# Cross-Entropy Method in Application to the SIRC Model
## 2 Optimization of Control for SIRC Model
### Derivation of Optimization Functions  
An approach to optimize this model was proposed by [PERSON] and [PERSON] (2016). In this article one can find the suggestion concerning performing the calculations as outlined by [PERSON] et al. (2016). Two functions were proposed in the approach as the parameter or controls. One relates to the susceptible people and the other describes the number of sick people. The method described by [PERSON] and [PERSON] (2016) was used to determine the optimal solution. In order to apply it, the set of Equation (1) should be updated accordingly. After these modifications, it has the following form:  
\[\begin{split}&\frac{dS}{dt}=\mu(1-S)-\beta SI+\gamma C-g(S(t),u(t)), \\ &\frac{dI}{dt}=\beta S(t)I(t)+\sigma\beta C(t)I(t)-(\mu+\alpha)I(t )-h(I(t),v(t)),\\ &\frac{dR}{dt}=(1-\sigma)\beta C(t)I(t)+\alpha I(t)-(\mu+\delta)R (t)+g(S(t),u(t))+h(I(t),v(t)),\\ &\frac{dC}{dt}=\delta R(t)-\beta C(t)I(t)-(\mu+\gamma)C(t),\end{split} \tag{3}\]  
where  
\[\begin{split}& g(S(t),u(t))=\rho_{1}S(t)u(t),\\ & h(I(t),v(t))=\rho_{2}I(t)v(t),\end{split} \tag{4}\]  
where \(u(t)\) represent the percentage of those susceptible who have been taken care of thanks to using control on population and \(v(t)\) represent the percentage of infected people with the same description, respectively. \(\rho_{1}\) and \(\rho_{2}\) are weights that optimize the proportion of given control options. Both functions \(g(S(t),u(t))\)\(h(I(t),v(t))\) and in order they can be interpreted as actions performed on  
Figure 1: Schematic representation of the Susceptible–Infectious–Recovered–Cross-immune (SIRC) model.  
people susceptible to the disease and infected. In addition, two new conditions related to optimization functions are added to the initial conditions.  
\[u_{min}\leqslant u(t)\leqslant u_{max},\,v_{min}\leqslant v(t)\leqslant v_{max}. \tag{5}\]  
They describe the limits on what part of the population care can be given to. The smallest values \(u_{min}\) and \(v_{min}\) is zero. The existence of the solution is shown by [PERSON] et al. [31] and they are functions that minimize the following cost index:  
\[J(u,v)=\int_{t_{1}}^{t_{2}}\varphi(S(t),I(t),u(t),v(t))dt=\int_{t_{1}}^{t_{2}} \left[\alpha_{1}S(t)+\alpha_{2}I(t)+\frac{1}{2}\tau_{1}u^{2}(t)+\frac{1}{2} \tau_{2}v^{2}(t)\right]dt, \tag{6}\]  
\(\alpha_{1},\alpha_{2}\) are used to maintain the balance in the susceptible and infected group. \(\tau_{1},\tau_{2}\) is interpreted as weighting in the cost index. \(t_{1},t_{2}\) determine the time interval. The square next to the functions \(u(t)\) and \(v(t)\) indicates increasing intensity of functions [32]. Thus the function \(J(u,v)\) reflects the human value of susceptible and infected persons, taking into account the growing value of funds used over a specified period of time.  
The objective function can be changed as long as there is a minimum. The CE method does not impose any restrictions here. However, to compare the results with [31], the function proposed by them is used also here. There, the objective function must be square in order for the quadratic programming techniques to be used. Usually, the quadratic function is good in mathematics, but not in practice. Changing the objective function and impact of it can be considered further.