Download An Introduction to Mathematical Epidemiology by Maia Martcheva PDF

By Maia Martcheva

The booklet is a comprehensive, self-contained creation to the mathematical modeling and research of infectious ailments. It contains model building, becoming to information, neighborhood and international research innovations. numerous varieties of deterministic dynamical types are thought of: traditional differential equation types, delay-differential equation types, distinction equation types, age-structured PDE versions and diffusion versions. It comprises numerous strategies for the computation of the elemental replica quantity in addition to ways to the epidemiological interpretation of the copy quantity. MATLAB code is integrated to facilitate the information becoming and the simulation with age-structured models.

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For any nonequilibrium point (x0 , y0 ) in the phase plane, we can compute the expression dy g(x, y) |(x0 ,y0 ) = dx f (x, y) which gives the slope of the trajectory in the (x, y)-plane, with tangent vector ( f (x0 , y0 ), g(x0 , y0 ))T . This vector also gives the direction of the trajectory. The tangent vector is not defined at the equilibria, since the flow stops at those points and they are fixed points. 0 Fig. 4 The vector field of the dimensionless SIR model alongside solutions of the model for several initial conditions The collection of tangent vectors defines a direction field.

In most cases, however, populations live in an environment that has a finite capacity to support only a certain population size. When the population size approaches this capacity, the per capita growth rate declines or becomes negative. This property of the environment to limit population growth is captured by the logistic model. The logistic model was developed by the Belgian mathematician Pierre Verhulst (1838), who suggested that the per capita growth rate of the population may be a decreasing function of population density: N 1 N (t) = r 1 − , N(t) K which gives the classical logistic model that we studied in Chap.

If μ is the natural death rate, then 1/ μ should be the average lifespan of an individual human being. 8 × 1011 years—quite unrealistic. If the lifespan is limited to biologically realistic values, such as a lifespan of 65 years, then the fit becomes worse. 2 The SIR Model with Demography To incorporate the demographics into the SIR epidemic model, we assume that all individuals are born susceptible. Individuals from each class die at a per capita death rate μ , so the total death rate in the susceptible class is μ S, while in the infective class, it is μ I, and in the removed class, it is μ R.

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