A small group of infected individuals is introduced into a large(fixed) population N, who are suspectible to this contagious disease(e.g., smallpox). The number of infected, I(t), increases at a rate dI/dt that is proportional to the product of the number of infected and the number of susceptibles S(t). Some of the infected recover from the disease, and this confers immunity, while some die from the disease and are therefore no longer infectious. We count both of these as R(t): the recovered, who are immune, and the dead, who can no longer transmit the disease. Write down the three coupled differential equations for dI/dt, dS/dt, and dR/dt.
There should only be two independent proportionality constants you can use in modeling this situation.
and also for the SIR model from previous, draw the direction field with arrows Indicating the direction of increase I(t) vs. S(t). Discuss what happens in time to the introduction of some infected individuals to a large population of susceptibles, and infer the condition under which an epidemic would develop.