Propagation dynamics of a reaction-diffusion two-group SIR epidemic model with constant recruitment. (English) Zbl 1542.35073
Summary: In this paper, we focus on asymptotic speeds of spread for a reaction-diffusion two-group SIR epidemic model with constant recruitment, which lacks the comparison principle. More precisely, if \(R_0 <1\), then the solution of the system converges to the disease-free equilibrium as \(t\to \infty\) and if \(R_0 >1\), there exists a critical speed \(c^\ast\) such that the solution of the system is uniformly persistent with \(|x|\leq ct\), \(\forall c\in [0,c^\ast)\) and the infection dies out with \(|x|\geq ct\) for any \(c> c^\ast\). Finally, some numerical experiments are presented to modeling the propagation dynamics of the system.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 34K30 | Functional-differential equations in abstract spaces |
| 35K45 | Initial value problems for second-order parabolic systems |
| 35K57 | Reaction-diffusion equations |
Keywords:
reaction-diffusion system; two-group SIR epidemic model; constant recruitment; asymptotic speeds of spreadReferences:
| [1] | Abi Rizk, L., Burie, J. and Ducrot, A., Asymptotic speed of spread for a nonlocal evolutionary-epidemic system, Discrete Contin. Dyn. Syst.41 (2021) 4959-4985. · Zbl 1473.35585 |
| [2] | Anderson, R. M. and May, R. M., Infectious Diseases of Humans: Dynamics and Control (Oxford University Press, Oxford, 1991). |
| [3] | Aronson, D. G., The asymptotic speed of propagation of a simple epidemic, Res. Notes Math.14 (1977) 1-23. · Zbl 0361.35011 |
| [4] | Aronson, D. G. and Weinberger, H. F., Nonlinear Diffusion in Population Genetics, Combustion, and Nerve Pulse Propagation (Springer, Berlin, 1975). · Zbl 0325.35050 |
| [5] | Aronson, D. G. and Weinberger, H. F., Multidimensional nonlinear diffusion arising in population dynamics, Adv. Math.30 (1978) 33-76. · Zbl 0407.92014 |
| [6] | Chen, X. and Tsai, J.-C., Spreading speed in a farmers and hunter-gatherers model arising from Neolithic transition in Europe, J. Math. Pures Appl.143 (2020) 192-207. · Zbl 1450.35106 |
| [7] | Ducrot, A., Convergence to generalized transition waves for some Holling-Tanner prey-predator reaction-diffusion system, J. Math. Pures Appl.100 (2013) 1-15. · Zbl 1284.35066 |
| [8] | Ducrot, A., Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differ. Equations260 (2016) 8316-8357. · Zbl 1338.35241 |
| [9] | Ducrot, A., Giletti, T. and Matano, H., Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type, Calc. Var. Partial Differ. Equations58 (2019) 1-34. · Zbl 1418.35230 |
| [10] | Ducrot, A., Giletti, T., Guo, J.-S. and Shimojo, M., Asymptotic spreading speeds for a predator-prey system with two predators and one prey, Nonlinearity34 (2021) 669-704. · Zbl 1458.35106 |
| [11] | Ducrot, A., Guo, J.-S., Lin, G. and Pan, S., The spreading speed and the minimal wave speed of a predator-prey system with nonlocal dispersal, Z. Angew. Math. Phys.70 (2019) 146. · Zbl 1422.35115 |
| [12] | Ducrot, A., Manceau, D. and Sylla, A., Spreading speed for an epidemic system modelling plant disease with adaptation, Discrete Contin. Dyn. Syst. Ser. B28 (2023) 2011-2043. · Zbl 1502.35016 |
| [13] | Guo, H., Li, M. Y. and Shuai, Z., Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q.14 (2006) 259-284. · Zbl 1148.34039 |
| [14] | Guo, J.-S., Pohand, A. and Shimojo, M., The spreading speed of an SIR epidemic model with nonlocal dispersal, Asymptot. Anal.120 (2020) 163-174. · Zbl 1460.35352 |
