zbMATH — the first resource for mathematics

Basic reproduction numbers for a class of reaction-diffusion epidemic models. (English) Zbl 1448.92361
Summary: We study the basic reproduction numbers for a class of reaction-diffusion epidemic models that are developed from autonomous ODE systems. We present a general numerical framework to compute such basic reproduction numbers; meanwhile, the numerical formulation provides useful insight into their characterizations. Using matrix analysis, we show that the basic reproduction numbers are the same for these PDE models and their associated ODE models in several important cases that include, among others, a single infected compartment, constant diffusion rates, uniform diffusion patterns among the infected compartments, and partial diffusion in the system.
92D30 Epidemiology
35P15 Estimates of eigenvalues in context of PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
Full Text: DOI
[1] Allen, LJS; Bolker, BM; Lou, Y.; Nevai, AL, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin Dyn Syst, 21, 1-20 (2008) · Zbl 1146.92028
[2] Bertuzzo, E.; Casagrandi, R.; Gatto, M.; Rodriguez-Iturbe, I.; Rinaldo, A., On spatially explicit models of cholera epidemics, J R Soc Interface, 7, 321-333 (2010)
[3] Cantrell, RS; Cosner, C., The effects of spatial heterogeneity in population dynamics, J Math Biol, 29, 315-338 (1991) · Zbl 0722.92018
[4] Cantrell, RS; Cosner, C., Spatial ecology via reaction-diffusion equations (2003), Hoboken: Wiley, Hoboken · Zbl 1059.92051
[5] Chen, S.; Shi, J., Asymptotic profiles of basic reproduction number for epidemic spreading in heterogeneous environment, SIAM J Appl Math, 80, 1247-1271 (2020) · Zbl 1442.35467
[6] Diekmann, O.; Heesterbeek, JAP; Metz, AJ, On the definition and the computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous population, J Math Biol, 28, 365-382 (1990) · Zbl 0726.92018
[7] Ge, J.; Lei, C.; Lin, Z., Reproduction numbers and the expanding fronts for a diffusion-advection SIS model in heterogeneous time-periodic environment, Nonlinear Anal Real World Appl, 33, 100-120 (2017) · Zbl 1352.35201
[8] Golub, GH; Van Loan, CF, Matrix computations (1996), Baltimore: Johns Hopkins University Press, Baltimore
[9] Horn, RA; Johnson, CR, Matrix analysis (1985), Cambridge: Cambridge University Press, Cambridge
[10] Kim, KI; Lin, Z.; Zhang, Q., An SIR epidemic model with free boundary, Nonlinear Anal Real World Appl, 14, 1992-2001 (2013) · Zbl 1310.92054
[11] Lou, Y.; Zhao, X-Q, A reaction-diffusion malaria model with incubation period in the vector population, J Math Biol, 62, 543-568 (2011) · Zbl 1232.92057
[12] Magal, P.; Webb, GF; Wu, Y., On the basic reproduction number of reaction-diffusion epidemic models, SIAM J. Appl. Math., 79, 284-304 (2019) · Zbl 1407.35113
[13] Mukandavire, Z.; Liao, S.; Wang, J.; Gaff, H.; Smith, DL; Morris, JG, Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proc. Nat. Acad. Sci. USA, 108, 8767-8772 (2011)
[14] Peng, R.; Zhao, X-Q, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25, 1451-1471 (2012) · Zbl 1250.35172
[15] Posny, D.; Wang, J., Modeling cholera in periodic environments, J Biol Dyn, 8, 1, 1-19 (2014)
[16] Rinaldo, A.; Bertuzzo, E.; Mari, L.; Righetto, L.; Blokesch, M.; Gatto, M.; Casagrandi, R.; Murray, M.; Vesenbeckh, SM; Rodriguez-Iturbe, I., Reassessment of the 2010-2011 Haiti cholera outbreak and rainfall-driven multiseason projections, Proc Nat Acad Sci USA, 109, 6602-6607 (2012)
[17] Richtmyer, RD; Morton, KW, Difference methods for initial-value problems (1994), Malabar: Krieger Publication Company, Malabar
[18] Saad, Y., Numerical methods for large eigenvalue problems (2011), Philadelphia: SIAM, Philadelphia · Zbl 1242.65068
[19] Sauty, JP, An analysis of hydrodispersive transfer in aquifers, Water Resour Res, 16, 145-158 (1980)
[20] Song, P.; Lou, Y.; Xiao, Y., A spatial SEIRS reaction-diffusion model in heterogeneous environment, J Differ Equ, 267, 5084-5114 (2019) · Zbl 1440.35101
[21] Sposito, GW; Jury, WA; Gupta, VK, Fundamental problems in the stochastic convection-dispersion model of solute transport in aquifer and field soils, Water Resour Res, 22, 77-88 (1986)
[22] Taylor, GI, Dispersion of solute matter in solvent flowing through a tube, Proc R Soc Ser A, 219, 186-203 (1953)
[23] Thieme, HR, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J Appl Math, 70, 188-211 (2009) · Zbl 1191.47089
[24] Thomas, JW, Numerical partial differential equations: finite difference methods (1995), New York: Springer, New York
[25] Tien, JH; Earn, DJ, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bull Math Biol, 72, 1506-1533 (2010) · Zbl 1198.92030
[26] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math Biosci, 180, 29-48 (2002) · Zbl 1015.92036
[27] Wang, F-B; Shi, J.; Zou, X., Dynamics of a host-pathogen system on a bounded spatial domain, Commun Pure Appl Anal, 14, 2535-2560 (2015) · Zbl 1328.35104
[28] Wang, X.; Gao, D.; Wang, J., Influence of human behavior on cholera dynamics, Math Biosci, 267, 41-52 (2015) · Zbl 1371.92132
[29] Wang, X.; Posny, D.; Wang, J., A reaction-convection-diffusion model for cholera spatial dynamics, Discrete Contin Dyn Syst Ser B, 21, 2785-2809 (2016) · Zbl 1347.35132
[30] Wang, W.; Zhao, X-Q, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J Appl Dyn Syst, 11, 1652-1673 (2012) · Zbl 1259.35120
[31] Wu, Y.; Zou, X., Dynamics and profile of a diffusive host-pathogen system with distinct dispersal rates, J Differ Equ, 264, 4989-5024 (2018) · Zbl 1387.35362
[32] Yamazaki, K.; Wang, X., Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model, Discrete Contin Dyn Syst Ser B, 21, 1297-1316 (2016) · Zbl 1346.35200
[33] Yang, C.; Lolika, P.; Mushayabasa, S.; Wang, J., Modeling the spatiotemporal variations in brucellosis transmission, Nonlinear Anal Real World Appl, 38, 49-67 (2017) · Zbl 1428.92125
[34] Yu, X.; Zhao, X-Q, A nonlocal spatial model for Lyme disease, J Differ Equ, 261, 340-372 (2016) · Zbl 1341.35173
[35] Zhao, L.; Wang, Z-C; Ruan, S., Traveling wave solutions in a two-group SIR epidemic model with constant recruitment, J Math Biol, 77, 1871-1915 (2018) · Zbl 1406.35445
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.