Abstract
In this paper, with the assumptions that an infectious disease in a population has a fixed latent period and the latent individuals of the population may diffuse, we formulate an SIR model with a simple demographic structure for the population living in a spatially continuous environment. The model is given by a system of reaction-diffusion equations with a discrete delay accounting for the latency and a spatially non-local term caused by the mobility of the individuals during the latent period. We address the existence, uniqueness, and positivity of solution to the initial-value problem for this type of system. Moreover, we investigate the traveling wave fronts of the system and obtain a critical value c * which is a lower bound for the wave speed of the traveling wave fronts. Although we can not prove that this value is exactly the minimal wave speed, numeric simulations seem to suggest that it is. Furthermore, the simulations on the PDE model also suggest that the spread speed of the disease indeed coincides with c *. We also discuss how the model parameters affect c *.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Anderson, R.M., Trewhella, W.J., 1985. Population dynamics of the badger (Meles meles) and the epidemiology of bovine tuberculosis (mycobacterium bovis). Philos. Trans. R. Soc. Lond. B 310, 327–381.
Arino, J., van den Driessche, P., 2003a. A multi-city epidemic model. Math. Popul. Stud. 10, 175–193.
Arino, J., van den Driessche, P., 2003b. The basic reproduction number in a multi-city compartmental epidemic model. LNCIS 294, 135–142.
Arino, J., van den Driessche, P., 2006. Metapopulation epidemic models, a survey. Fields Inst. Commun. 48, 1–12.
Barlow, N.D., 2000. Non-linear transmission and simple models for bovine tuberculosis. J. Anim. Ecol. 69, 703–713.
Brauer, F., van den Driessche, P., 2001. Models for transmission of disease with immigration of infectives. Math. Biosci. 171, 143–154.
Brauer, F., van den Driessche, P., Wu, J., 2008. Mathematical Epidemiology. Lecture Notes in Mathematics. Springer, Berlin/Heidelberg.
Castillo-Chavez, C., Yakubu, A.A., 2001. Dispersal, disease and life-history evolution. Math. Biosci. 173, 35–53.
Daners, D., Medina, P.K., 1992. Abstract evolution equations, periodic problems and applications. In: Pitman Research Notes in Mathematics, vol. 279. Longman, Harlow.
Dunbar, S., 1981. Travelling wave solutions of diffusive Lotka–Volterra interaction equations. Ph.D. thesis, Univ. Minnesota, Minneapolis.
Dunbar, S., 1983. Travelling wave solutions of diffusive Lotka–Volterra equations. J. Math. Biol. 17, 11–32.
Dunbar, S., 1984. Traveling wave solutions of diffusive Lotka–Volterra equations: a heteroclinic connection in R 4. Trans. Am. Math. Soc. 286, 557–594.
Gardner, R.A., 1984. Existence of travelling wave solutions of predator-prey systems via the connection index. SIAM J. Appl. Math. 44, 56–79.
Garnett, B.T., Delahay, R.J., Roper, T.J., 2002. Use of cattle farm resources by badgers (Meles meles) and risk of bovine tuberculosis (mycobacterium bovis) transmission to cattle. Proc. R. Soc. B 269, 1487–1491.
Hethcote, H.W., 2000. The mathematics of infectious diseases. SIAM Rev. 42, 599–653.
Hsieh, Y.-H., van den Driessche, P., Wang, L., 2007. Impact of travel between patches for spatial spread of disease. Bull. Math. Biol. 69, 1355–1375.
Huang, J., Lu, G., Ruan, S., 2003. Existence of traveling wave solutions in a diffusive predator-prey model. J. Math. Biol. 46, 132–152.
Li, J., Zou, X., 2009a. Generalization of the Kermack-McKendrick SIR model to a patchy environment for a disease with latency. Math. Model. Nat. Phenom. 4(2), 92–118.
Li, J., Zou, X., 2009b. An epidemic model with non-local infections on a patchy environment. J. Math. Biol. doi:10.1007/s00285-009-0280-9, in press.
Liang, X., Zhao, X.-Q., 2007. Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Commun. Pure Appl. Math. 60, 1–40.
Martin Jr, R.H., Smith, H.L., 1990. Abstract functional-differential equations and reaction-diffusion systems. Trans. Am. Math. Soc. 321, 1–44.
Metz, J.A.J., Diekmann, O. (Eds.), 1986. The Dynamics of Physiologically Structured Populations. Springer, New York.
Murray, J.D., 2002. Mathematical Biology, 3rd edn. Springer, New York.
Salmani, M., van den Driessche, P., 2006. A model for disease transmission in a patchy environment. Discrete Contin. Dyn. Syst. B 6, 185–202.
Thoen, C.O., Karlson, A.G., Himes, E.M., 1984. Mycobacterium tuberculosis complex. In: The Mycobacteria, A Sourcebook, pp. 1209–1235. Dekker, New York.
van den Driessche, P., Wang, L., Zou, X., 2007. Modelling disease with latency and replase. Math. Biosci. Eng. 4, 205–219.
Wang, W., Mulone, G., 2003. Threshold of disease transmission on a patch environment. J. Math. Anal. Appl. 285, 321–335.
Wang, W., Zhao, X.-Q., 2004. An epidemic model in a patchy environment. Math. Biosci. 190, 97–112.
Wang, W., Zhao, X.-Q., 2005. An age-structured epidemic model in a patchy environment. SIAM J. Appl. Math. 65, 1597–1614.
Wang, W., Zhao, X.-Q., 2006. An epidemic model with population dispersal and infection period. SIAM J. Appl. Math. 66, 1454–1472.
Wu, J., 1996. Theory and Applications of Partial Functional Differential Equations. Applied Mathematical Science, vol. 119. Springer, Berlin.
Wu, J., Zou, X., 2001. Traveling wave fronts of reaction-diffusion systems with delay. J. Dyn. Differ. Equ. 13, 651–687.
Zou, X., 2002. Delay induced traveling wave fronts in reaction diffusion equations of KPP-Fisher type. J. Comput. Appl. Math. 146, 309–321.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported by NSERC, by NCE-MITACS of Canada and by PREA of Ontario.
Rights and permissions
About this article
Cite this article
Li, J., Zou, X. Modeling Spatial Spread of Infectious Diseases with a Fixed Latent Period in a Spatially Continuous Domain. Bull. Math. Biol. 71, 2048–2079 (2009). https://doi.org/10.1007/s11538-009-9457-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-009-9457-z