# An SIRS Epidemic Model Supervised by a Control System for Vaccination and Treatment Actions Which Involve First-Order Dynamics and Vaccination of Newborns

^{*}

## Abstract

**:**

## 1. Introduction

## 2. SIRS Epidemic Model under Vaccination and Treatment Controls

- b is the natural birth rate of the host population;
- $\mu $ is the natural death rate of the host population;
- $\beta $ is the transmission rate of the disease within the host population;
- $\rho $ is the immunity loss rate within the recovered subpopulation, whose individuals become susceptible to the disease after losing the immunity;
- $\alpha $ is the death rate by causes related to the disease;
- $\gamma $ is the recovery rate of the infectious subpopulation.

- $q\in \left[0,1\right]$ is the proportion of newborn individuals who are vaccinated;
- ${c}_{i}$, for $i\in \left\{1,2,3,4\right\}$, are the parameters for designing the vaccination of the susceptible subpopulation. Such control parameters allow us to weight the vaccination rate according to the state of the disease propagation considering the number of susceptible individuals S, infectious ones I, and/or the probability of contacts between them $\frac{S\left(t\right)I\left(t\right)}{N\left(t\right)}$ at each time t. Such parameters provide the availability of giving more importance to one term of (4) against the other ones in the design of the law for the vaccination $v\left(t\right)$. The unit of the parameter ${c}_{1}$ is (time)
^{−1}, usually (day)^{−1}, and that of the parameters ${c}_{i}$, for $i\in \left\{2,3,4\right\}$ is (time)^{−2}for coherence in (4). - ${c}_{i}$, for $i\in \left\{5,6\right\}$, are the parameters for designing the treatment of the infectious subpopulation. Such control parameters shape the law for the treatment ${t}_{r}\left(t\right)$ according to the number of infectious individuals. However, the values allowed for such parameters are constrained for the potency of the available medicaments. In this sense, a larger $\frac{{c}_{6}}{{c}_{5}}$ is less the recovery time interval for the treated infectious individuals. The unit of the parameter ${c}_{5}$ is (time)
^{−1}, usually (day)^{−1}, and that of the parameters ${c}_{6}$ is (time)^{−2}for coherence in (5).

#### 2.1. Positivity of the Controlled SIRS Epidemic Model

**Theorem**

**1**

**(positivity of the model).**

**Remark**

**1.**

- (i)
- The conditions (i)–(iv) of Theorem 1 are only sufficient conditions, since the solutions of the model can be non-negative even if some of these conditions are not satisfied by the control parameters.
- (ii)
- The birth rate of a host population $b$ is close to its mortality rate $\mu N\left(t\right)$ for any $t\ge 0$ under normal conditions (in absence of a lethal disease). Then, $b-\mu N\left(0\right)$ is close to zero at the beginning of the propagation of an infectious disease. Then, typically $b-\left(\mu +\beta \right)N\left(0\right)\approx -\beta N\left(0\right)<0$. Moreover, in the first stage of any epidemic disease propagation, the infectious and recovered subpopulations are much smaller than the susceptible one. Then, $N\left(0\right)\approx S\left(0\right)$ and, as a consequence, the condition (ii) is satisfied for any $q\in \left[0,1\right]$.
- (iii)
- The condition (iii) of Theorem 1 depends on the maximum value reached by the infectious subpopulation during the propagation of the disease. Such a value cannot be known ‘a priori’, and then, one cannot appropriately choose the values of the parameters ${c}_{3}$ and $q$ to satisfy it. However, such a condition is fulfilled if ${c}_{3}=0$ for any $q\in \left[0,1\right]$. Then, from continuity arguments, the condition is fulfilled for ${c}_{3}\in \left[0,{\overline{c}}_{3}\right]$ with ${\overline{c}}_{3}={\overline{c}}_{3}\left(q,{I}_{max}\right)=\frac{b\left(1-q\right)\left(\mu +\beta \right)}{{I}_{max}}$. In summary, a value for ${c}_{3}$ small enough has to be chosen in order to satisfy the condition (iii) if a vaccination provided by (4) is proposed. Note that if $q=1$, which means a vaccination of all the newborns, then ${c}_{3}=0$ has to be taken to guarantee the non-negativity of all the model variables for all the time.
- (iv)
- In a real situation, the control actions to fight against an epidemic outbreak are taken after the disease is detected in the infectious individuals. Then, the constraint$v\left(0\right)={t}_{r}\left(0\right)=0$in the initial condition for guaranteeing the non-negativity of the variables of the controlled epidemic model is coherent with such a fact.

