#
Optimal Control of Virus Spread under Different Conditions of Resources Limitations^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- -
- the class of susceptible subjects (S) that are the healthy individuals that may contract the virus;
- -
- the class of the infected individuals (I) not aware of their condition;
- -
- the class of the pre-AIDS patients (P);
- -
- the class of the AIDS patients (A).

## 2. The Mathematical Model, the Optimal Control Problem Formulation and Its Solution

#### 2.1. The HIV/AIDS Model

- -
- $\beta $ regulates the interaction responsible of the infectious propagation;
- -
- $\gamma $ takes into account the fact that a wise individual in ${S}_{2}\left(t\right)$ can, accidentally, assume a incautious behaviour as the ${S}_{1}\left(t\right)$ persons;
- -
- $\delta $ weights the natural rate of $I\left(t\right)$ subjects becoming aware of their status;
- -
- $\alpha $ characterizes the natural rate of transition from $P\left(t\right)$ to $A\left(t\right)$ due to the evolution of the infectious disease;
- -
- $\psi $ determines the effect of the test campaign on the unaware individuals $I\left(t\right)$;
- -
- $\varphi $ is the fraction of individuals in $I\left(t\right)$ which become, after test, classified as $P\left(t\right)$ ($\varphi $) or $A\left(t\right)$ ((1 − $\varphi $));
- -
- $\epsilon $ is the fraction of individuals $I\left(t\right)$ which discover to be in the pre-AIDS condition or in the AIDS one;
- -
- d is responsible of the natural death rate, assumed the same for all the classes, while $\mu $ is the additional death factor for the individuals $A\left(t\right)$.

#### 2.2. The Optimal Control

- i.
- each control ${u}_{i}\ge 0$ is bounded, at each time instant, by an upper limit;
- ii.
- the instantaneous total control effort, given by the sum, at each time instant, of all the controls, is bounded by a total upper limit;
- iii.
- the total control effort of each input, measured by its integral over all the action time $[0,{t}_{f}]$, is equal to a prefixed value;
- iv.
- the total control effort of all the inputs over the action time interval $[0,{t}_{f}]$ is equal to a global value.

#### 2.3. Problem Formulation for Case i

**Theorem**

**1.**

**Proof.**

#### 2.4. Problem Formulation for Case ii

- A
- ${\eta}_{1}\ne 0$, ${\eta}_{2}\ne 0$, ${\eta}_{3}\ne 0$. In this case no solution is possible, since it requires the fulfilment of all the constraints at the same time, clearly impossible.
- B
- ${\eta}_{1}\ne 0$, ${\eta}_{2}\ne 0$, ${\eta}_{3}=0$. This implies that ${u}_{1}=0$ and ${u}_{2}=0$, that is, both the controls are not acting.
- C
- ${\eta}_{1}\ne 0$, ${\eta}_{2}=0$, ${\eta}_{3}\ne 0$. Being that the first and the third constraint are satisfied on the boundary, one has ${u}_{1}=0$ and ${u}_{2}=U$.
- D
- ${\eta}_{1}=0$, ${\eta}_{2}\ne 0$, ${\eta}_{3}\ne 0$. This case is equivalent to C, with the controls exchanged: ${u}_{2}=0$ and ${u}_{1}=U$.
- E
- ${\eta}_{1}=0$, ${\eta}_{2}=0$, ${\eta}_{3}\ne 0$. In this case the third constraint is verified on its boundary, giving for the solution ${u}_{1}+{u}_{2}=U$.
- F
- ${\eta}_{1}=0$, ${\eta}_{2}\ne 0$, ${\eta}_{3}=0$. This situation corresponds to ${u}_{2}=0$ while ${u}_{1}$ can assume any value in $(0,U)$.
- G
- ${\eta}_{1}\ne 0$, ${\eta}_{2}=0$, ${\eta}_{3}=0$. This is the dual case of F, so that one has ${u}_{1}=0$ while ${u}_{2}$ can assume any value in $(0,U)$.
- H
- ${\eta}_{1}=0$, ${\eta}_{2}=0$, ${\eta}_{3}=0$. These conditions correspond to both ${u}_{1}$ and ${u}_{2}$ in the interval $(0,U)$.

