Journal of Mathematical Biology

, Volume 65, Issue 6–7, pp 1387–1410 | Cite as

Threshold dynamics of an infective disease model with a fixed latent period and non-local infections

  • Zhiming Guo
  • Feng-Bin Wang
  • Xingfu ZouEmail author


In this paper, we derive and analyze an infectious disease model containing a fixed latency and non-local infection caused by the mobility of the latent individuals in a continuous bounded domain. The model is given by a spatially non-local reaction–diffusion system carrying a discrete delay associated with the zero-flux condition on the boundary. By applying some existing abstract results in dynamical systems theory, we prove the existence of a global attractor for the model system. By appealing to the theory of monotone dynamical systems and uniform persistence, we show that the model has the global threshold dynamics which can be described either by the principal eigenvalue of a linear non-local scalar reaction diffusion equation or equivalently by the basic reproduction number \({\mathcal{R}_0}\) for the model. Such threshold dynamics predicts whether the disease will die out or persist. We identify the next generation operator, the spectral radius of which defines basic reproduction number. When all model parameters are constants, we are able to find explicitly the principal eigenvalue and \({\mathcal{R}_0}\). In addition to computing the spectral radius of the next generation operator, we also discuss an alternative way to compute \({\mathcal{R}_0}\).


Infectious disease model Reaction–diffusion equation Non-local infection Delay Principal eigenvalue Basic reproduction number 

Mathematics Subject Classification (2000)

35K57 37N25 92D30 


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.School of Mathematics and Information SciencesGuangzhou UniversityGuangzhouPeople’s Republic of China
  2. 2.Department of Applied MathematicsUniversity of Western OntarioLondonCanada

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