**Corollary**

**1.**

**Proof.**

**Assumption**

**1.**

**Remark**

**2.**

#### 2.2. Control Reproduction Number and Equilibrium Points of the Controlled SIRS Model

**Theorem**

**2**

**(Non-existence of the EE point).**

**Proof.**

**Remark**

**3.**

#### 2.3. Local Stability of the Disease-Free Equilibrium Point

**Theorem**

**3**

**(Local stability of the DFE point).**

**Proof.**

**Remark**

**4.**

#### 2.4. Global Stability of the Disease-Free Equilibrium Point

**Theorem**

**4**

**(Global stability of the disease-free equilibrium point).**

**Proof.**

**Remark**

**5.**

- (i)
- From Theorems 1–4, the model is positive and only has an equilibrium point, namely, the DFE point, which is locally and globally asymptotically stable provided that the control parameters fulfil the conditions of Theorem 1 and that ${R}_{c}<{\overline{R}}_{c}$ or, equivalently, ${R}_{0}<{\overline{R}}_{0}$.
- (ii)
- The control parameters${c}_{1}$and${c}_{2}$influence the components of both the DFE and EE points, see (10) and (14), (15) respectively.
- (iii)
- The control parameters${c}_{3}$,${c}_{4}$,${c}_{5}$, and${c}_{6}$influence the components of the EE point.
- (iv)
- The control parameters${c}_{1}$,${c}_{2}$,${c}_{5}$, and${c}_{6}$influence the stability of the DFE point according to (19) and, also, they can imply the non-existence of the EE point under an appropriate choice of their values according to (16).
- (v)
- The threshold value ${\overline{R}}_{c}$ given in (16) depends on the control parameters ${c}_{5}$ and ${c}_{6}$ associated with the treatment effort, while the control reproduction number ${R}_{c}$ given in (13) depends on the control parameters $q$, ${c}_{1}$, and ${c}_{2}$ associated with the vaccination efforts so that the non-existence of the EE point can be guaranteed by a treatment strategy adapted to a designed vaccination campaign.
- (vi)
- Neither${R}_{c}$nor${\overline{R}}_{c}$depend on the control parameters${c}_{3}$and ${c}_{4}$associated with the effort of the vaccination of susceptible individuals so that such parameters are not relevant for eradicating the disease. Such parameters affect the values of the subpopulations at the EE point if such a point is reached, which is intended to be avoided.
- (vii)
- The expression for ${R}_{c}$ can be equivalently written as ${R}_{c}=\frac{\beta \left[\mu \left(1-q\right)+\rho \right]}{\left(\mu +\alpha +\gamma \right)\left[\mu +\rho +\frac{{c}_{2}}{{c}_{1}}\right]}=\frac{\mu \left(1-q\right)+\rho}{\mu +\rho +\frac{{c}_{2}}{{c}_{1}}}{R}_{0}$. Then, one can see that ${R}_{c}$ is inversely proportional to $\frac{{c}_{2}}{{c}_{1}}$ so that an increment in the value of $\frac{{c}_{2}}{{c}_{1}}$ results in a decrement of ${R}_{c}$, implying a small incidence of the infectious disease. In this context, a large value for $\frac{{c}_{2}}{{c}_{1}}$ is interesting for reducing the incidence of the disease on the host population. A large value for the relation $\frac{{c}_{2}}{{c}_{1}}$ can be obtained by considering small values for ${c}_{1}$. However, the condition (i) of Theorem 1 requires a lower bound for ${c}_{1}$, namely, ${c}_{1}>\mu +\beta +2\sqrt{{c}_{2}+{c}_{4}}$, for some prescribed values for $\mu $, $\beta $, ${c}_{2}$, and ${c}_{4}$, in order to guarantee the non-negativity of the solutions of the controlled model under any non-negative initial condition. Then, the only way of increasing the relation $\frac{{c}_{2}}{{c}_{1}}$ is by means of an increment of ${c}_{2}$, which also implies an increment of ${c}_{1}$ to guarantee the condition ${c}_{1}>\mu +\beta +2\sqrt{{c}_{2}+{c}_{4}}$ for some prescribed values for $\mu $, $\beta $, and ${c}_{4}$. In summary, the only practical way of increasing the relation $\frac{{c}_{2}}{{c}_{1}}$ is via increasing simultaneously the value of both parameters ${c}_{1}$ and ${c}_{2}$. However, a large value for ${c}_{2}$ can imply large values for the vaccination control effort, since it affects directly the forced term of Equation (4) for the dynamics of the vaccination law. In fact, the vaccination can be constrained to a number of available vaccines in a practical situation, which implies upper-bounds for the control parameters ${c}_{2}$, ${c}_{3}$, and ${c}_{4}$ of the forced terms of (4).
- (viii)
- One can see that ${\overline{R}}_{c}$ is directly proportional to $\frac{{c}_{6}}{{c}_{5}}$ so that an increment in the value of $\frac{{c}_{6}}{{c}_{5}}$ implies an increase of ${\overline{R}}_{c}$. Moreover, note that the EE point of the controlled model does not exist if ${R}_{c}<{\overline{R}}_{c}$ so that an increment of ${\overline{R}}_{c}$ can be interesting in order to guarantee the non-existence of such an EE point for a prescribed value for ${R}_{c}$ adjusted by values for $q$, ${c}_{1}$, and ${c}_{2}$ adapted to the number of available vaccines.
- (ix)
- The influence of the parameter $q\in \left[0,1\right]$ on the value of the control reproduction number ${R}_{c}$ is negligible if the value of the natural death rate $\mu $ is very small, as it happens in the case of humans. In such a case, the influence of $q\in \left[0,1\right]$ on the eigenvalues ${\lambda}_{1}$ and ${\lambda}_{2}$ of the matrix ${C}_{0}$ as well as on the function $F\left(q\right)$, both defined in the proof of Theorem 4, is also negligible. Such a fact implies that the DFE point is globally asymptotically stable $\forall q\in \left[0,1\right]$, since ${q}_{c}>1$, provided that the control parameters ${c}_{i}$, for $i\in \left\{1,2,\dots ,6\right\}$, are chosen such that Assumption 1 and ${R}_{c}<{\overline{R}}_{c}$ are satisfied.

