# On a Controlled Se(Is)(Ih)(Iicu)AR Epidemic Model with Output Controllability Issues to Satisfy Hospital Constraints on Hospitalized Patients

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## Abstract

**:**

## 1. Introduction

## 2. The SE(Is)(Ih)(Iicu)AR Epidemic Model

- ${b}_{1}$ is the recruitment rate,
- ${b}_{2}$ is the natural average death rate,
- $\beta \left(t\right),{\beta}_{hr}\beta \left(t\right),{\beta}_{icu}\beta \left(t\right),{\beta}_{ar}\beta \left(t\right)$ are the transmission rates to the susceptible from the respective slight (un-hospitalized) symptomatic infectious, serious (hospitalized) symptomatic infectious, intensive care unit hospitalized infectious and asymptomatic infectious subpopulations,
- $\eta $ is a parameter such that $1/\eta $ is the average duration of the immunity period reflecting a transition from the recovered to the susceptible,
- $\gamma $ is the transition rate from the exposed to all (i.e., both symptomatic and asymptomatic) infectious,
- $\alpha $ is the average extra mortality associated with the symptomatic seriously infectious subpopulation,
- ${\alpha}_{icu}$ is the average extra mortality over the above one registered in the subpopulation of the intensive care unit,
- ${\tau}_{0}$ is the natural immune response rate for the whole infectious subpopulation (i.e., $A+I$), respectively, ${p}_{s},{p}_{h},{p}_{icu},{p}_{a}$ are the fractions of the exposed which become slight symptomatic infectious, serious symptomatic infectious and asymptomatic infectious, respectively, whose sum equalizes unity.
- $1/\mu $ is the average period of infectiousness after death,
- $V\left(t\right)={k}_{V}\left(t\right)S\left(t\right)$ and ${T}_{h}\left(t\right)={k}_{Th}\left(t\right){I}_{h}\left(t\right)$, ${T}_{icu}\left(t\right)={k}_{Ticu}\left(t\right){I}_{icu}\left(t\right)$ are, respectively, the vaccination and antiviral treatment linear feedback controls on the susceptible, (non-intensive care) hospitalized infectious and intensive care hospitalized infectious, respectively, of feedback gains ${k}_{V},{k}_{Th},{k}_{Ticu}:{R}_{0+}\to {R}_{0+}$. The vaccination control is applied to the susceptible individuals while the treatment controls to the hospitalized with no intensive care and those in the intensive care unit are, in general different and have different degrees of intensity.

**Remark**

**1.**

**Remark**

**2.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

## 3. Output Controllability Concerns and Basic Control Design Algorithms for Targeting Intensive Care Unit Levels

**Remark**

**3.**

**Remark**

**4.**

- (1)
- Assume that one zeroes the vaccination binary indicator, i.e.,${\rho}_{V}\left(t\right)=0$for$t\in \left[0,T\right]$and${k}_{V}\left(t\right)$is not identically zero for$t\in \left[0,T\right]$. Then, a vaccination control$V(t)={k}_{V}\left(t\right)S\left(t\right)$is applied on$\left[0,T\right]$and it is either prefixed in the model or, if supervised or monitored, it is not designed via hospitalization targeting objectives, i.e., it is designed without further manipulation of its gain from the auxiliary control$v\left(t\right)$of the system (8)–(10). By this reason, its effect in the model is included in the matrix of dynamics$A(\left(x\left(t\right),t\right)$of the auxiliary dynamic system (8)–(10) and, in parallel, it is removed from the control matrix$B\left(x\left(t\right),t\right)$.
- (2)
- Assume that${k}_{V}\left(t\right)$is being designed for$t\in \left[0,T\right]$, from the auxiliary control$v\left(t\right)$of the system (8), as a manipulated variable to achieve an hospitalization targeting objective. Thus, the vaccination indicator is fixed to unity, i.e.,${\rho}_{V}\left(t\right)=1$for$t\in \left[0,T\right]$. By this reason, the vaccination effect in the model is removed from the matrix of dynamics$A(\left(x\left(t\right),t\right)$of the auxiliary dynamic system (8)–(10) and, in parallel, it is considered in the control matrix$B\left(x\left(t\right),t\right)$.
- (3)
- If no vaccination is applied then${k}_{V}\left(t\right)=0$for$t\in \left[0,T\right]$and the binary indicator${\rho}_{V}\left(t\right)$can be fixed to zero or one since it does not influence the whole dynamics.
- (4)
- Similar considerations apply mutatis-mutandis for the remaining controls and their associate indicators.

