ANALYSIS OF A DEGENERATED DIFFUSION SVEQIRV EPIDEMIC MODEL WITH GENERAL INCIDENCE IN A SPACE HETEROGENEOUS ENVIRONMENT

Yantao Luo; Yunqiang Yuan; Zhidong Teng

doi:10.11948/20230346

2024 Volume 14 Issue 5

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Yantao Luo, Yunqiang Yuan, Zhidong Teng. ANALYSIS OF A DEGENERATED DIFFUSION SVEQIRV EPIDEMIC MODEL WITH GENERAL INCIDENCE IN A SPACE HETEROGENEOUS ENVIRONMENT[J]. Journal of Applied Analysis & Computation, 2024, 14(5): 2704-2732. doi: 10.11948/20230346

Citation:

Yantao Luo, Yunqiang Yuan, Zhidong Teng. ANALYSIS OF A DEGENERATED DIFFUSION SVEQIRV EPIDEMIC MODEL WITH GENERAL INCIDENCE IN A SPACE HETEROGENEOUS ENVIRONMENT[J]. Journal of Applied Analysis & Computation, 2024, 14(5): 2704-2732. doi: 10.11948/20230346

ANALYSIS OF A DEGENERATED DIFFUSION SVEQIRV EPIDEMIC MODEL WITH GENERAL INCIDENCE IN A SPACE HETEROGENEOUS ENVIRONMENT

1.
College of Mathematics and Systems Science, Xinjiang University, Urumqi 830017, China
2.
College of Science, National University of Defense Technology, Changsha 410073, China

Author Bio: Email: yyq070100@163.com(Y. Yuan); Email: zhidong1960@163.com(Z. Teng)
Corresponding author: Email: luoyantaoxj@163.com(Y. Luo)
Fund Project: The authors were supported Natural Science Foundation of Xinjiang Uygur Autonomous Region (2022D01C64), National Natural Science Foundation of P. R. China (12201540), the project of University Scientific Research of Xinjiang (XJEDU2021Y001), the Doctoral Research Initiation Fund of Xinjiang University (620320024)

Abstract

Abstract

Considering the comprehensive impact of vaccination, quarantine and spatial heterogeneity on diseases dynamics, we formulates an SVEQIRV model with degenerate diffusion. Firstly, we discuss the well-posedness of the model solution. Then, we analyze the dynamic properties of model by using the semigroup theory and the global exponential attractor theory. We use the threshold feature $\lambda^{*}$ which is the principal eigenvalue of the eigenvalue problem associated with the linearized system at the disease free equilibrium, to describe the transmission dynamics of epidemics. The results show that the disease-free equilibrium is globally asymptotically stable when $\lambda^{*}<0$ and the system is uniformly persistent when $\lambda^{*}>0$. Finally, some numerical simulations and the sensitivity analysis are conducted to visualize the theoretical results and the effect of vaccination rate on disease dynamics.
- Dynamical model /
- spatial heterogeneity /
- global exponential attractor /
- numerical simulations
MSC: 92D30, 35B35, 35K57, 37N25

