Chapter 7.P1, Problem 3P

If epidemics are identified with solution trajectories in which the number of infected individuals initially increases, reaches a maximum, and then decreases, use a nullclineanalysis to show that an epidemic occurs if and only if $S(0)>\rho =v/\beta$. Assume that $v/\beta <N$. Thus $\rho =v/\beta$ is, ineffect, a threshold value of susceptibles separating $\Gamma$ into anepidemic region and a nonepidemic region. Explain how thesize of the nonepidemic region depends on contact rate andlength of infection period.