Mathematical Methods I
Mathematical Methods I AME 60611
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Date Created: 11/01/15
Technical Review Lee G Neuharth AME 60611 8 October 2007 In their recent article Wang and Zhao1 discuss a spatialtemporal model for the spread of contagious disease This model is of particular interest in current our current global society where a simple plane ride can spark a fresh outbreak of a disease thousands of kilometers from the previous outbreak In their model Wang and Zhao consider the following basic factors immigration and emigration based upon patches age demographics birth rates and death rates Patches are de ned as any geographic region a city nation speci c coordinate ranges on a map etc The age demo graphics here focus on a juveniles changing to adults in a given patch 239 Given the basic factors listed above the spread of disease can be represented by the system dd BiAitAit MiJz O RN 21 Ci j b dd Rit diSi iSiIi 711139 21 aiijv 1 2 iSiIi di 701139 21 bijIjz where Ji Si and Ii are respectively the total numbers of juveniles persons susceptible to disease and infected persons Additionally A is the adult population ai is the juvenile death rate R is the rate of change of juveniles to adults di is the adult death rate is the rate of contact between the infected and the uninfected 1 and Si and 71 is the recovery rate for the infected Regarding the summations cij aij and 2139 represent immigration rates for respectively susceptible persons infected persons and juveniles The components of each term represent immigration from the jth to the ith patch with J S and j representing the same values for the jth patch that their 2th patch counterparts represent Note that juveniles are left out of the susceptible pool the birth rate is actually a function of adult population and adult population is the sum of the susceptible and infected poo s The greatest accomplishment of this paper lies in the generation of birth rate functions which then allows them to create a two patch mo el An extensive derivation shows how birth rates were calculated by the authors using a disease free equilibrium condition This derivation owed transparently and systematically from step to step The example posed most often is that of the spread of sexually transmitted diseases Indeed the assumption to leave the juveniles out of the susceptible persons pool is reasonable in this case However using this model poses a system that can be easily manipulated to portray disease diffusion as variant with other age demographics The elderly for instance are more susceptible to many diseases If a new Ri is de ned for the rate of change from regular adulthood to elderly status risks to elderly people or any demographic could be estimated As the current state of the world has more tightly de ned demographics moving among more i and j patches with greater frequency the diffusion of disease becomes harder to model This makes the work done by Wang and Zhao potentially helpful to epidemiologists and healthcare leaders The results of such analytical which can then be numerical schemes would also be helpful to politicians and legislators as they deal with immigration and travel regulations As the authors conclude this paper they highlight the need to model greater complexity including more age demographic steps the relationship between juvenile dispersion and adult dispersion by family association and more patches These conclusions come mostly from their test case which most prominently showed a pulsation of epidemic as juvenile dispersion increased Pulsation or persistence and extinction cycling seems counterintuitive and begs further modelling 1Wang W and Zhao X 2005 An AgeStructured Epidemic Model in a Patchy Environment SIAM Journal of Applied Mathematics Vol 65 No 5 pp 159771614