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Complex dynamics of a reaction-diffusion epidemic model. (English) Zbl 1327.92069

Summary: In this paper, we investigate the complex dynamics of a reaction-diffusion \(S-I\) model incorporating demographic and epidemiological processes with zero-flux boundary conditions. By the method of Lyapunov function, the global stability of the disease free equilibrium and the epidemic equilibrium was established. In addition, the conditions of Turing instability were obtained and the Turing space in the parameters space were given. Based on these results, we present the evolutionary processes that involves organism distribution and their interaction of spatially distributed population with local diffusion, and find that the model dynamics exhibits a diffusion-controlled formation growth to “holes, holes-stripes, stripes, spots-stripes and spots” pattern replication. Furthermore, we indicate that the diseases’ spread is getting smaller with \(R_{0}\) increasing, and the increasing the diffusion of infectious will increase the speed of diseases spreading. Our results indicate that the diffusion has a great influence on the spread of the epidemic and extend well the finding of spatio-temporal dynamics in the epidemic model.

MSC:

92D30 Epidemiology
35K57 Reaction-diffusion equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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[1] Kermack, W. O.; McKendrick, A. G., A contribution to the mathematical theory of epidemics, Proc. R. Soc. A, 115, 700-721 (1927) · JFM 53.0517.01
[2] Ma, Z.; Zhou, Y.; Wu, J., Modeling and Dynamics of Infectious Diseases (2009), Higher Education Press: Higher Education Press Beijing · Zbl 1180.92081
[3] Smith, H. L., Subharmonic bifurcation in an SIR epidemic model, J. Math. Biol., 17, 2, 163-177 (1983) · Zbl 0578.92023
[4] Beretta, E.; Takeuchi, Y., Global stability of an SIR epidemic model with time delays, J. Math. Biol., 33, 3, 250-260 (1995) · Zbl 0811.92019
[5] Hethcote, H. W., The mathematics of infectious diseases, SIAM Rev., 42, 4, 599-653 (2000) · Zbl 0993.92033
[6] Fan, M.; Li, M. Y.; Wang, K., Global stability of an SEIS epidemic model with recruitment and a varying total population size, Math. Biosci., 170, 2, 199-208 (2001) · Zbl 1005.92030
[7] Li, M. Y.; Smith, H. L.; Wang, L., Global dynamics of an SEIR epidemic model with vertical transmission, SIAM J. Appl. Math., 62, 1, 58-69 (2001) · Zbl 0991.92029
[8] Ruan, S.; Wang, W., Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations, 188, 1, 135-163 (2003) · Zbl 1028.34046
[9] Berezovsky, F.; Karev, G.; Song, B.; Castillo-Chavez, C., A simple epidemic model with surprising dynamics, Math. Biosci. Eng., 2, 133-152 (2005) · Zbl 1061.92052
[10] Wang, W.; Ruan, S., Bifurcations in an epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl., 291, 2, 775-793 (2004) · Zbl 1054.34071
[11] Zhou, Y.; Ma, Z.; Brauer, F., A discrete epidemic model for SARS transmission and control in China, Math. Comp. Model., 40, 13, 1491-1506 (2004) · Zbl 1066.92046
[12] Xiao, D.; Ruan, S., Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208, 2, 419-429 (2007) · Zbl 1119.92042
[13] Okubo, A.; Levin, S., Diffusion and Ecological Problems: Modern Perspectives (2001), Springer: Springer New York · Zbl 1027.92022
[14] Neuhauser, C., Mathematical challenges in spatial ecology, Notices Amer. Math. Soc., 48, 11, 1304-1314 (2001) · Zbl 1128.92328
[15] Cantrell, R.; Cosner, C., Spatial Ecology via Reaction-Diffusion Equations (2003), Wiley · Zbl 1059.92051
[16] Murray, J. D., Mathematical Biology (2003), Springer: Springer New York
[17] Holmes, E. E.; Lewis, M. A.; Banks, J. E.; Veit, R. R., Partial differential equations in ecology: spatial interactions and population dynamics, Ecology, 75, 17-29 (1994)
[18] Rass, L.; Radcliffe, J., Spatial Deterministic Epidemics (2003), American Mathematical Society · Zbl 1018.92028
[19] Hosono, Y.; Ilyas, B., Traveling waves for a simple diffusive epidemic model, Math. Model. Method. Appl. Sci., 5, 935-966 (1995) · Zbl 0836.92023
[20] Cruickshank, I.; Gurney, W.; Veitch, A., The characteristics of epidemics and invasions with thresholds, Theor. Popu. Biol., 56, 3, 279-292 (1999) · Zbl 0963.92034
[21] Ferguson, N. M.; Donnelly, C. A.; Anderson, R. M., The foot-and-mouth epidemic in Great Britain: pattern of spread and impact of interventions, Science, 292, 5519, 1155 (2001)
[22] Grenfell, B. T.; Bjornstad, O. N.; Kappey, J., Travelling waves and spatial hierarchies in measles epidemics, Nature, 414, 6865, 716-723 (2001)
[23] He, D.; Stone, L., Spatio-temporal synchronization of recurrent epidemics, Proc. R. Soc. Lond. B, 270, 1523, 1519-1526 (2003)
