In 1927 Kermack and McKendrick introduced and analyzed a rather general epidemic model (nota bene : their model takes the form of a nonlinear renewal equation and the familiar SIR model is but a very special case !). The aim of this lecture is to revive the spatial variant of this model, as studied in the late seventies by Horst Thieme and myself (see the AMS book 'Spatial Deterministic Epidemics' by L. Rass and J. Radcliffe, 2003).
The key result is a characterization of c_0 , the lowest possible speed of travelling waves and the proof that c_0 is also the asymptotic speed of epidemic propagation.
We develop spatial models of vector-borne disease dynamics on a network of patches to examine how the movement of humans in heterogeneous environments affects transmission. We show that the movement of humans between patches is sufficient to maintain disease persistence in patches with zero transmission. We construct two classes of models using different approaches: (i) Lagrangian that mimics human commuting behavior and (ii) Eulerian that mimics human migration. We determine the basic reproduction number R0 for both modeling approaches and study the transmission dynamics in terms of R0. We also study the dependence of R0 on some parameters such as the travel rate of the infectives.
Mosquitoes can rapidly develop resistance to insecticides, which is a big problem in malaria control. Current insecticides kill rapidly on contact, but this leads to intense selection for resistance because young adults are killed. Of considerable current interest is the possibility of slowing down or even halting the evolution of resistance. Biologists believe that much weaker selection for resistance can be achieved if insecticides target only old mosquitoes that have already laid most of their eggs. This strategy aims to exploit the fact that most mosquitoes do not live long enough to transmit malaria, due to a long latency stage for the malaria parasite in the mosquito. I will present the results of some mathematical work using stage structured population models that can make predictions about the delayed onset of resistance in the mosquito population when they are subjected to an insecticide that only acts late in life. I will also summarise some ongoing work that includes the malaria disease dynamics and also the consequences of mosquito control using larvicides. Larvae can become resistant to larvicides, but the evolutionary cost of this acquired resistance may be reduced longevity as adults, which reduces the likelihood of the parasites completing their developmental stages and thus can actually benefit malaria control.
This is a joint collaboration with Rongsong Liu, Chuncheng Wang and Jianhong Wu.
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The basic reproduction number and its computation formulae are established for epidemic models with reaction-diffusion structures. It is proved that the basic reproduction number provides the threshold value for disease invasion in the sense that the disease-free steady state is asymptotically stable if the basic reproduction number is less than unity and the disease is uniformly persistent if it is greater than unity. On the basis of these theoretical results, three epidemic models for rabies, lyme disease and West Nile transmissions are analyzed to reveal the better strategies for these diseases. With the aid of numerical simulations, we find that the reduction of heterogenous infection is beneficial because the more heterogenous infection leads to the higher value of basic reproduction numbers. Moreover, influences from spatial configurations of disease infection and diffusion coefficients are investigated. This is a joint work with Xiaoqiang Zhao.