**Programme**

**1.**

**Programme**

**2.**

**Remark**

**5.**

**Remark**

**6.**

**Remark**

**7.**

## 4. Extended Algorithms for Multi-Interval and Multi-Objective Control Designs Related to Mixed Non-Intensive Care and Intensive Care Hospitalization Constraints

**Programme**

**3.**

**Programme**

**4.**

**Programme**

**5.**

**Programme**

**6.**

## 5. Intervention Rules and No Partial Fulfilment of Behaviour Rules Influencing the Transmission Rate

**Assumption**

**1.**

- (1)
- ${N}_{A}$ actions in the set of actions or “events” $A=\left\{{A}_{1},{A}_{2},\dots ,{A}_{{N}_{}}\right\}$ are in favor of decreasing ${\beta}_{c}\left(t\right)$ if fulfilled. Each action ${A}_{i}$ influences a fraction ${\lambda}_{{A}_{i}}$ of the chosen time unity time and has a maximum degree of influence (or effectiveness) per unit of time ${r}_{{A}_{i}}$ in decreasing ${\beta}_{c\mathit{max}}$ provided that the population in average fulfils with it in a positive way. To evaluate this, we introduce the degree of fulfilment ${d}_{{A}_{i}}$ of ${A}_{i}$ per unity of time. The fraction per unity of time ${\lambda}_{{A}_{i}}\in \left[0,1\right]$ of ${A}_{i}$ has a clear sense. For instance, assume that the use of masks out of home is decreed and it is labelled as ${A}_{1}$. We can estimate that the averaged period for people staying out of home is 12 h per day. If the unity of time used is days, then ${\lambda}_{{A}_{1}}=0.5$.Each action of the set of events $A$ contributes to decrease ${\beta}_{c}\left(t\right)$ compared to its maximum value ${\beta}_{c\mathit{max}}$ in its respective amount ${\lambda}_{{A}_{i}}{r}_{{A}_{i}}{d}_{{A}_{i}}$. In general, the above amounts depend on time along long periods since the intervention decrees are adapted to the epidemiological situation and regularly updated accordingly.
- (2)
- ${N}_{B}$ actions in the set of events $B=\left\{{B}_{1},{B}_{2},\dots ,{B}_{{N}_{B}}\right\}$ contribute intrinsically to an increase in the disease spread but this increase is more significant as the average degree of either non-fulfilment by the population increases as, for instance, the removal or masks in risk situations. Such events also include the situations of removal of restrictions in some risk situations, for instance, the removal of masks in restaurants along lunch/dinner time. For instance, assume that the action ${B}_{1}$ is the removal of masks at the restaurant but the complementary rule is that not more than four people should sit at the same table. If people stay at restaurant in average two hours per day, then ${\lambda}_{{B}_{1}}=1/12$ (fraction of time per unity of time “day” of the action ${B}_{1}$). The theoretical degree of fulfilment ${d}_{{B}_{1}}$ of the action increases as the norm is violated. For instance, if the average of people would sit in groups of six at the same of table then ${d}_{{B}_{1}}\to {d}_{{B}_{1}^{\xb4}}=\left(3/2\right){d}_{{B}_{1}}$.

**Assumption**

**2.**

## 6. Worked Numerical Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

## 7. Conclusions and Potential Future Research

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Evolution of all population when Programme 2 is employed and $\beta $ is a control variable.