| [15] | Lam, K.-Y. and Lou, Y., Asymptotic behavior of the principal eigenvalue for cooperative elliptic systems and applications, J. Dyn. Differ. Equations28 (2016) 29-48. · Zbl 1334.35122 |
| [16] | Liang, X., Yi, Y. and Zhao, X.-Q., Spreading speeds and traveling waves for periodic evolution systems, J. Differ. Equations231 (2006) 57-77. · Zbl 1105.37017 |
| [17] | Liang, X. and Zhao, X.-Q., Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal.259 (2010) 857-903. · Zbl 1201.35068 |
| [18] | Lin, G., Spreading speeds of a Lotka-Volterra predator-prey system: The role of the predator, Nonlinear Anal.74 (2011) 2448-2461. · Zbl 1211.35138 |
| [19] | Liu, Q., Liu, S. and Lam, K.-Y., Stacked invasion waves in a competition-diffusion model with three species, J. Differ. Equations271 (2021) 665-718. · Zbl 1454.35216 |
| [20] | Murray, J. D., Mathematical Biology (Springer-Verlag, Berlin, 1989). · Zbl 0682.92001 |
| [21] | Pan, S., Asymptotic spreading in a Lotka-Volterra predator-prey system, J. Math. Anal. Appl.407 (2013) 230-236. · Zbl 1306.92049 |
| [22] | Rass, L. and Radcliffe, J., Spatial Deterministic Epidemics (American Mathematical Society, Providence, RI, 2003). · Zbl 1018.92028 |
| [23] | Ruan, S. and Wu, J., Modeling spatial spread of communicable diseases involving animal hosts, in Spatial Ecology (Chapman & Hall/CRC, Boca Raton, FL, 2009), pp. 293-316. · Zbl 1183.92074 |
| [24] | Thieme, H. and Zhao, X.-Q., Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differ. Equations195 (2003) 430-470. · Zbl 1045.45009 |
| [25] | Weinberger, H., On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol.45 (2002) 511-548. · Zbl 1058.92036 |
| [26] | Weinberger, H. F., Some deterministic models for the spread of genetic and other alterations, in Biological Growth and Spread (Springer, Berlin, New York, 1980), pp. 320-349. · Zbl 0444.92008 |
| [27] | Weinberger, H. F., Long-time behavior of a class of biological models, SIAM J. Math. Anal.13 (1982) 353-396. · Zbl 0529.92010 |
| [28] | Weinberger, H., Kawasaki, K. and Shigesada, N., Spreading speeds of spatially periodic integro-difference models for populations with non-monotone recruitment functions, J. Math. Biol.57 (2008) 387-411. · Zbl 1141.92041 |
| [29] | Wu, S.-L., Pang, L. and Ruan, S., Propagation dynamics in periodic predator-prey systems with nonlocal dispersal, J. Math. Pures Appl.170 (2023) 57-95. · Zbl 1507.35069 |
| [30] | Van den Driessche, P. and Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci.180 (2002) 29-48. · Zbl 1015.92036 |
| [31] | Yang, X. and Lin, G., Spreading speeds and traveling waves for a time periodic DS-I-A epidemic model, Nonlinear Anal. Real World Appl.66 (2022) 103515. · Zbl 1485.35110 |
| [32] | Zhao, L., Wang, Z.-C. and Ruan, S., Traveling wave solutions in a two-group SIR epidemic model with constant recruitment, J. Math. Biol.77 (2018) 1871-1915. · Zbl 1406.35445 |
| [33] | Zhao, L., Wang, Z.-C. and Zhang, L., Propagation dynamics for a time-periodic reaction-diffusion SI epidemic model with periodic recruitment, Z. Angew. Math. Phys.72 (2021) 142. · Zbl 1466.35086 |
| [34] | Zhao, L. and Huo, H., Spatial propagation for a reaction-diffusion SI epidemic model with vertical transmission, Math. Biosci. Eng.18 (2021) 6012-6033. · Zbl 1501.92208 |
| [35] | Zhao, M., Yuan, R., Ma, Z. and Zhao, X., Spreading speeds for the predator-prey system with nonlocal dispersal, J. Differ. Equations316 (2022) 552-598. · Zbl 1486.35121 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
