Fast diffusion inhibits disease outbreaks

Gao, Daozhou; Dong, Chao-Ping

doi:10.1090/proc/14868

Fast diffusion inhibits disease outbreaks
HTML articles powered by AMS MathViewer

by Daozhou Gao and Chao-Ping Dong PDF

Proc. Amer. Math. Soc. 148 (2020), 1709-1722 Request permission

Abstract:

We show that the basic reproduction number of an SIS patch model with standard incidence is either strictly decreasing and strictly convex with respect to the diffusion coefficient of infected subpopulation if the patch reproduction numbers of at least two patches in isolation are distinct or constant otherwise. Biologically, it means that fast diffusion of infected people reduces the risk of infection. This completely solves and generalizes a conjecture by Allen et al. [SIAM J. Appl. Math., 67 (2007) pp. 1283–1309]. Furthermore, a substantially improved and reachable lower bound on the multipatch reproduction number, a generalized monotone result on the spectral bound of the Jacobian matrix of the model system at the disease-free equilibrium, and the limit of the endemic equilibrium as the diffusion coefficient goes to infinity are obtained. The approach and results can be applied to a class of epidemic patch models where only one class of infected compartments migrate between patches and one transmission route is involved.

References

L. J. S. Allen, B. M. Bolker, Y. Lou, and A. L. Nevai, Asymptotic profiles of the steady states for an $SIS$ epidemic patch model, SIAM J. Appl. Math. 67 (2007), no. 5, 1283–1309. MR 2341750, DOI 10.1137/060672522
L. J. S. Allen, B. M. Bolker, Y. Lou, and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. 21 (2008), no. 1, 1–20. MR 2379454, DOI 10.3934/dcds.2008.21.1
Lee Altenberg, The evolutionary reduction principle for linear variation in genetic transmission, Bull. Math. Biol. 71 (2009), no. 5, 1264–1284. MR 2515970, DOI 10.1007/s11538-009-9401-2
Lee Altenberg, Resolvent positive linear operators exhibit the reduction phenomenon, Proc. Natl. Acad. Sci. USA 109 (2012), no. 10, 3705–3710. MR 2903373, DOI 10.1073/pnas.1113833109
Lee Altenberg, On the ordering of spectral radius product $r(\textbf {A})\,r(\textbf {AD})$ versus $r(\textbf {A}^2\textbf {D})$ and related applications, SIAM J. Matrix Anal. Appl. 34 (2013), no. 3, 978–998. MR 3073650, DOI 10.1137/130906179
Pierre Auger, Etienne Kouokam, Gauthier Sallet, Maurice Tchuente, and Berge Tsanou, The Ross-Macdonald model in a patchy environment, Math. Biosci. 216 (2008), no. 2, 123–131. MR 2476998, DOI 10.1016/j.mbs.2008.08.010
Abraham Berman and Robert J. Plemmons, Nonnegative matrices in the mathematical sciences, Classics in Applied Mathematics, vol. 9, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. Revised reprint of the 1979 original. MR 1298430, DOI 10.1137/1.9781611971262
Joel E. Cohen, Convexity of the dominant eigenvalue of an essentially nonnegative matrix, Proc. Amer. Math. Soc. 81 (1981), no. 4, 657–658. MR 601750, DOI 10.1090/S0002-9939-1981-0601750-2
C. Cosner, J. C. Beier, R. S. Cantrell, D. Impoinvil, L. Kapitanski, M. D. Potts, A. Troyo, and S. Ruan, The effects of human movement on the persistence of vector-borne diseases, J. Theoret. Biol. 258 (2009), no. 4, 550–560. MR 2973264, DOI 10.1016/j.jtbi.2009.02.016
Renhao Cui and Yuan Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equations 261 (2016), no. 6, 3305–3343. MR 3527631, DOI 10.1016/j.jde.2016.05.025
Keng Deng and Yixiang Wu, Dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), no. 5, 929–946. MR 3569144, DOI 10.1017/S0308210515000864
O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28 (1990), no. 4, 365–382. MR 1057044, DOI 10.1007/BF00178324