- A
- not admissible, so no solutions can be obtained.
- B
- in this case one has ${\eta}_{3}\left(t\right)=0$ and$$u\left(t\right)=\left(\right)open="("\; close=")">\begin{array}{c}{u}_{1}\left(t\right)\\ {u}_{2}\left(t\right)\end{array}$$so that (46) becomes$$\left(\right)open="("\; close=")">\begin{array}{c}0\\ 0\end{array}$$$$\left(\right)open="("\; close=")">\begin{array}{c}{\eta}_{1}\left(t\right)\\ {\eta}_{2}\left(t\right)\end{array}$$
- C
- this case corresponds to the solutions ${\eta}_{2}\left(t\right)=0$, with$$u\left(t\right)=\left(\right)open="("\; close=")">\begin{array}{c}{u}_{1}\left(t\right)\\ {u}_{2}\left(t\right)\end{array}$$$$\left(\right)open="("\; close=")">\begin{array}{c}0\\ U\end{array}$$$$\left(\right)open="("\; close=")">\begin{array}{c}{\eta}_{1}\left(t\right)\\ {\eta}_{3}\left(t\right)\end{array}+{G}^{T}\left(x\left(t\right)\right)\lambda $$
- D
- since this case is equivalent to C once ${\eta}_{1}\left(t\right)$ and ${\eta}_{2}\left(t\right)$ as well as ${u}_{1}\left(t\right)$ and ${u}_{2}\left(t\right)$ are exchanged, the controls are defined as$$u\left(t\right)=\left(\right)open="("\; close=")">\begin{array}{c}{u}_{1}\left(t\right)\\ {u}_{2}\left(t\right)\end{array}$$$$\left(\right)open="("\; close=")">\begin{array}{c}{\eta}_{2}\left(t\right)\\ {\eta}_{3}\left(t\right)\end{array}+{G}^{T}\left(x\left(t\right)\right)\lambda \left(t\right)$$
- E
- In this case the known functions are ${\eta}_{1}\left(t\right)=0$, ${\eta}_{2}\left(t\right)=0$ and ${u}_{1}\left(t\right)+{u}_{2}\left(t\right)=U$ or, equivalently, ${u}_{2}\left(t\right)=U-{u}_{1}\left(t\right)$. Equation (46) becomes$$\left(\right)open="("\; close=")">\begin{array}{c}{u}_{1}\left(t\right)\\ U-{u}_{1}\left(t\right)\end{array}{u}_{1}\left(t\right)+\left(\right)open="("\; close=")">\begin{array}{c}0\\ U\end{array}$$$$\left(\right)open="("\; close=")">\begin{array}{c}{u}_{1}\left(t\right)\\ {\eta}_{3}\left(t\right)\end{array}\left(\right)open="("\; close=")">R\left(\right)open="("\; close=")">\begin{array}{c}0\\ U\end{array}$$$$\begin{array}{ccc}\hfill {u}_{1}\left(t\right)& =& -\left(\begin{array}{cc}1& 0\end{array}\right){\left(\begin{array}{c}R\left(\right)open="("\; close=")">\begin{array}{c}1\\ -1\end{array}\\ {m}_{3}\end{array}\right)}^{}-1\left(\right)open="("\; close=")">R\left(\right)open="("\; close=")">\begin{array}{c}0\\ U\end{array}\hfill & +{G}^{T}\left(x\left(t\right)\right)\lambda \left(t\right)\end{array}$$$$\begin{array}{ccc}\hfill {u}_{2}\left(t\right)& =& U-{u}_{1}\left(t\right)=\frac{{r}_{1}}{{r}_{1}+{r}_{2}}U+\frac{1}{{r}_{1}+{r}_{2}}\left(\right)open="("\; close=")">{g}_{1}^{T}\left(x\left(t\right)\right)-{g}_{2}^{T}\left(x\left(t\right)\right)\lambda \hfill \end{array}$$
- F
- making use of the values ${\eta}_{1}\left(t\right)=0$, ${\eta}_{3}\left(t\right)=0$ and$${u}_{2}\left(t\right)=0$$$$\left(\right)open="("\; close=")">\begin{array}{c}{u}_{1}\left(t\right)\\ 0\end{array}{u}_{1}\left(t\right)=-{R}^{-1}{G}^{T}\left(x\left(t\right)\right)\lambda \left(t\right)-{R}^{-1}{m}_{2}{\eta}_{2}\left(t\right)$$$$\left(\right)open="("\; close=")">\begin{array}{c}{u}_{1}\left(t\right)\\ {\eta}_{2}\left(t\right)\end{array}{G}^{T}\left(x\left(t\right)\right)\lambda \left(t\right)$$$$\begin{array}{ccc}\hfill {u}_{1}\left(t\right)& =& -\left(\begin{array}{cc}1& 0\end{array}\right){\left(\begin{array}{c}R\left(\right)open="("\; close=")">\begin{array}{c}1\\ 0\end{array}\\ {m}_{2}\end{array}\right)}^{}-1{G}^{T}\left(x\left(t\right)\right)\lambda \left(t\right)=\hfill \end{array}$$
- G
- since ${\eta}_{2}\left(t\right)=0$, ${\eta}_{3}\left(t\right)=0$ and$${u}_{1}\left(t\right)=0$$$$\left(\right)open="("\; close=")">\begin{array}{c}0\\ {u}_{2}\left(t\right)\end{array}{u}_{2}\left(t\right)=-{R}^{-1}{G}^{T}\left(x\left(t\right)\right)\lambda \left(t\right)-{R}^{-1}{m}_{1}{\eta}_{1}\left(t\right)$$$$\left(\right)open="("\; close=")">\begin{array}{c}{u}_{2}\left(t\right)\\ {\eta}_{1}\left(t\right)\end{array}{G}^{T}\left(x\left(t\right)\right)\lambda \left(t\right)$$From this expression it is possible to compute$$\begin{array}{ccc}\hfill {u}_{2}\left(t\right)& =& -\left(\begin{array}{cc}1& 0\end{array}\right){\left(\begin{array}{c}R\left(\right)open="("\; close=")">\begin{array}{c}0\\ 1\end{array}\\ {m}_{1}\end{array}\right)}^{}-1{G}^{T}\left(x\left(t\right)\right)\lambda \left(t\right)=\hfill \end{array}$$
- H
- since ${\eta}_{1}\left(t\right)=0$, ${\eta}_{2}\left(t\right)=0$ and ${\eta}_{3}\left(t\right)=0$, (46) directly gives the solution$$\left(\right)open="("\; close=")">\begin{array}{c}{u}_{1}\left(t\right)\\ {u}_{2}\left(t\right)\end{array},$$

