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Mathematical analysis and pattern formation for a partial immune system modeling the spread of an epidemic disease. (English) Zbl 1252.35053
The paper concerns a mathematical model describing the spatial propagation of an epidemic disease through a population. The pathogen diversity is structured here into two clusters and then the population is divided into eight classes which permits to distinguish between the infected/uninfected population with respect to clusters. Some weak and the global existence results of the solutions for the considered reaction-diffusion system with Neumann boundary condition are proved. Next, a mathematical Turing formulation and numerical simulations are introduced to show the pattern formation for such systems.
MSC:
35B36Pattern formation in solutions of PDE
92C15Developmental biology, pattern formation
35K57Reaction-diffusion equations
35K55Nonlinear parabolic equations
92D30Epidemiology
35K51Second-order parabolic systems, initial bondary value problems
References:
[1]Allen, L.J.S., Langlais, M., Philipps, C.J.: The dynamics of two viral infections in a single host population with applications to hantavirus. Math. Biosci. 186(2), 191–217 (2003) · Zbl 1033.92029 · doi:10.1016/j.mbs.2003.08.002
[2]Bailey, N.T.J.: The Mathematical Theory of Infectious Diseases and Its Applications, 2nd edn. Hafner, New York (1975)
[3]Barrio, R.A., Verea, C., Aragon, J.L., Maini, P.K.: A two-dimensional numerical study of spatial pattern formation in interaction systems. Bull. Math. Biol. 61, 43–505 (1999) · doi:10.1006/bulm.1998.0093
[4]Busenberg, S., Cooke, K.: Vertically Transmitted Diseases. Biomathematics, vol. 23. Springer, Berlin (1993)
[5]Cattaneo, C.: Sur une forme de l’equation de la chaleur elinant le paradoxe d’une propagation instantance. C. R. Acad. Sci. 247, 431–432 (1958)
[6]Diekmann, O., Hessterbeek, J.A.P.: Mathematical Epidemiology of Infectious Diseases. Mathematical and Computational Biology. Wiley, Chichester (2000)
[7]Gomes, M.G.M., Medley, G.F., Nokes, D.J.: On the determinants of population structure in antigentically diverse pathogens. Proc. R. Soc. Lond. B 269, 227–233 (2002) · doi:10.1098/rspb.2001.1869
[8]Gupta, S., Maiden, M.C.J.: Exploring the evolution of diversity in pathogen populations. Trends Microbiol. 9, 181–185 (2001) · doi:10.1016/S0966-842X(01)01986-2
[9]Eden, A., Michaux, B., Rakotoson, J.M.: Doubly nonlinear parabolic equations as dynamical systems. J. Dyn. Differ. Equ. 3(1), 87–131 (1991) · Zbl 0802.35011 · doi:10.1007/BF01049490
[10]Fitzgibbon, W.E., Langlais, M., Morgan, J.J.: A mathematical model for indirectly transmitted diseases. Math. Biosci. 206(2), 233–248 (2007) · Zbl 1124.35326 · doi:10.1016/j.mbs.2005.07.005
[11]Fromont, E., Pontier, D., Langlais, M.: Dynamics of a feline retrovirus (FeLV) in host populations with variable spatial structure. Proc. R. Soc. Lond. B 265, 1097–1104 (1998) · doi:10.1098/rspb.1998.0404
[12]Hollis, S., Martin, R.H., Pierre, M.: Global existence and boundedness in reaction diffusion systems. SIAM J. Math. Anal. 18, 744–761 (1987) · Zbl 0655.35045 · doi:10.1137/0518057
[13]Ladyzhenskaya, O.A., Solonnikov, V., Ural’ceva, N.: Linear and Quasi-Linear Equations of Parabolic Type. Transl. AMS, vol. 23. Am. Math. Soc., Providence (1968)
[14]Morgan, J.J.: Global existence for semilinear parabolic systems. SIAM J. Math. Anal. 20, 1128–1144 (1989) · Zbl 0692.35055 · doi:10.1137/0520075
[15]Morgan, J.J.: Boundedness and decay results for reaction diffusion systems. SIAM J. Math. Anal. 21, 1172–1184 (1990) · Zbl 0723.35039 · doi:10.1137/0521064
[16]Morgan, J.J., Hollis, S.L.: The existence of periodic solutions to reaction-diffusion systems with periodic data. SIAM J. Math. Anal. 26, 1225–1232 (1995) · Zbl 0849.35052 · doi:10.1137/S0036141093257179
[17]Murray, J.D.: Mathematical Biology I: An Introduction, 3rd edn. Springer, Berlin (2003)
[18]Murray, J.D.: Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd edn. Springer, Berlin (2003)
[19]Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B 237, 37–72 (1952) · doi:10.1098/rstb.1952.0012
[20]Simon, J.: Compact sets in the space L p (0,T;B). Ann. Math. Pura Appl. 65–96 (1989)