Deterministic Threshold Models in the Theory of Epidemics by P. Waltman

By P. Waltman

These notes correspond to a collection of lectures given on the Univer­ sity of Alberta through 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 offers a few compu­ tational effects and makes an attempt to tie the version with different arithmetic. In all of the final 3 sections a separate, really expert subject is gifted. the writer needs to thank Professor F. Hoppensteadt for making to be had preprints of 2 of his papers and for studying 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 collage of Alberta used to be a really 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 the 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 college 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 decision of the depth of an outbreak. 10 21 three. A Threshold version. four. A Threshold version with transitority Immunity. 34 five. a few specific circumstances and a few Numerical Examples forty eight A inhabitants Threshold version . sixty two 6.

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Markus, On the Nonlinear Difference Differential Equation y'(t) =(A-B(t-T))y(t). Contributions to the Theory of Nonlinear Oscillations IV S. ), Princeton, 1958. In F. Hoppensteadt and P. Waltman, A Problem in the Theory of Epidemics II, Math. 3). 4) were taken from graphs computed by J. Mosevich using the numerical technique outlined in this section. More detailed computations and an explanation of this technique are given in J. Mosevich, A Numerical Method for Approximating Solutions to the Functional Equations Arising in the EPidemic Models of Hoppensteadt and Waltman, in preparation.

In particular it is not known whether has a solution with rence property). 1) (5. 3) periodic (or with any other specific recur- We will return to this particular question at the end of this section with some numerical evidence. We look first at some special cases of the model in order to relate i t to other known work. Suppose that and Wi thout the threshold class w= 0 E (instant recovery). has no members while w=0 m= 0 (no threshold), makes class T( t) ;: t R and empty as well. Schematically this is represented by S-+I-+S.

6 ) S '( t) -r(t)S(t)[IO(t) +SO - eVS(t)], -r(t)S(t)ev [S(t-cr) - S(t)], S(o) = sO. This is a differential difference equation which could be solved by the method of steps once the initial function is given. t S(t) =SO exp -Jor(x)Io(X)dX. On [to'cr], S(t) On [O,t O]' is the solution of the Ricatti equation S '( t) Thereafter solutions can be found successively on intervals [jo(j+l)cr], j =1,2,···, by the method of steps. tion can also be found. 3, some sample solutions are given to illustrate S(t) and I(t) for a variety of parameter values.

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