×

Control strategies for a multi-strain epidemic model. (English) Zbl 1478.92208

Summary: This article studies a multi-strain epidemic model with diffusion and environmental heterogeneity. We address the question of a control strategy for multiple strains of the infectious disease by investigating how the local distributions of the transmission and recovery rates affect the dynamics of the disease. Our study covers both full model (in which case the diffusion rates for all subgroups of the population are positive) and the ODE-PDE case (in which case we require a total lock-down of the susceptible subgroup and allow the infected subgroups to have positive diffusion rates). In each case, a basic reproduction number of the epidemic model is defined and it is shown that if this reproduction number is less than one then the disease will be eradicated in the long run. On the other hand, if the reproduction number is greater than one, then the disease will become permanent. Moreover, we show that when the disease is permanent, creating a common safety area against all strains and lowering the diffusion rate of the susceptible subgroup will result in reducing the number of infected populations. Numerical simulations are presented to support our theoretical findings.

MSC:

92D30 Epidemiology
35F20 Nonlinear first-order PDEs
35K57 Reaction-diffusion equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ackleh, AS; Allen, LJS, Competitive exclusion and coexistence for pathogens in an epidemic model with variable population size, J Math Biol, 47, 153-168 (2003) · Zbl 1023.92022 · doi:10.1007/s00285-003-0207-9
[2] Ackleh, AS; Allen, LJS, Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality, Discrete Contin Dyn Syst Ser B, 5, 2, 175-188 (2005) · Zbl 1080.34034 · doi:10.3934/dcdsb.2005.5.175
[3] Ackleh, AS; Deng, K.; Wu, Y., Competitive exclusion and coexistence in a two-strain pathogen model with diffusion, Math Biosci Eng, 13, 1, 1-18 (2016) · Zbl 1326.92064 · doi:10.3934/mbe.2016.13.1
[4] Allen, LSJ; Bolker, BM; Lou, Y.; Nevai, AL, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin Dyn Syst, 21, 1, 1-20 (2008) · Zbl 1146.92028 · doi:10.3934/dcds.2008.21.1
[5] Bremermann HJ, Thieme HR (1989) A competitive exclusion principle for pathogen virulence. J Math Biol 27: 179-190 · Zbl 0715.92027
[6] Cantrell, RS; Cosner, C., Spatial ecology via reaction-diffusion equations. Series in mathematical and computational biology (2003), Chichester: Wiley, Chichester · Zbl 1059.92051
[7] Cui, R.; Lam, K-Y; Lou, Y., Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J Differ Equ, 263, 4, 2343-2373 (2017) · Zbl 1388.35086 · doi:10.1016/j.jde.2017.03.045
[8] Cui, R.; Lou, Y., A spatial SIS model in advective heterogeneous environments, J Differ Equ, 261, 4, 3305-3343 (2016) · Zbl 1342.92231 · doi:10.1016/j.jde.2016.05.025
[9] Deng, K., Asymptotic behavior of an SIR reaction-diffusion model with a linear source, Discrete Contin Dyn Syst Ser B, 24, 5945-5957 (2019) · Zbl 1422.35114
[10] Deng, K.; Wu, Y., Dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model, Proc R. Soc Edinb Sect A, 146, 929-946 (2016) · Zbl 1353.92094 · doi:10.1017/S0308210515000864
[11] Gao, D., Travel frequency and infectious disease, SIAM J Appl Math, 79, 1581-1606 (2019) · Zbl 1421.92032 · doi:10.1137/18M1211957
[12] Ge J, Kim KI, Lin Z, Zhu H (2015) An SIS reaction-diffusion-advection model in a low-risk and high-risk domain. J Differ Equ 259(10):5486-5509 · Zbl 1341.35171
[13] Henry, D., Geometric theory of semilinear parabolic equations (1981), Berlin: Springer, Berlin · Zbl 0456.35001 · doi:10.1007/BFb0089647
[14] Húska, J., Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains, J Differ Equ, 226, 541-557 (2006) · Zbl 1102.35027 · doi:10.1016/j.jde.2006.02.008
[15] Hess P (1991) Periodic-parabolic boundary value problems and positivity. Pitman Research Notes in Mathematics Series, Wiley · Zbl 0731.35050
[16] Levin, SA; Pimentel, D., Selection of intermediate rates of increase in parasite-hosts systems, Am Nat, 117, 308-29 (1981) · doi:10.1086/283708
[17] Li, H.; Peng, R.; Wang, F-B, Varying total population enhances disease persistence: qualitative analysis on a diffusive SIS epidemic model, J Differ Equ, 262, 885-913 (2017) · Zbl 1355.35107 · doi:10.1016/j.jde.2016.09.044
[18] Li, H.; Peng, R.; Xiang, T., Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion, Eur J Appl Math, 31, 25-56 (2020) · Zbl 1504.35072 · doi:10.1017/S0956792518000463
[19] Liu S, Lou Y (2021) Classifying the level sets of principal eigenvalue for time-periodic parabolic operators and applications (under review)
[20] Mena-Lorca J, Velasco-Hernandez JX (1995) Superinfection, virulence and density dependent mortality in an epidemic model, Technical Report BU 1299-M. Cornell University, Biometrics Unit · Zbl 0937.92028
[21] Peng, R., Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model I, J Differ Equ, 247, 1096-1119 (2009) · Zbl 1165.92035 · doi:10.1016/j.jde.2009.05.002
[22] Peng, R.; Liu, S., Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Anal, 71, 239-247 (2009) · Zbl 1162.92037 · doi:10.1016/j.na.2008.10.043
[23] Peng, R.; Yi, F., Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: effects of epidemic risk and population movement, Phys D, 259, 8-25 (2013) · Zbl 1321.92076 · doi:10.1016/j.physd.2013.05.006
[24] Peng, R.; Zhao, XQ, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25, 1451-1471 (2012) · Zbl 1250.35172 · doi:10.1088/0951-7715/25/5/1451
[25] Quittner, P.; Souplet, P., Superliner parabolic problems: blow-up, global existence and steady states (2007), Basel: Birkhauser, Basel · Zbl 1128.35003
[26] Song, P.; Lou, Y.; Xiao, Y., A spatial SEIRS reaction-diffusion model in heterogeneous environment, J Differ Equ, 267, 5084-5114 (2018) · Zbl 1440.35101 · doi:10.1016/j.jde.2019.05.022
[27] Tuncer, N.; Martcheva, M., Analytical and numerical approaches to coexistence of strains in a two-strain SIS model with diffusion, J Biol Dyn, 6, 406-439 (2012) · Zbl 1447.92478 · doi:10.1080/17513758.2011.614697
[28] Wu, Y.; Tuncer, N.; Martcheva, M., Coexistence and competitive-exclusion in SIS model with standard incidence and diffusion, Discrete Contin Dyn Syst Ser B, 22, 7, 1167-1187 (2017) · Zbl 1360.35307
[29] Wu, Y.; Zou, X., Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J Differ Equ, 261, 4424-4447 (2016) · Zbl 1346.35199 · doi:10.1016/j.jde.2016.06.028
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.