By Maia Martcheva

The booklet is a comprehensive, self-contained advent to the mathematical modeling and research of infectious ailments. It contains model building, becoming to info, neighborhood and international research thoughts. quite a few varieties of deterministic dynamical versions are thought of: usual differential equation versions, delay-differential equation versions, distinction equation versions, age-structured PDE types and diffusion versions. It comprises numerous strategies 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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**Example text**

Taking an exponent, we obtain I(0) rt I = e . 3 A Simple SIS Epidemic Model 21 where B = I(0)/(K − I(0)). We see from this that lim I(t) = K, t→∞ and the disease remains in the population indefinitely. The threshold condition r > 0 can be rewritten as R0 > 1, where R0 = βN α is called basic reproduction number of the disease. Mathematically, the reproduction number plays the role of a threshold value for the dynamics of the system and the disease. If R0 > 1, the disease remains in the population, and the number of infecteds stabilizes around K.

Right: no intersections of the function α g(I) and the horizontal line y = . Thus, there are no positive equilibria β Fig. 2 Bistability To decide the stability of equilibria, we have to derive the sign of f (I ∗ ) for each equilibrium I ∗ . That may not be an easy task to do analytically. Fortunately, the stability of the equilibria can be read off the graph of the function f (I) for each of the three cases above. If R0 < 1 and there are no nontrivial equilibria, then all solutions of Eq. 15) are attracted by the disease-free equilibrium.

5 Phase-plane analysis of the dimensionless SIR model. Nullclines and the direction of the vector field along them y=0 x= 1 R0 On the nullcline y = 0, f (x0 , y0 ) = ρ (1 − x0). We have f (x0 , y0 ) > 0 if x0 < 1, and the tangent vector points to the right. Furthermore, we have f (x0 , y0 ) < 0 if x0 > 1, and the tangent vector points to the left. On the y-nullcline x = R10 , we have f (x0 , y0 ) = ρ 1 − R10 − y0 . We have that y0 > ρ 1 − R10 if the point (x0 , y0 ) is on the y-nullcline 44 3 The SIR Model with Demography: General Properties of Planar Systems above the intersection of the y-nullcline with the x-nullcline.