duffy, Genetic and serologic properties of Zika virus associated with an epidemic, Yap State, Micronesia, 2007, Emerg. Infect. Dis., 2008, 14, 1232–1239. doi: 10.3201/eid1408.080287 CrossRef Google Scholar
[16]	J. P. Romero-Leiton, E. K. E. Laisón, R. Alfaro, E. J. Parmley, J. Arino, K. R. Acharya and B. Nasri, Exploring Zika's dynamics: A scoping review journey from epidemic to equations through mathematical modelling, Infect. Dis. Model., 2025, 10, 536–558. Google Scholar
[17]	F. Li and X. -Q. Zhao, Global dynamics of a reaction–diffusion model of Zika virus transmission with seasonality, Bull. Math. Biol., 2021, 83, 43. Google Scholar
[18]	E. R. Krow-Lucal, B. J. Biggerstaff and J. E. Staples, Estimated incubation period for Zika virus disease, Emerg. Infect. Dis., 2017, 23, 841–845. doi: 10.3201/eid2305.161715 CrossRef Google Scholar
[19]	F. N. MacNamara, Zika virus: A report on three cases of human infection during an epidemic of jaundice in Nigeria, Trans. R. Soc. Trop. Med. Hyg., 1954, 48, 139–145. doi: 10.1016/0035-9203(54)90006-1 CrossRef Google Scholar
[20]	A. R. Plourde and E. M. Bloch, A literature review of Zika virus, Emerg. Infect. Dis., 2016, 22, 1185–1192. doi: 10.3201/eid2207.151990 CrossRef Google Scholar
[21]	R. Rakkiyappan, V. Latha, R. Preethi and F. A. Rihan, A fractional-order model for Zika virus infection with multiple delays, Complexity, 2019, 2019, 4178073. doi: 10.1155/2019/4178073 CrossRef Google Scholar
[22]	R. F. Relich and M. Loeffelholz, Zika virus, Clin. Lab. Med., 2017, 37, 253–267. doi: 10.1016/j.cll.2017.01.002 CrossRef Google Scholar
[23]	Y. Saito, T. Hara and W. Ma, Necessary and sufficient conditions for permanence and global stability of a Lotka–Volterra system with two delays, J. Math. Anal. Appl., 1999, 236, 534–556. doi: 10.1006/jmaa.1999.6464 CrossRef Google Scholar
[24]	D. P. Shutt, C. A. Manore, S. Pankavich, A. T. Porter and S. Y. Del Valle, Estimating the reproductive number, total outbreak size, and reporting rates for Zika epidemics in South and Central America, Epidemics, 2017, 21, 63–79. doi: 10.1016/j.epidem.2017.06.005 CrossRef Google Scholar
[25]	H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, 1995. Google Scholar
[26]	H. Song, S. Jin, G. J. Yang, N. W. H. Lim, J. T. Lim and B. L. Dickens, Modelling the impact and cost-effectiveness of upcoming Zika virus vaccines on congenital Zika syndrome, Vaccine, 2025, 48, 126728. doi: 10.1016/j.vaccine.2025.126728 CrossRef Google Scholar
[27]	J. Tognarelli, S. Ulloa, E. Villagra, et al., A report on the outbreak of Zika virus on Easter Island, South Pacific, 2014, Arch. Virol., 2016, 161, 665–668. doi: 10.1007/s00705-015-2695-5 CrossRef Google Scholar
[28]	J. Tran and W. A. Woldegerima, Singular perturbation analysis of a two-time scale model of vector-borne disease: Zika virus model as a case study, Chaos Solitons Fractals, 2025, 194, 116209. doi: 10.1016/j.chaos.2025.116209 CrossRef Google Scholar
[29]	X. Wang, M. Shen, Y. Xiao and L. Rong, Optimal control and cost-effectiveness analysis of a Zika virus infection model with comprehensive interventions, Appl. Math. Comput., 2019, 359, 165–185. Google Scholar
[30]	Y. Wang, Q. Yan and L. Liu, Multi-scale modeling and analysis of vaccine prevention and control strategies for Chikungunya disease, Adv. Contin. Discret. Models, 2026, 2026, 4. Google Scholar
[31]	Z. Wang, S. Cai, G. Chen, et al., Dynamics of dengue fever transmission model with multi-stage and fluctuations, Mathematics, 2024, 12, 2491. doi: 10.3390/math12162491 CrossRef Google Scholar
[32]	Y. Wu, Z. Zhang, L. Song and C. Xia, Global stability analysis of two strains epidemic model with imperfect vaccination and immunity waning in a complex network, Chaos Solitons Fractals, 2024, 179, 114414. doi: 10.1016/j.chaos.2023.114414 CrossRef Google Scholar
[33]	W. Zhang and C. T. Sandamali, Modeling and analysis of low-level transmission ZIKV dynamics via a Poisson point process on sexual transmission route, J. Appl. Anal. Comput., 2025, 13, 1044–1069. Google Scholar
[34]	X. -Q. Zhao, Dynamical Systems in Population Biology, Second Edition, Springer-Verlag, New York, 2017. Google Scholar