By Maia Martcheva
The booklet is a comprehensive, self-contained advent to the mathematical modeling and research of infectious illnesses. It contains model building, becoming to information, neighborhood and worldwide research recommendations. numerous sorts of deterministic dynamical versions are thought of: traditional differential equation types, delay-differential equation types, distinction equation types, age-structured PDE versions and diffusion versions. It comprises numerous options for the computation of the elemental replica quantity in addition to techniques to the epidemiological interpretation of the replica quantity. MATLAB code is integrated to facilitate the knowledge becoming and the simulation with age-structured models.
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C4 are arbitrary constants. Clearly, in this case, u → 0 and v → 0 if and only if λ1 < 0 and λ2 < 0. The eigenvalues of the Jacobian are real and equal, say λ . 22) where C1 , . . ,C4 are arbitrary constants. Clearly, in this case, u → 0 and v → 0 if and only if λ < 0. The eigenvalues of the Jacobian are complex conjugates, say λ1 = ξ + η i and λ2 = ξ − η i. 3 Analysis of Two-Dimensional Systems 47 where C1 , . . ,C4 are arbitrary constants. Clearly, in this case, u → 0 and v → 0 if and only if ξ < 0; that is, the eigenvalues have negative real part.
Similar reasoning as above implies that I(t) → K. We can further investigate the concavity of the solutions by looking at the second derivative: d2I 2I dI 2I I = r2 1 − = r 1− I 1− . 10) 2 dt K dt K K For solutions in the interval 0 < I(t) < K, the second derivative changes sign when I(t) crosses the horizontal line y = K2 . 3 A Simple SIS Epidemic Model 23 I(t) 350 300 250 200 150 100 50 5 10 15 20 25 30 t Fig. 9) converge to the endemic equilibrium second derivative of I is positive, and I(t) is concave up.
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. So the diseasefree equilibrium is globally stable in this case. For each of the other two cases, we graph the function f (I) in Fig. 9. Looking at Fig. 9, we see that in the case R0 > 1 (left figure), we have f (0) > 0.