# Cross-Entropy Method in Application to the SIRC Model
## 2 Optimization of Control for SIRC Model
### SIRC Model  
The subject of the work is to solve the problem of optimizing the spread of the disease, which can be modeled with the SIRC model. This model was proposed by [PERSON] et al. [9]. Its creation was intended to create a better model that would describe the course of influenza type A. Contrary to appearances, this is an important topic because the disease, although it is widely regarded as weak, is a huge problem for healthcare. In the US alone, the cost associated with the influenza epidemic in the season is estimated to exceed USD 10 billion (v. [26]) and the number of deaths is over 21,000 per season (v. [27]).  
Various articles have previously proposed many different mathematical models to describe a pandemic of influenza type A. An overview of such models was made by [PERSON] [28]. In general, it came down to combining SIR models with the use of cross-resistance parameters (cf. details in the papers by [PERSON] et al. [29] and [PERSON] et al. [30]). The authors of the article write that the main disadvantage of this approach is the difficulty of the analysis and calculations with a large number of strains. The results of their research showed that the classic SIR model cannot be used to model and study influenza epidemics. Instead, they proposed the extension of the SIR model to a new class C that will simulate a state between a susceptible state and a fully protected state. This extension aims to cover situations where vulnerable carriers are exposed to similar stresses as they had before. As a result, their immune system is stronger when fighting this disease.  
The SIRC model divides the community into four groups:  
* S--persons susceptible to infection, who have not previously had contact with this disease and have no immune defense against this strain
* I--people infected with the current disease
* R--people who have had this disease and are completely immune to this strain
* C--people partially resistant to the current strain (e.g., vaccinated or those who have had a different strain)  
Figure 1 contains a general scheme of this model. The four rhombi represent the four compartments of individuals, the movement between the compartments is indicated by the continuous arrows. The main advantage of this model over other SIR users is the fact that after recovering from a given strain, in addition to being completely resistant to this strain, they are also partly immune to a new virus that will appear later. This allows you to model the resistance and response of people in the group to different types of diseases.  
Parameters \(\alpha,\delta\) i \(\gamma\) can be interpreted as the reciprocal of the average time spent by a person in order of ranges \(I,R,C\). Parameters \(\mu\) with indices \(S,I,R,C\) represent the natural mortality in each group, respectively. In some versions of the model, additional mortality rates are considered for the group of infected people. Here we assume that this factor is not affected by the disease and we will denote it as \(\mu\) in later formulas (\(\mu_{S}=\mu_{I}=\mu_{R}=\mu_{C}=\mu\)). The next one \(\sigma\) is the likelihood of reinfection of a person who has cross-resistance while the parameter \(\beta\) describes the contact indicator.  
The SIRC model is represented as a set of four ordinary differential equations. Let \(S(t)\), \(I(t)\) be the number of people in adequate compartments. We have (v. [9])  
\[\begin{split}\frac{dS}{dt}&=\mu(1-S(t))-\beta S(t)I (t)+\gamma C(t),\\ \frac{dI}{dt}&=\beta S(t)I(t)+\sigma\beta C(t)I(t)-( \mu+\alpha)I(t),\\ \frac{dR}{dt}&=(1-\sigma)\beta C(t)I(t)+\alpha I(t) -(\mu+\delta)R(t),\\ \frac{dC}{dt}&=\delta R(t)-\beta C(t)I(t)-(\mu+ \gamma)C(t),\end{split} \tag{1}\]with initial conditions  
\[S(0)=S_{0},\,I(0)=I_{0},\,R(0)=R_{0},\,C(0)=C_{0}. \tag{2}\]