Mathematical Modeling in Epidemiology by James C. Frauenthal (auth.)

By James C. Frauenthal (auth.)

The textual content of this booklet is derived from classes taught by way of the writer within the division of utilized arithmetic and statistics on the nation collage of recent York at Stony Brook. The viewers for those classes was once composed virtually totally of fourth yr undergraduate scholars majoring within the mathematical sciences. the scholars had mostly accomplished 4 semesters of calculus and one among likelihood. Few had any earlier event with differential equations, stochastic procedures, or epidemiology. It additionally turns out prudent to say that the author's historical past is in engineering and utilized arithmetic and never in epidemiology; it truly is was hoping that this isn't painfully visible. the themes coated during this publication have at times been transformed from the way in which they have been initially provided. although, care has been taken to incorporate an appropriate volume of fabric for a one semester direction; the temptation so as to add gratuitous subject material has been resisted. equally, while a call among readability and rigor was once to be had, the extra simply understood exposition used to be chosen. by means of having a look in basic terms on the desk of contents, the informal reader will be simply misled into pondering that the most drawback of this booklet is with epidemiology. this isn't the case. the aim of this ebook is to demonstrate the method of formulating and fixing mathematical models.

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Act that this same initial susceptible avoids getting the disease from the initial susceptible who becomes infected. The '2' results from the fact that this can occur in two exclusive ways. Thus, p{212} = I-p{012} - P{l12} Let us continue one more step before generalizing. P{013} Q3 0 P{l13} 3P{llllQ~ p{213} 3P{21 2 }QoQ2 POI3} I - P{013} - p{113} - p{213} q 2 The procedure should by now be obvious. For general SO: Let So = 3. 49 CHAIN BINOMIAL MODELS S -R (:~) p{RooiRoo}Qo 0 p{RoofSo} 00 q Roo (SO-Roo) SO-l 2: 1 - p{SoiSo} p{RooiSo} R00=0 The chief advantage to method III over the other two methods is that one calculates the most interesting quantity concerning the epidemic 2 directly.

Use this mechanism to construct a mathematical model for estimating the clock time duration of a rumor and carry out the solution as far as you are able. 2. With ~k = (sk,ik,r k ) as defined in the rumor model, derive an expression for r k , and show that sk + i k + r k = N + 1. 24 3. Find an approximate expression for the size of the error which is introduced when the expression: + 2 - ! N is replaced by 1* ~ 2 - 2exp {-1*}. Your answer should be a series (polynomial) in powers of N. Find the coefficient of the leading term.

B. Although this equation is almost identical in form to the one for the trivial model, it is now nonlinear in the index i. causes considerable difficulty. This fact In fact, it does not appear to be possible to write down a simple, closed-form expression for either the Pitt) or for the moments. For this reason, it is unproductive to continue our analysis any further. All methods at our disposal lead to disappointingly messy and uninformative expressions. There is, however, one interesting question which we can answer without undue complication.

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