| Contents |
| The SIR (susceptible-infected-removed) model for the spread of epidemics in a
population of a variable size $ N $ is considered. We control the spread of the infection by three
bounded controls: $ u(t) $ is vaccination of the susceptible, $ w(t) $ is treatment of the
infected, and $ v(t) $ is other "indirect" strategies aimed at a reduction of the incidence rate. The model equations are:
$$
\left\{
\begin{array}{lll}
S'(t) & = & \sigma N(t) - \frac{\displaystyle{\beta}}{\displaystyle{1 + v(t)}}
\frac{\displaystyle{S(t)I(t)}}{\displaystyle{N(t)}} - (\mu + u(t))S(t), \;\; t \in [0,T], \\
I'(t) & = & \frac{\displaystyle{\beta}}{\displaystyle{1 + v(t)}}
\frac{\displaystyle{S(t)I(t)}}{\displaystyle{N(t)}} - (\gamma + \delta + \mu + w(t))I(t), \\
R'(t) & = & u(t)S(t) + (\gamma + w(t))I(t) - \mu R(t),
\end{array}
\right.
$$
where $ \sigma $ and $ \mu $ are birth and natural mortality rates, $ \delta $ is infection induced
mortality rate, $ \beta $ and $ \gamma $ are effective contact and recovery rates, respectively.
Since $ N(t) = S(t) + I(t) + R(t) $, then the equation for a size of a population
$$
N'(t) = (\sigma - \mu)N(t) - \delta I(t),
$$
is added into the considered system. Excluding $ R(t) $, we then study properties of the system of the equations for $ N(t) $, $ S(t) $, and $ I(t) $. This very nonlinear system is more complex than one for a constant population size. We state the optimal control problem of minimizing the number of the infected at the terminal time $ T $. The corresponding optimal solutions are obtained with the use of the Pontryagin maximum principle. The types of optimal controls are investigated analytically. Numerical results illustrate the optimal solutions. |
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