#### 2.5. Problem Formulation for Case iii

#### 2.6. Problem Formulation for Case iv

## 3. Numerical Results

#### 3.1. Case i

#### 3.2. Case ii

#### 3.3. Case iii

#### 3.4. Case iv

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Case i: time evolution of the ${S}_{1}\left(t\right)$ subjects with and without optimal control actions.

**Figure 3.**Case i: time evolution of the ${S}_{2}\left(t\right)$ subjects with and without optimal control actions.

**Figure 4.**Case i [17]: time evolution of the $I\left(t\right)$ subjects with and without optimal control actions.

**Figure 5.**Case i [17]: time evolution of the $P\left(t\right)$ subjects with and without optimal control actions.

**Figure 6.**Case i [17]: time evolution of the $A\left(t\right)$ subjects with and without optimal control actions.

**Figure 7.**Case i [17]: optimal controls.

**Figure 8.**Case i [17]: time evolution of the total number of infected, with and without the control actions.

**Figure 9.**Case i [17]: time evolution of all the population, with and without the control actions.

**Figure 10.**Case ii: time evolution of the ${S}_{1}\left(t\right)$ subjects with and without optimal control actions.

**Figure 11.**Case ii: time evolution of the ${S}_{2}\left(t\right)$ subjects with and without optimal control actions.

**Figure 12.**Case ii: time evolution of the transient for infected $I\left(t\right)$, $P\left(t\right)$ and $I\left(t\right)$ subjects under optimal control actions.

**Figure 14.**Case ii: time evolution of the total number of infected, with and without the control actions.

**Figure 19.**Case iii: time evolution of the ${S}_{1}\left(t\right)$ subjects with and without optimal control actions.

**Figure 20.**Case iii: time evolution of the ${S}_{2}\left(t\right)$ subjects with and without optimal control actions.

**Figure 21.**Case iii: time evolution of the transient for infected $I\left(t\right)$, $P\left(t\right)$ and $I\left(t\right)$ subjects under optimal control actions.

**Figure 22.**Case iii: time evolution of the total number of infected, with and without the control actions.

**Figure 27.**Case iv: time evolution of the ${S}_{1}\left(t\right)$ subjects with and without optimal control actions.

**Figure 28.**Case iv: time evolution of the ${S}_{2}\left(t\right)$ subjects with and without optimal control actions.

**Figure 29.**Case iv: time evolution of the transient for infected $I\left(t\right)$, $P\left(t\right)$ and $I\left(t\right)$ subjects under optimal control actions.

**Figure 30.**Case iv: time evolution of the total number of infected, with and without the control actions.

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**MDPI and ACS Style**

Di Giamberardino, P.; Iacoviello, D.
Optimal Control of Virus Spread under Different Conditions of Resources Limitations. *Information* **2019**, *10*, 214.
https://doi.org/10.3390/info10060214

**AMA Style**

Di Giamberardino P, Iacoviello D.
Optimal Control of Virus Spread under Different Conditions of Resources Limitations. *Information*. 2019; 10(6):214.
https://doi.org/10.3390/info10060214

**Chicago/Turabian Style**

Di Giamberardino, Paolo, and Daniela Iacoviello.
2019. "Optimal Control of Virus Spread under Different Conditions of Resources Limitations" *Information* 10, no. 6: 214.
https://doi.org/10.3390/info10060214