Dynamical properties of a new SIR epidemic model

Lei Li; Wenjie Ni; Mingxin Wang

doi:10.3934/dcdss.2023076

Article Contents

2024, Volume 17, Issue 2: 690-707. Doi: 10.3934/dcdss.2023076

This issue Previous Article Existence and asymptotic behavior of square-mean $ S $-asymptotically periodic solutions of fractional stochastic evolution equations Next Article The linear stability and basic reproduction numbers for autonomous FDEs

Dynamical properties of a new SIR epidemic model

1.
College of Science, Henan University of Technology, Zhengzhou 450001, China
2.
School of Science and Technology, University of New England, Armidale, NSW 2351, Australia
3.
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454003, China

^*Corresponding author: Mingxin Wang
^*Corresponding author: Mingxin Wang

Received: November 2022

Revised: March 2023

Early access: April 7, 2023

Published online: April 7, 2023

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Abstract

Taking into account the depletion of food supply by all individuals, and the fact that chronic infectious diseases will not cause the infected individuals to lose their fertility completely, we first propose a new SIR epidemic model of ODE. For this model, we derive its basic reproduction number $ \mathcal{R}_0 $, and show that the disease-free equilibrium point is globally asymptotically stable when $ \mathcal{R}_0\le1 $, while the unique positive equilibrium point is globally asymptotically stable when $ \mathcal{R}_0>1 $. Then we incorporate the spatial dispersion and free boundary condition into this ODE model. The well-posedness and longtime behaviors are obtained. Particularly, we find a spreading-vanishing dichotomy in which the basic reproduction number $ \mathcal{R}_0 $ plays a crucial role.

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Mathematics Subject Classification: 35K57, 35R35, 92D30.

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References

[1]	G. Bunting, Y. H. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Networks and Heterogeneous Media, 7 (2012), 583-603. doi: 10.3934/nhm.2012.7.583.
[2]	Y. J. Chen and M. X. Wang, A nonlocal SIR epidemic problem with double free boundaries, Appl. Math. Lett., 133 (2022), 108259, 7 pp. doi: 10.1016/j.aml.2022.108259.
[3]	H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356. doi: 10.1016/0025-5564(76)90132-2.
[4]	H. M. Huang and M. X. Wang, The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. B, 20 (2015), 2039-2050. doi: 10.3934/dcdsb.2015.20.2039.
[5]	H. M. Huang and M. X. Wang, A nonlocal SIS epidemic problem with double free boundaries, Z. Angew. Math. Phys., 70 (2019), Paper No. 109, 19 pp. doi: 10.1007/s00033-019-1156-5.
[6]	W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. London A, 115 (1927), 700-721. doi: 10.1098/rspa.1927.0118.
[7]	K. I. Kim, Z. G. Lin and Q. Y. Zhang, An SIR epidemic model with free boundary, Nonlinear Anal. Real World Appl., 14 (2013), 1992-2001. doi: 10.1016/j.nonrwa.2013.02.003.
[8]	T. Kuniya and J. L. Wang, Lyapunov functions and global stability for a spatially diffusive SIR epidemic model, Appl. Anal., 96 (2017), 1935-1960. doi: 10.1080/00036811.2016.1199796.
[9]	L. Li, S. Y. Liu and M. X. Wang, A viral propagation model with a nonlinear infection rate and free boundaries, Sci. China Math., 64 (2021), 1971-1992. doi: 10.1007/s11425-020-1680-0.
[10]	J. D. Murray, E. A. Stanley and D. L. Brown, On the spatial spread of rabies among foxes, Proc. Roy. Soc. London B, 229 (1986), 111-150. doi: 10.1098/rspb.1986.0078.
[11]	P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.
[12]	M. X. Wang, Note on the Lyapunov functional method, Appl. Math. Lett., 75 (2018), 102-107. doi: 10.1016/j.aml.2017.07.003.
[13]	M. X. Wang, Existence and uniqueness of solutions of free boundary problems in heterogeneous environments, Discrete Cont. Dyn. Syst. B, 24 (2019), 415-421. doi: 10.3934/dcdsb.2018179.
[14]	M. X. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with different free boundaries, J. Differential Equations, 264 (2018), 3527-3558. doi: 10.1016/j.jde.2017.11.027.
[15]	M. X. Wang and J. F. Zhao, A free boundary problem for the predator-prey model with double free boundaries, J. Dyn. Diff. Equat., 29 (2017), 957-979. doi: 10.1007/s10884-015-9503-5.
[16]	Z.-C. Wang, L. Zhang and X.-Q. Zhao, Time periodic traveling waves for a periodic and diffusive SIR epidemic model, J. Dyn. Diff. Equat., 30 (2018), 379-403. doi: 10.1007/s10884-016-9546-2.
[17]	G. Y. Yang and M. X. Wang, An SIR epidemic model with nonlocal diffusion and free boundaries, Discrete Contin. Dyn. Syst. B, 28 (2023), 4221-4230. doi: 10.3934/dcdsb.2023007.
[18]	G. Y. Yang, S. W. Yao and M. X. Wang, An SIR epidemic model with nonlocal diffusion, nonlocal infection and free boundaries, J. Math. Anal. Appl., 518 (2023), 126731, 17 pp. doi: 10.1016/j.jmaa.2022.126731.