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References

[1]	A. Alshehri and M. El. Hajji, Mathematical study for Zika virus transmission with general incidence rate, AIMS Math., 7(4), 7117–7142. doi: 10.3934/math.2022397 CrossRef Google Scholar
[2]	X. An and X. Song, A spatial SIS model in heterogeneous environments with vary advective rate, Math. Biosci. Eng., 2021, 18(5), 5449–5477. doi: 10.3934/mbe.2021276 CrossRef Google Scholar
[3]	S. Annas, M. I. Pratama, M. Rifandi, et al., Stability analysis and numerical simulation of SEIR model for pandemic COVID-19 spread in Indonesia, Chaos, Solitons & Fractals, 2020, 139, 110072. Google Scholar
[4]	E. Avila-Vales and Á. G. C. Pérez, Dynamics of a reaction–diffusion SIRS model with general incidence rate in a heterogeneous environment, Z. Angew. Math. Phys., 2022, 73(1), 9. doi: 10.1007/s00033-021-01645-0 CrossRef Google Scholar
[5]	Y. Cai, X. Lian, Z. Peng, et al., Spatiotemporal transmission dynamics for influenza disease in a heterogenous environment, Nonlinear Anal. Real World Appl., 2019, 46, 178–194. doi: 10.1016/j.nonrwa.2018.09.006 CrossRef Google Scholar
[6]	A. d'Onofrio, Globally stable vaccine-induced eradication of horizontally and vertically transmitted infectious diseases with periodic contact rates and disease-dependent demographic factors in the population, Appl. Math. Comput., 2003, 140(2–3), 537–547. Google Scholar
[7]	Z. Du and R. Peng, A priori $L^{\infty}$ estimates for solutions of a class of reaction-diffusion systems, J. Math. Biol, 2016, 72(6), 1429–1439. doi: 10.1007/s00285-015-0914-z CrossRef $L^{\infty}$ estimates for solutions of a class of reaction-diffusion systems" target="_blank">Google Scholar
[8]	L. Duan, L. Huang and C. Huang, Spatial dynamics of a diffusive SIRI model with distinct dispersal rates and heterogeneous environment, Commun. Pure Appl. Anal., 2021, 20(10), 3539, 3560. Google Scholar
[9]	M. El. Hajji and A. H. Albargi, A mathematical investigation of an "SVEIR" epidemic model for the measles transmission, Math. Biosci. Eng., 2022, 19, 2853–2875. doi: 10.3934/mbe.2022131 CrossRef Google Scholar
[10]	H. F. Huo, K. D. Cao and H. Xiang, Modelling the effects of the vaccination on seasonal influenza in Gansu, China, J. Appl. Anal. Comput., 2022, 12(1), 407–435. Google Scholar
[11]	Q. Ji, R. Wu and Z. Feng, Dynamics of the nonlocal diffusive vector-disease model with delay and spatial heterogeneity, Z. Angew. Math. Phys., 2023, 74(5), 183. doi: 10.1007/s00033-023-02077-8 CrossRef Google Scholar
[12]	J. Jiao, Z. Liu and S. Cai, Dynamics of an SEIR model with infectivity in incubation period and homestead-isolation on the susceptible, Appl. Math. Lett., 2020, 107, 106442. doi: 10.1016/j.aml.2020.106442 CrossRef Google Scholar
[13]	S. Kejriwal, S. Sheth and P. S. Silpa, Attaining herd immunity to a new infectious disease through multi-stage policies incentivising voluntary vaccination, Chaos, Solitons & Fractals, 2022, 154, 111710. Google Scholar
[14]	W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. London Ser. A, 1927, 115(772), 700–721. doi: 10.1098/rspa.1927.0118 CrossRef Google Scholar
[15]	A. Kumar and P. K. Srivastava, Vaccination and treatment as control interventions in an infectious disease model with their cost optimization, Commun. Nonlinear Sci. Numer. Simul., 2017, 44, 334–343. doi: 10.1016/j.cnsns.2016.08.005 CrossRef Google Scholar
[16]	A. Kumar, P. K. Srivastava and R. P. Gupta, Nonlinear dynamics of infectious diseases via information-induced vaccination and saturated treatment, Math. Comput. Simu., 2019, 157, 77–99. doi: 10.1016/j.matcom.2018.09.024 CrossRef Google Scholar
[17]	Q. Li, M. C. Li, L. Lv, et al., A new prediction model of infectious diseases with vaccination strategies based on evolutionary game theory, Chaos, Solitons & Fractals, 2017, 104, 51–60. Google Scholar
[18]	Y. Li, H. Zhao and K. Wang, Dynamics of an impulsive reaction-diffusion mosquitoes model with multiple control measures, Math. Biosci. Eng., 2023, 20(1), 775–806. Google Scholar
[19]	Y. Luo, L. Zhang, Z. Teng, et al., Analysis of a general multi-group reaction-diffusion epidemic model with nonlinear incidence and temporary acquired immunity, Math. Comput. Simu., 2021, 182, 428–455. doi: 10.1016/j.matcom.2020.11.002 CrossRef Google Scholar
[20]	A. J. May and E. J. A. Vales, Global dynamics of a two-strain flu model with a single vaccination and general incidence rate, Math. Biosci. Eng., 2020, 17(6), 7862–7891. doi: 10.3934/mbe.2020400 CrossRef Google Scholar
[21]	F. Ndaïrou, M. Khalighi and L. Lahti, Ebola epidemic model with dynamic population and memory, Chaos, Solitons & Fractals, 2023, 170, 113361. Google Scholar
[22]	R. D. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem, Berlin, Heidelberg: Springer Berlin Heidelberg, 2006, 309–330. Google Scholar
