Effects of heterogeneity and global dynamics of weakly connected subpopulations

Derdei M. Bichara¹ ; Abderrahman Iggidr² ; Souad Yacheur^2, 3

¹ Department of Mathematics, California State University, Fullerton, CA 92831, USA.
² Université de Lorraine, CNRS, Inria, IECL, 57000 Metz, France.
³ Department of Mathematics, University of Tlemcen, Tlemcen 13000, Algeria.

Mathematical modelling of natural phenomena, Tome 16 (2021), article no. 44.

Résumé

We develop a method that completely characterizes the global dynamics of models with multiple subpopulations that are weakly interconnected. The method is applied on two classes of models with multiple subpopulations: an epidemic model that involves multiple host species and multiple vector species and a patchy vector-borne model. The method consists of two main steps: reducing the system using tools of large scale systems and studying the dynamics of an auxiliary system related the original system. The developed method determines the underlying dynamics and the “weight” of each subpopulations with respect to the dynamics of the whole population, and how the topology of the connectivity matrix alters the dynamics of the overall population. The method provides global stability results for all types of equilibria, namely trivial, boundary or interior equilibria.

Détail

DOI : 10.1051/mmnp/2021034

Bibliographie

[1] R.M. Anderson and R.M. May, Infectious Diseases of Humans. Dynamics and Control. Oxford Science Publications (1991).

[2] J. Arino, Diseases in metapopulations, in Modeling and Dynamics of Infectious Diseases, edited by Z. Ma, Y. Zhou, and J. Wu. Volume 11 of Series in Contemporary Applied Mathematics. World Scientific (2009) 65–123. Also CDM Preprint Series report 2008-04.

[3] J. Arino and P. van den Driessche, Disease spread in metapopulations, in Nonlinear dynamics and evolution equations, edited by X.-O Zhao and X. Zou. Vol. 48 of Fields Institute Communications. AMS, Providence, R.I. (2006) 1–13.

[4] J. Arino, A. Ducrot, P. Zongo A metapopulation model for malaria with transmission-blocking partial immunity in hosts J. Math. Biol 2012 423 448

[5] P. Auger, E. Kouokam, G. Sallet, M. Tchuente, B. Tsanou The Ross-Macdonald model in a patchy environment Math. Biosci 2008 123 131

[6] A. Berman and R.J. Plemmons, Nonnegative matrices in the mathematical sciences. SIAM (1994).

[7] D.M. Bichara Effects of migration on vector-borne diseases with forward and backward stage progression Discr. Continu. Dyn. Syst. Ser. B 2019 6297 6323

[8] D.M. Bichara Global analysis of multi-host and multi-vector epidemic models J. Math. Anal. Appl 2019 1532 1553

[9] D.M. Bichara, A. Iggidr Multi-patch and multi-group epidemic models: a new framework J. Math. Biol 2018 107 134

[10] D.M. Bichara, Y. Kang, C. Castillo-Chavez, R. Horan, C. Perrings Sis and sir epidemic models under virtual dispersal Bull. Math. Biol 2015 2004 2034

[11] D.M. Bichara, C. Castillo-Chavez Vector-borne diseases models with residence times – a Lagrangian perspective Math. Biosci 2016 128 138

[12] B. Bonzi, A. Fall, A. Iggidr, G. Sallet Stability of differential susceptibility and infectivity epidemic models J. Math. Biol 2011 39 64

[13] C. Cosner, J.C. Beier, R.S. Cantrell, D. Impoinvil, L. Kapitanski, M.D. Potts, A. Troyo, S. Ruan The effects of human movement on the persistence of vector-borne diseases J. Theor. Biol 2009 550 560

[14] G. Cruz-Pacheco, L. Esteva, C. Vargas Multi-species interactions in West Nile virus infection J. Biol. Dyn 2012 281 298

[15] O. Diekmann, J.A.P. Heesterbeek, J.A.J. Metz On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations J. Math. Biol 1990 365 382

