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A vector-borne disease is caused by a range of pathogens and transmitted to hosts through vectors. To investigate the multiple effects of the spatial heterogeneity, the temperature sensitivity of extrinsic incubation period and intrinsic incubation period, and the seasonality on disease transmission, we propose a nonlocal reaction–diffusion model of vector-borne disease with periodic delays. We introduce the basic reproduction number $$\mathfrak {R}_0$$ for this model and then establish a threshold-type result on its global dynamics in terms of $$\mathfrak {R}_0$$ . In the case where all the coefficients are constants, we also prove the global attractivity of the positive constant steady state when $$\mathfrak {R}_0>1$$ . Numerically, we study the malaria transmission in Maputo Province, Mozambique.

Bat-borne rabies is a viral infection caused by lyssaviruses. To explore the complex interaction between bats and cattle, we develop a periodic compartment system with seasonality, temperature-dependent incubation period and different control measures. We derive the basic reproduction number R 0 and show it determines the global dynamics. In particular, we prove the non-existence of positive periodic solutions in the critical case where R 0 = 1 . Numerically, we observe that sufficient long time-varying incubation periods could potentially increase the disease outbreak risk.

In this paper, we study the propagation dynamics for a class of integrodifference competition models in a periodic habitat. An interesting feature of such a system is that multiple spreading speeds can be observed, which biologically means different species may have different spreading speeds. We show that the model system admits a single spreading speed, and it coincides with the minimal wave speed of the spatially periodic traveling waves. A set of sufficient conditions for linear determinacy of the spreading speed is also given., arXiv admin note: text overlap with arXiv:1410.4591, arXiv:1504.03788