[23]	A. Ouakka, A. El. Azzouzi and Z. Hammouch, Global dynamic behavior of a vaccination–age SVIR model with treatment and general nonlinear incidence rate, J. Comput. Appl. Math., 2023, 422, 114848. doi: 10.1016/j.cam.2022.114848 CrossRef Google Scholar
[24]	C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Springer Science & Business Media, 2012. Google Scholar
[25]	R. Potluri, A. Kumar, V. Oriol-Mathieu, et al., Model-based evaluation of the impact of prophylactic vaccination applied to Ebola epidemics in Sierra Leone and Democratic Republic of Congo, BMC Infect. Dis., 2022, 22(1), 769. doi: 10.1186/s12879-022-07723-6 CrossRef Google Scholar
[26]	M. H. Protter, H. F. Weinberger, Maximum Principles in Differential Equations, Springer Science & Business Media, 2012. Google Scholar
[27]	Y. Shan, X. Wu and J. Gao, Analysis of a degenerate reaction-diffusion host-pathogen model with general incidence rate, J. Math. Anal. Appl., 2021, 502(2), 125256. doi: 10.1016/j.jmaa.2021.125256 CrossRef Google Scholar
[28]	Y. Sheng, J. A. Cui and S. Guo, The modeling and analysis of the COVID-19 pandemic with vaccination and isolation: A case study of Italy, Math. Biosci. Eng., 2023, 20(3), 5966–5992. doi: 10.3934/mbe.2023258 CrossRef Google Scholar
[29]	H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Soc., 1995. Google Scholar
[30]	P. Song, Y. Lou and Y. Xiao, A spatial SEIRS reaction-diffusion model in heterogeneous environment, J. Diff. Eqs., 2019, 267(9), 5084–5114. doi: 10.1016/j.jde.2019.05.022 CrossRef Google Scholar
[31]	C. Tadmon, B. Tsanou and A. F. Feukouo. Avian–human influenza epidemic model with diffusion, nonlocal delay and spatial homogeneous environment, Nonlinear Anal. Real World Appl., 2022, 67, 103615. doi: 10.1016/j.nonrwa.2022.103615 CrossRef Google Scholar
[32]	H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 1992, 30(7), 755–763. Google Scholar
[33]	A. Viguerie, A. Veneziani, G. Lorenzo, et al., Diffusion–reaction compartmental models formulated in a continuum mechanics framework: Application to COVID-19, mathematical analysis, and numerical study, Comput. Mech., 2020, 66, 1131–1152. doi: 10.1007/s00466-020-01888-0 CrossRef Google Scholar
[34]	I. I. Vrabie, Co-Semigroups and Applications, Elsevier, 2003. Google Scholar
[35]	J. Wang and B. Dai, Qualitative analysis on a reaction-diffusion host-pathogen model with incubation period and nonlinear incidence rate, J. Math. Anal. Appl., 2022, 514(2), 126322. doi: 10.1016/j.jmaa.2022.126322 CrossRef Google Scholar
[36]	J. Wang and H. Lu, Dynamics and profiles of a degenerated reaction–diffusion host-pathogen model with apparent and inapparent infection period, Commun. Nonlinear Sci. Numer. Simul., 2023, 107318. Google Scholar
[37]	J. Wang, W. Wu and T. Kuniya, Analysis of a degenerated reaction–diffusion cholera model with spatial heterogeneity and stabilized total humans, Math. Comput. Simulation, 2022, 198, 151–171. doi: 10.1016/j.matcom.2022.02.026 CrossRef Google Scholar
[38]	J. Wang, F. Xie and T. Kuniya, Analysis of a reaction-diffusion cholera epidemic model in a spatially heterogeneous environment, Commun. Nonlinear Sci. Numer. Simul., 2020, 80, 104951. doi: 10.1016/j.cnsns.2019.104951 CrossRef Google Scholar
[39]	S. Wang, Introduction to Sobolev Spaces and Partial Differential Equations, Beijing: Science Press, 2009. Google Scholar
[40]	R. Wu and X. Q. Zhao, A reaction–diffusion model of vector-borne disease with periodic delays, J. Nonlinear Sci., 2019, 29, 29–64. doi: 10.1007/s00332-018-9475-9 CrossRef Google Scholar
[41]	Y. Zha and W. Jiang, Global dynamics and asymptotic profiles for a degenerate Dengue fever model in heterogeneous environment, J. Diff. Eqs., 2023, 348, 278–319. doi: 10.1016/j.jde.2022.12.012 CrossRef Google Scholar
[42]	F. Zhang, W. Cui, Y. Dai, et al., Bifurcations of an SIRS epidemic model with a general saturated incidence rate, Math. Biosci. Eng., 2022, 19(11), 10710–10730. doi: 10.3934/mbe.2022501 CrossRef Google Scholar
[43]	J. Zhang, P. E. Kloeden, M. Yang, et al., Global exponential $\kappa-$dissipative semigroups and exponential attraction, Discrete Contin. Dynam. Systems, 2017, 37, 3487–3502. $\kappa-$dissipative semigroups and exponential attraction" target="_blank">Google Scholar
[44]	T. Zheng, L. Nie, H. Zhu, et al., Role of seasonality and spatial heterogeneous in the transmission dynamics of avian influenza, Nonlinear Anal. Real World Appl., 2022, 67, 103567. Google Scholar
[45]	C. C. Zhu and J. Zhu, Dynamic analysis of a delayed COVID-19 epidemic with home quarantine in temporal-spatial heterogeneous via global exponential attractor method, Chaos, Solitons & Fractals, 2021, 143, 110546. Google Scholar
[46]	C. C. Zhu, J. Zhu and X. L. Liu, Influence of spatial heterogeneous environment on long-term dynamics of a reaction-diffusion SVIR epidemic model with relapse, Math. Biosci. Eng., 2019, 16, 5897–5922. Google Scholar