[16] P. Du, M.Y. Li Impact of network connectivity on the synchronization and global dynamics of coupled systems of differential equations Physica D 2014 32 42

[17] A. Fenton, D.G. Streicker, O.L. Petchey, A.B Pedersen Are all hosts created equal? Partitioning host species contributions to parasite persistence in multihost communities Am. Nat 2015 610 622

[18] A.K. Githeko, J.M. Ayisi, P.K. Odada, F.K. Atieli, B.A. Ndenga, J.I. Githure, G. Yan Topography and malaria transmission heterogeneity in western Kenya highlands: prospects for focal vector control Malaria J 2006

[19] H. Guo, M.Y. Li, Z. Shuai Global stability of the endemic equilibrium of multigroup SIR epidemic models. Can. Appl. Math. Quart. 2006 259 284

[20] H. Guo, M.Y. Li, Z. Shuai A graph-theoretic approach to the method of global Lyapunov functions Proc. Am. Math. Soc 2008 2793 2802

[21] I. Hanski, Metapopulation Ecology. Oxford University Press (1999).

[22] R.A. Horn and C.R. Johnson, Matrix analysis. Cambridge University Press (1985).

[23] A. Iggidr, G. Sallet, B. Tsanou Global stability analysis of a metapopulation SIS epidemic model Math. Popul. Stud 2012 115 129

[24] A. Iggidr, G. Sallet, M.O. Souza On the dynamics of a class of multi-group models for vector-borne diseases J. Math. Anal. Appl 2016 723 743

[25] W.B. Karesh, A. Dobson, J.O. Lloyd-Smith, J. Lubroth, M.A. Dixon, M. Bennett, S. Aldrich, T. Harrington, P. Formenty, E.H. Loh, C.C. Machalaba Ecology of zoonoses: natural and unnatural histories The Lancet 2012 1936 1945

[26] W.O. Kermack, A.G. Mckendrick Contributions to the mathematical theory of epidemics–I. 1927 Bull. Math. Biol 1991 33 55

[27] A. Lajmanovich, J.A. Yorke A deterministic model for gonorrhea in a nonhomogeneous population Math. Biosci. 1976 221 236

[28] D.J. Rodríguez, L. Torres-Sorando Models of infectious diseases in spatially heterogeneous environments Bull. Math. Biol 2001 547 571

[29] N.W. Ruktanonchai, D.L. Smith, P. De Leenheer Parasite sources and sinks in a patched Ross-MacDonald malaria model with human and mosquito movement: implications for control Math. Biosci 2016 90 101

[30] M. Salmani, P. Van Den Driessche A model for disease transmission in a patchy environment Discr. Continu. Dyn. Syst. B 2006 185 202

[31] D.L. Smith, J. Dushoff, F. Ellis Mckenzie The risk of a mosquito-borne infection in a heterogeneous environment PLoS Biol 2004 1957 1964

[32] H.L. Smith, Monotone dynamical systems. An introduction to the theory of competitive and cooperativ systems. AMS, Providence, R.I. (1995).

[33] P. Van Den Driessche, J. Watmough Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission Math. Biosci. 2002 29 48

[34] P. Van Den Driessche, Y.-H Hsieh, L. Wang Impact of travel between patches for spatial spread of disease Bull. Math. Biol 2007 1355 1375

[35] M. Vidyasagar Decomposition techniques for large-scale systems with nonadditive interactions: stability and stabilizability IEEE Trans. Autom. Control 1980 773 779

[36] J.P. Webster, A. Borlase, J.W. Rudge Who acquires infection from whom and how? disentangling multi-host and multi-mode transmission dynamics in the ‘elimination’ era Philos. Trans. Roy. Soc. B 2017

[37] People, pathogens and our planet, volume 1: Towards a one health approach for controlling zoonotic diseases World Bank, Agriculture and Rural Development Health, Nutrition and Population 2010