ABSTRACT
In this lab, we use optimal control techniques to find a vaccination schedule for an epidemic disease. A micro-parasitic infectious disease is considered. Permanent immunity to the disease can be achieved through natural recovery or immunization. Immunity is not passed on during birth, so that everyone is born susceptible. Our goal is to minimize the number of infectious persons and the overall cost of the vaccine during a fixed time period. To model the dynamics of the disease in a population, we use a standard
SEIR (or SEIRN) model. Let S(t), I(t), and R(t) represent number of susceptible, infectious, and recovered (immune) individuals at time t. The model allows for an incubation period for the disease inside its host, where an infected person remains latent for some time before becoming infectious, creating an exposed class. Let E(t) be the number of exposed or latent individuals at time t. Let N(t) be the total number of people in the population, so that N(t) = S(t) + E(t) + I(t) +R(t). Let u(t), the control, be the percentage of susceptible individuals being
vaccinated per unit of time. As vaccination of the entire susceptible population is impossible, we bound the control with 0 ≤ u(t) ≤ 0.9. Let b be the natural exponential birth rate of the population and d the natural exponential death rate. The incidence of the disease is described by the term cS(t)I(t). The parameter e is the rate at which the exposed individuals become infectious, and g is the rate at which infectious individuals recover. Therefore, 1e is the mean latent period, and 1g is the mean infectious period before recovery, if recovery occurs. The death rate due to the disease in infectious individuals is a. The optimal control problem is as follows,
min u
AI(t) + u(t)2 dt
subject to S′(t) = bN(t)− dS(t)− cS(t)I(t)− u(t)S(t), S(0) = S0 ≥ 0, E′(t) = cS(t)I(t)− (e+ d)E(t), E(0) = E0 ≥ 0, I ′(t) = eE(t)− (g + a+ d)I(t), I(0) = I0 ≥ 0, R′(t) = gI(t)− dR(i) + u(t)S(t), R(0) = R0 ≥ 0, N ′(t) = (b− d)N(t)− aI(t), N(0) = N0, 0 ≤ u(t) ≤ 0.9.
