Problem 4. The equilibrium points are I* = Problem 5. I and I* = The equilibrium point I* = is stable when and unstable when The equilibrium point I* = is stable when and unstable when Problem 6. The disease is endemic when Disease Spread With Recovery Consider the far more likely scenario in which a disease is circulating in a population, in such a way that individuals recover from the disease unharmed but are susceptible to reinfection. Again, let I, denote the fraction of infected individuals in the population at time t. A discrete-time dynamical system which models this scenario is given by I+=+ble(1-1)-kl₁ where the positive parameter b represents the per capita rate at which susceptible member of the population is infected, and the positive parameter k represents the rate at which infected individuals recover. Note that both b> 0 and k > 0. 4. Again let b = 0.5, and find the equilibrium points for this new system algebraically (your answers will depend on the parameter k). 5. Use the slope criterion to test the stability of the equilibrium points you found in Problem 4. For what values of k are each of the equilibria stable? unstable? According to the Centre for Disease Control, "endemic refers to the constant presence and/or usual prevalence of a disease or infectious agent in a population within a geographic area" in other words, a disease is considered endemic if the dynamical system describing it has a positive equilibrium that is stable. Consider the original system with recovery, i.e. = I+1 1+bl(1-1)-kl₁ where b> 0 and k > 0. The equilibria of this system are I=0 k I=1- b so that the system will only have a positive equilibrium if 1->0, which we can rewrite as b>k, or b - k > 0. 6. Using the slope criterion, find a condition on b-k so that the positive equilibrium I=1- is stable, meaning that the disease being modelled is endemic.