By Maia Martcheva

ISBN-10: 1489976116

ISBN-13: 9781489976116

ISBN-10: 1489976124

ISBN-13: 9781489976123

The e-book is a comprehensive, self-contained creation to the mathematical modeling and research of infectious illnesses. It contains model building, becoming to facts, neighborhood and international research innovations. quite a few forms of deterministic dynamical types are thought of: traditional differential equation types, delay-differential equation versions, distinction equation versions, age-structured PDE types and diffusion types. It contains quite a few suggestions for the computation of the fundamental replica quantity in addition to ways to the epidemiological interpretation of the replica quantity. MATLAB code is integrated to facilitate the information becoming and the simulation with age-structured models.

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N(t) 7000 6000 5000 4000 3000 1960 1970 1980 1990 2000 2010 t Fig. 2 World population data alongside logistic model predictions. 0247. 3 A Simplified Logistic Model The third model of population growth is a simplified version of the logistic model. It assumes constant birth rate, independent of population size. It also assumes constant per capita death rate. The model becomes N (t) = Λ − μ N. Here Λ is the total birth rate, and μ is the per capita natural death rate. Then μ N is the total death rate.

If only one eigenvector corresponds to the double eigenvalue, the degenerate node is called improper. The origin is a saddle if the eigenvalues are real and of opposite sign. A saddle is always unstable. The origin is a spiral (or focus) if the eigenvalues are complex with nonzero real part. If the real part is negative, the focus is a stable focus; if the real part is positive, the focus is an unstable focus. 48 3 The SIR Model with Demography: General Properties of Planar Systems Center The origin is a center if the eigenvalues are complex with zero real part (purely imaginary).

Furthermore, solutions that start from a value above K stay above K: I(0) > K =⇒ I(t) > K. If 0 < I(t) < K, then f (I) > 0, which means that dI dt > 0. This means that the solutions in that interval are increasing functions of time. Since I(t) is increasing and bounded, it follows that I(t) converges to a finite limit as t → ∞. To deduce the behavior of the derivative, we use the following corollary. 1 (Thieme [151]). Assume that f (t) converges as t → ∞. Assume also that f (t) is uniformly continuous.

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