By P. Waltman
These notes correspond to a collection of lectures given on the Univer sity of Alberta throughout the spring semester, 1973. the 1st 4 sec tions current a scientific improvement of a deterministic, threshold version for the spraad of infection. part five provides a few compu tational effects and makes an attempt to tie the version with different arithmetic. In all the final 3 sections a separate, really good subject is gifted. the writer needs to thank Professor F. Hoppensteadt for making to be had preprints of 2 of his papers and for interpreting and remark ing on a initial model of those notes. He additionally needs to thank Professor J. Mosevich for offering the graphs in part five. The stopover at on the college of Alberta used to be a truly friendly one and the writer needs to specific his appreciation to Professors S. Ghurye and J. Macki for the invitation to go to there. ultimately, thank you are as a result of very useful secretarial employees on the college of Alberta for typing the unique draft of the lecture notes and to Mrs. Ada Burns of the collage of Iowa for her very good typescript of the ultimate model. desk OF CONTENTS 1. an easy Epidemic version with everlasting elimination . . . • . . . 1 2. A extra basic version and the selection of the depth of a scourge. 10 21 three. A Threshold version. four. A Threshold version with transitority Immunity. 34 five. a few exact circumstances and a few Numerical Examples forty eight A inhabitants Threshold version . sixty two 6.
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Example text
3) (IO(t) and Il(t) + So - IO(t) + JT(t-a-w) r(x)S (x) I(x) dx, t JT(T(t) r(x)S(x)I(x)dx. ) The questions of the existence, uniqueness, and continuous dependence of solutions were resolved so the model is mathematically sensible, but many other questions of interest in studying epidemics were left open. Foremost among these are questions involving limiting behavior and the development of numerical techniques for computing solutions. Since the supply of susceptibles is replenished from the removed class, some sort of recurrence is not unexpected.
The first mathematical question of interest is whether or not there exists a solution to such equations, and if so, whether it is unique. One would also like to know something about the ultimate (limiting) behavior of the solutions, in particular, what sort of recurrence properties can be established. find the solutions numerically. It is also of interest to In this section we consider only the existence and uniqueness question-most of the rest remains 5). 1. Pl ~ IO(t) ~ 0, Pz >0 and Set) w, and m.
Uous dependence of solutions was Existence, uniqueness, and continestablished~ The limiting value of the infective population when rand p are constant was first determined by H. Hethcote, Note on Determining the Limiting Susceptible Population in an Epidemic Model, Math. Biosciences 9(1970), 161-163. A complete analysis of this special case of the_model was presented in L. O. Wilson, An Epidemic Model Involving a Threshold, Math. Biosciences 15(1972), 109-121, where an exact solution was found and an asymptotic estimate provided of the approach to the limit, and the changes with respect to parameters investigated.