In order to more accurately predict the spread trend of infectious diseases and seek the best strategy to control the development of diseases. In this paper, a spatial diffusion SVEIQR epidemic model with a saturated incidence rate and temporary immunity is presented. We define the basic reproduction number R0 and establish the threshold criteria for the global dynamics. By constructing a Lyapunov function, we ascertain that the disease-free equilibrium point is globally asymptotically stable when R0<1, and when R0>1, the endemic equilibrium point exhibits the same property. Finally, the correctness of the theoretical derivation was verified by numerical simulation and it was found that the diffusion effect will not fundamentally change the global threshold stability of the epidemic model. However, a large diffusion coefficient will accelerate the spread of the epidemic and thus accelerate the speed of the epidemic to reach a stable state.
WANG Haixia
,
LU Yanling
,
MENG Ziyi
. Dynamics Analysis of a SVEIQR Epidemic Model with Reaction-diffusion[J]. Complex Systems and Complexity Science, 2026
, 23(4)
: 43
-50
.
DOI: 10.13306/j.1672-3813.2026.04.006
[1] Kermack W O, Mckendrick A G. Contributions to the mathematical theory of epidemics-I[J]. Bulletin of Mathematical Biology, 1991, 53(1): 33-55.
[2] Wang L W, Liu Z J, Guo C H, et al. New global dynamical results and application of several SVEIS epidemic models with temporary immunity[J]. Applied Mathematics and Computation, 2021, 390: 125648.
[3] Wei F Y, Chen L H. Extinction and stationary distribution of an epidemic model with partial vaccination and nonlinear incidence rate[J]. Physica A: Statistical Mechanics and Its Applications, 2020, 545: 122852.
[4] Zhao Y, Jiang D Q. The threshold of a stochastic SIS epidemic model with vaccination[J]. Applied Mathematics and Computation, 2014, 243: 718-727.
[5] Kokomo E, Emvudu Y. Mathematical analysis and numerical simulation of an age-structured model of cholera with vaccination and demographic movements[J]. Nonlinear Analysis: Real World Applications, 2019, 45: 142-156.
[6] Liu Q, Jiang D Q, Shi N Z, et al. The threshold of a stochastic SIS epidemic model with imperfect vaccination[J]. Mathematical Models and Computer Simulations, 2018, 144: 78-90.
[7] Liu X B, Yang L J. Stability analysis of an SEIQV epidemic model with saturated incidence rate[J]. Nonlinear Analysis. Real World Applications, 2012, 13(6): 2671-2679.
[8] Trisilowati, Darti I, Musafir R R, et al. Dynamics of a fractional-order COVID-19 epidemic model with quarantine and standard incidence rate[J]. Axioms, 2023, 12(6):591.
[9] Guo S B, He M, Cui J A. Global stability of a time-delayed malaria model with standard incidence rate[J]. Acta Mathematicae Applicatae Sinica-English Series, 2023, 39: 211-221.
[10] Capasso V, Serio G. A generalization of the Kermack-McKendrick deterministic epidemic model[J]. Mathematical Biosciences, 1978, 42(1/2): 43-61.
[11] Li Y F, Cui J G. The effect of constant and pulse vaccination on SIS epidemic models incorporating media coverage[J]. Communications in Nonlinear Science and Numerical Simulation, 2009, 14: 2353-2365.
[12] Meng L, Zhu W. Analysis of SEIR epidemic patch model with nonlinear incidence rate, vaccination and quarantine strategies[J]. Mathematics and Computers in Simulation, 2022, 200: 489-503.
[13] Liu Q, Jiang D Q, Hayat T, et al. A stochastic SIRS epidemic model with logistic growth and general nonlinear incidence rate[J]. Physica A: Statistical Mechanics and Its Applications, 2020, 551: 124512.
[14] Yorke J A, London W P. Recurrent outbreaks of measles, chickenpox and mumps II systematic differences in contact rates and stochastic effects[J]. American Journal of Epidemiology, 1973, 98(6): 469-482.
[15] Li H Q, Guo X R. Dynamics of a delayed stochastic epidemic model with double epidemic hypothesis and saturated incidence[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 26(11): 1250-1271.
[16] Zhu L H, Huang X Y, Liu Y, et al. Spatiotemporal dynamics analysis and optimal control method for an SI reaction-diffusion propagation model[J]. Journal of Mathematical Analysis and Applications, 2021, 493(2): 124539.
[17] Yang Y, Zou L, Zhou J L, et al. Dynamics of a waterborne pathogen model with spatial heterogeneity and general incidence rate[J]. Nonlinear Analysis: Real World Applications, 2020, 53: 103065.
[18] Cai Y L, Lian X Z, Peng Z H, et al. Spatiotemporal transmission dynamics for influenza disease in a heterogenous environment[J]. Nonlinear Analysis: Real World[19] Wang J P, Teng Z D, Dai B X. Qualitative analysis of a reaction-diffusion SIRS epidemic model with nonlinear incidence rate and partial immunity[J]. Infectious Disease Modelling, 2023, 8(3): 881-911.
[20] Liu S Y, Bi Y J, Liu Y W. Modeling and dynamic analysis of tuberculosis in mainland China from 1998 to 2017: the effect of DOTS strategy and further control[J]. Theoretical Biology and Medical Modelling, 2020, 17(1): 6.
[21] Yang Y L, Tang S Y, Ren X H, et al. Global stability and optimal control for a tuberculosis model with vaccination and treatment[J]. Discrete and Continuous Dynamical Systems Ser B, 2016, 21(3): 1009-1022.
[22] Abubakar I, Pimpin L, Ariti C, et al. Systematic review and meta-analysis of the current evidence on the duration of protection by bacillus Calmette-Guerin vaccination against tuberculosis[J]. British Medical Joural, 2002, 325(7374): 1233.
[23] Guo Z Y, Xiao D, Wang X H, et al. Epidemiological characteristics of pulmonary tuberculosis in mainland China from 2004 to 2015: a model-based analysis[J]. BMC Public Health, 2019, 19: 1-11.
[24] Sulayman F, Amit M. A SEIRS model of tuberculosis infection model with vital dynamics, early treatment for latent patients and treatment of infective[J]. International Journal of Mathematical Sciences and Computing, 2020, 6(3): 42-52.
[25] 国家统计局.中国统计年鉴2023[M]. 北京:中国统计出版社,2023.