S. Friedland and S. Karlin, Some inequalities for the spectral radius of non-negative matrices and applications, Duke Math. J. 42 (1975), no. 3, 459–490. MR 376717
Shmuel Friedland, Convex spectral functions, Linear and Multilinear Algebra 9 (1980/81), no. 4, 299–316. MR 611264, DOI 10.1080/03081088108817381
Daozhou Gao, Travel frequency and infectious diseases, SIAM J. Appl. Math. 79 (2019), no. 4, 1581–1606. MR 3996918, DOI 10.1137/18M1211957
Daozhou Gao and Shigui Ruan, An SIS patch model with variable transmission coefficients, Math. Biosci. 232 (2011), no. 2, 110–115. MR 2849207, DOI 10.1016/j.mbs.2011.05.001
Daozhou Gao, P. van den Driessche, and Chris Cosner, Habitat fragmentation promotes malaria persistence, J. Math. Biol. 79 (2019), no. 6-7, 2255–2280. MR 4031813, DOI 10.1007/s00285-019-01428-2
Peter Hess, Periodic-parabolic boundary value problems and positivity, Pitman Research Notes in Mathematics Series, vol. 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. MR 1100011
Roger A. Horn and Charles R. Johnson, Matrix analysis, 2nd ed., Cambridge University Press, Cambridge, 2013. MR 2978290
Ji Fa Jiang, On the global stability of cooperative systems, Bull. London Math. Soc. 26 (1994), no. 5, 455–458. MR 1308362, DOI 10.1112/blms/26.5.455
S. Karlin, Classifications of selection-migration structures and conditions for a protected polymorphism, In: M.K. Hecht, B. Wallace, and G.T. Prance (eds), Evolutionary Biology, Plenum, New York, 14:61–204, 1982.
Huicong Li, Rui Peng, and Feng-Bin Wang, Varying total population enhances disease persistence: qualitative analysis on a diffusive SIS epidemic model, J. Differential Equations 262 (2017), no. 2, 885–913. MR 3569410, DOI 10.1016/j.jde.2016.09.044
Huicong Li and Rui Peng, Dynamics and asymptotic profiles of endemic equilibrium for SIS epidemic patch models, J. Math. Biol. 79 (2019), no. 4, 1279–1317. MR 4019925, DOI 10.1007/s00285-019-01395-8
Michael Y. Li and Zhisheng Shuai, Global stability of an epidemic model in a patchy environment, Can. Appl. Math. Q. 17 (2009), no. 1, 175–187. MR 2681418
Rui Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. I, J. Differential Equations 247 (2009), no. 4, 1096–1119. MR 2531173, DOI 10.1016/j.jde.2009.05.002
Rui Peng and Xiao-Qiang Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity 25 (2012), no. 5, 1451–1471. MR 2914149, DOI 10.1088/0951-7715/25/5/1451
Mahin Salmani and P. van den Driessche, A model for disease transmission in a patchy environment, Discrete Contin. Dyn. Syst. Ser. B 6 (2006), no. 1, 185–202. MR 2172202, DOI 10.3934/dcdsb.2006.6.185
Hal L. Smith, Monotone dynamical systems, Mathematical Surveys and Monographs, vol. 41, American Mathematical Society, Providence, RI, 1995. An introduction to the theory of competitive and cooperative systems. MR 1319817
Pengfei Song, Yuan Lou, and Yanni Xiao, A spatial SEIRS reaction-diffusion model in heterogeneous environment, J. Differential Equations 267 (2019), no. 9, 5084–5114. MR 3991554, DOI 10.1016/j.jde.2019.05.022
Joseph H. Tien, Zhisheng Shuai, Marisa C. Eisenberg, and P. van den Driessche, Disease invasion on community networks with environmental pathogen movement, J. Math. Biol. 70 (2015), no. 5, 1065–1092. MR 3319566, DOI 10.1007/s00285-014-0791-x
P. van den Driessche and James Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), 29–48. John A. Jacquez memorial volume. MR 1950747, DOI 10.1016/S0025-5564(02)00108-6
Wendi Wang and Xiao-Qiang Zhao, An epidemic model in a patchy environment, Math. Biosci. 190 (2004), no. 1, 97–112. MR 2067829, DOI 10.1016/j.mbs.2002.11.001