×

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:

35B36 Pattern formations in context of PDEs
92C15 Developmental biology, pattern formation
35K57 Reaction-diffusion equations
35K55 Nonlinear parabolic equations
92D30 Epidemiology
35K51 Initial-boundary value problems for second-order parabolic systems
PDF BibTeX XML Cite
Full Text: DOI

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
[2] Bailey, N.T.J.: The Mathematical Theory of Infectious Diseases and Its Applications, 2nd edn. Hafner, New York (1975) · Zbl 0334.92024
[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) · Zbl 1323.92026
[4] Busenberg, S., Cooke, K.: Vertically Transmitted Diseases. Biomathematics, vol. 23. Springer, Berlin (1993) · Zbl 0837.92021
[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)
[8] Gupta, S., Maiden, M.C.J.: Exploring the evolution of diversity in pathogen populations. Trends Microbiol. 9, 181–185 (2001)
[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
[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
[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)
[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
[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) · Zbl 0174.15403
[14] Morgan, J.J.: Global existence for semilinear parabolic systems. SIAM J. Math. Anal. 20, 1128–1144 (1989) · Zbl 0692.35055
[15] Morgan, J.J.: Boundedness and decay results for reaction diffusion systems. SIAM J. Math. Anal. 21, 1172–1184 (1990) · Zbl 0723.35039
[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
[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) · Zbl 1006.92002
[19] Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B 237, 37–72 (1952) · Zbl 1403.92034
[20] Simon, J.: Compact sets in the space L p (0,T;B). Ann. Math. Pura Appl. 65–96 (1989) · Zbl 0629.46031
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.