[24] Lloyd, A. L.; Jansen, V. A., Spatiotemporal dynamics of epidemics: synchrony in metapopulation models, Math. Biosci., 188, 1-2, 1-16 (2004) · Zbl 1036.92029
[25] van Ballegooijen, W. M.; Boerlijst, M. C., Emergent trade-offs and selection for outbreak frequency in spatial epidemics, PNAS, 101, 52, 18246 (2004)
[26] Filipe, J. A.N.; Maule, M. M., Effects of dispersal mechanisms on spatio-temporal development of epidemics, J. Theoret. Biol., 226, 2, 125-141 (2004) · Zbl 1439.92171
[27] Funk, G.; Jansen, V.; Bonhoeffer, S.; Killingback, T., Spatial models of virus-immune dynamics, J. Theoret. Biol., 233, 2, 221-236 (2005) · Zbl 1442.92163
[28] Pascual, M.; Guichard, F., Criticality and disturbance in spatial ecological systems, TREE, 20, 2, 88-95 (2005)
[29] Festenberg, N. V.; Gross, T.; Blasius, B., Seasonal forcing drives spatio-temporal pattern formation in rabies epidemics, Math. Model. Nat. Phen., 2, 4, 63-73 (2007) · Zbl 1337.92207
[30] Mulone, G.; Straughan, B.; Wang, W., Stability of epidemic models with evolution, Studies in Appl. Math., 118, 2, 117-132 (2007)
[31] Hilker, F. M.; Langlais, M.; Petrovskii, S. V.; Malchow, H., A diffusive SI model with Allee effect and application to FIV, Math. Biosci., 206, 1, 61-80 (2007) · Zbl 1124.92044
[32] Wang, K.; Wang, W., Propagation of HBV with spatial dependence, Math. Biosci., 210, 1, 78-95 (2007) · Zbl 1129.92052
[33] Wang, K.; Wang, W.; Song, S., Dynamics of an HBV model with diffusion and delay, J. Theoret. Biol., 253, 1, 36-44 (2008) · Zbl 1398.92257
[34] Malchow, H.; Petrovskii, S. V.; Venturino, E., (Spatiotemporal Patterns in Ecology and Epidemiology—Theory, Models, and Simulation. Spatiotemporal Patterns in Ecology and Epidemiology—Theory, Models, and Simulation, Mathematical and Computational Biology Series (2008), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton) · Zbl 1298.92004
[35] Upadhyay, R. K.; Kumari, N.; Rao, V., Modeling the spread of bird flu and predicting outbreak diversity, Nonl. Anal.: RWA, 9, 4, 1638-1648 (2008) · Zbl 1154.92324
[36] Xu, R.; Ma, Z., An HBV model with diffusion and time delay, J. Theoret. Biol., 257, 3, 499-509 (2009) · Zbl 1400.92560
[37] B. Camara, Complexité de dynamiques de modèles proie-prédateur avec diffusion et applications, Ph.D Thesis, Lorient, Universitè du Havre, 2009.; B. Camara, Complexité de dynamiques de modèles proie-prédateur avec diffusion et applications, Ph.D Thesis, Lorient, Universitè du Havre, 2009.
[38] Turing, A. M., The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. Lond. B, 237, 641, 37-72 (1952) · Zbl 1403.92034
[39] Liu, Q.; Jin, Z., Formation of spatial patterns in an epidemic model with constant removal rate of the infectives, J. Stat. Mech., 2007, P05002 (2007)
[40] Sun, G.; Jin, Z.; Liu, Q. X.; Li, L., Pattern formation in a spatial \(S - I\) model with non-linear incidence rates, J. Stat. Mech., 2007, P11011 (2007)
[41] Sun, G.; Jin, Z.; Liu, Q. X.; Li, L., Chaos induced by breakup of waves in a spatial epidemic model with nonlinear incidence rate, J. Stat. Mech., 2008, P08011 (2008)
[42] Sun, G.; Jin, Z.; Liu, Q. X.; Li, L., Spatial pattern in an epidemic system with cross-diffusion of the susceptible, J. Biol. Sys., 17, 1, 1-12 (2009)
[43] Cai, Y.; Wang, W., Spatiotemporal dynamics of a reaction-diffusion epidemic model with nonlinear incidence rate, J. Stat. Mech., P02025 (2011)
[44] Wang, W.; Lin, Y.; Wang, H.; Liu, H.; Tan, Y., Pattern selection in an epidemic model with self and cross diffusion, J. Biol. Sys., 19, 19-31 (2011) · Zbl 1404.92206
[45] Bendahmane, M.; Saad, M., Mathematical analysis and pattern formation for a partial immune system modeling the spread of an epidemic disease, Acta Appl. Math., 115, 17-42 (2011) · Zbl 1252.35053
[46] Arcuri, P.; Murray, J. D., Pattern sensitivity to boundary and initial conditions in reaction-diffusion models, J. Math. Biol., 24, 2, 141-165 (1986) · Zbl 0595.92001
[47] Hwang, T. W.; Kuang, Y., Deterministic extinction effect of parasites on host populations, J. Math. Biol., 46, 1, 17-30 (2003) · Zbl 1015.92042
[48] Chen, W.; Wang, M., Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion, Math. Comput. Model., 42, 1-2, 31-44 (2005) · Zbl 1087.35053
[49] Henry, D., Geometric theory of semilinear parabolic equations, (Lecture Notes in Mathematics (1981), Springer-Verlag) · Zbl 0456.35001
[50] Baurmann, M.; Gross, T.; Feudel, U., Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J. Theoret. Biol., 245, 2, 220-229 (2007) · Zbl 1451.92248
[51] Wang, W.; Liu, Q.; Jin, Z., Spatiotemporal complexity of a ratio-dependent predator-prey system, Phys. Rev. E, 75, 5, 051913 (2007)
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