**Figure 8.**Evolution of all population when Programme 6 is employed with $\beta $ being a control variable.

**Figure 11.**Evolution of all population when Programme 6 is employed with $\beta $ being a control variable.

Parameter | Interpretation | Value | Source |
---|---|---|---|

b_{1} | Recruitment rate | 57,554 years^{−1} | [49]-year 2008 |

b_{2} | Natural average death rate | 1/85 years^{−1} | [49]-year 2008 |

$\beta $ | Transmission rate of symptomatic | 1/N(0) | [40], adjusted to provide a basic reproduction number between 5–6 |

${\beta}_{ar}$ | Specific transmission rate factor of asymptomatic | 1 | [50] |

${\beta}_{hr}$ | Specific transmission rate factor of severe cases (hospitalized) | 1/50 | Small due to higher protection degrees. |

${\beta}_{icu}$ | Specific transmission rate factor of (ICU) | 0 | Negligible due to higher protection degrees. |

$\gamma $ | Average incubation period | 1/5.5 days^{−1} | [38] |

$\eta $ | Average immunity loss rate | 0 | [38] |

$\alpha $ | Mortality rate for severe cases associated with disease | 12% | [51] |

${\alpha}_{icu}$ | Extra mortality rate for severe cases in ICU | $10\alpha $ | Ten times higher, [54,55] |

${\tau}_{0}$ | Average immune response rate | 1/10 days^{−1} | [38] |

p_{s} | Fraction of cases that are slight | 55% | [52,53] |

p_{h} | Fraction of cases that require hospitalization | 18% | [53] |

${p}_{icu}$ | Fraction of cases that require ICU | 2% | [53] |

Population | Value |
---|---|

S(0) | 6,778,382 |

E(0) | 1 |

I_{s}(0) | 0 |

I_{h}(0) | 0 |

A(0) | 0 |

R(0) | 0 |

N(0) | 6,778,383 |

**Table 4.**Average value of $N\left(0\right)\beta $ in fortnights. The value of $N\left(0\right)\beta $ is constrained to the interval [0.5, 1].

Fortnight 1 | 1 | Fortnight 2 | 1 | Fortnight 3 | 1 | Fortnight 4 | 1 |

Fortnight 5 | 1 | Fortnight 6 | 0.6528 | Fortnight 7 | 0.5000 | Fortnight 8 | 0.5000 |

Fortnight 9 | 0.5000 | Fortnight 10 | 0.5000 | Fortnight 11 | 0.5778 | Fortnight 12 | 0.6444 |

Fortnight 13 | 0.6917 | Fortnight 14 | 0.7444 | Fortnight 15 | 0.8040 | Fortnight 16 | 0.8764 |

Fortnight 17 | 0.9597 | Fortnight 18 | 1 | Fortnight 19 | 1 | Fortnight 20 | 1 |

Fortnight 21 | 1 | Fortnight 22 | 1 | Fortnight 23 | 1 | Fortnight 24 | 1 |

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**MDPI and ACS Style**

De la Sen, M.; Ibeas, A.
On a Controlled Se(Is)(Ih)(Iicu)AR Epidemic Model with Output Controllability Issues to Satisfy Hospital Constraints on Hospitalized Patients. *Algorithms* **2020**, *13*, 322.
https://doi.org/10.3390/a13120322

**AMA Style**

De la Sen M, Ibeas A.
On a Controlled Se(Is)(Ih)(Iicu)AR Epidemic Model with Output Controllability Issues to Satisfy Hospital Constraints on Hospitalized Patients. *Algorithms*. 2020; 13(12):322.
https://doi.org/10.3390/a13120322

**Chicago/Turabian Style**

De la Sen, Manuel, and Asier Ibeas.
2020. "On a Controlled Se(Is)(Ih)(Iicu)AR Epidemic Model with Output Controllability Issues to Satisfy Hospital Constraints on Hospitalized Patients" *Algorithms* 13, no. 12: 322.
https://doi.org/10.3390/a13120322