## 3. Simulation Results

#### 3.1. Example 1: SIRS Model without Vaccination and Treatment

#### 3.2. Example 2: SIRS Model with Vaccination and Treatment

#### 3.3. Example 3: SIRS Model with a Defective Vaccination and Treatment

#### 3.4. Example 4: Study of the Influence of the Control Parameter ${c}_{2}$ on the Behaviour of the Controlled Model

#### 3.5. Example 5: Study of the Influence of the Control Parameter ${c}_{6}$ on the Behaviour of the Controlled Model

- The evolution of the infectious population reaches a peak of 215 individuals, i.e., approximately 21% of the initial whole population.
- The vaccination cost during the transient supposes 428 vaccines, i.e., the percentage of the susceptible subpopulation to be vaccinated is around of the 43% of the initial population, assuming one vaccine per individual.
- The treatment cost during the transient is of 554 medicaments, i.e., the percentage of infectious subpopulation to be treated is around 55% of the initial population, assuming one medicament per individual.

#### 3.6. Example 6: Study of the Behaviour of the Controlled Model under Vaccination

- Infectious peak: 92;
- Transient duration: 235 days;
- Number of vaccines: 2559;
- ${S}_{DFE}=42$, ${R}_{DFE}=1008$ and ${v}_{DFE}=8$.

#### 3.7. Example 7: Study of the Behaviour of the Controlled Model under Treatment

#### 3.8. Example 8: Study of the Behaviour of the Controlled Model under Different Rates for the Loss of Immunity

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Kermack, W.O.; McKendrick, A.G. Contributions to the mathematical theory of epidemics, part 1. Proc. R. Soc. A
**1927**, 115, 700–721. [Google Scholar] - Li, J.; Ma, Z. Global analysis of SIS epidemic models with variable total population size. Math. Comput. Model.
**2004**, 39, 1231–1242. [Google Scholar] - Pang, J.; Cui, J.; Zhou, X. Dynamical behavior of a hepatitis B virus transmission model with vaccination. J. Theor. Biol.
**2010**, 265, 572–578. [Google Scholar] [CrossRef] - Meng, X.; Chen, L.; Cheng, H. Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination. Appl. Math. Comput.
**2007**, 186, 516–529. [Google Scholar] [CrossRef] - Cai, L.; Li, X.; Ghosh, M.; Guo, B. Stability analysis of an HIV/AIDS epidemic model with treatment. J. Comput. Appl. Math.
**2009**, 229, 313–323. [Google Scholar] [CrossRef] [Green Version] - De la Sen, M.; Alonso-Quesada, S.; Ibeas, A. On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules. Appl. Math. Comput.
**2015**, 270, 953–976. [Google Scholar] [CrossRef] - Trawicki, M.B. Deterministic SEIRS epidemic model for modeling vital dynamics, vaccinations, and temporary immunity. Mathematics
**2017**, 5, 7. [Google Scholar] [CrossRef] - Gbadamosi, B.; Ojo, M.M.; Oke, S.I.; Matadi, M.B. Qualitative analysis of a Dengue fever model. Math. Comput. Appl.
**2018**, 23, 33. [Google Scholar] [CrossRef] [Green Version] - Tulu, T.W.; Tian, B.; Wu, Z. Modeling the effect of quarantine and vaccination on Ebola disease. Adv. Differ. Equ.
**2017**, 2017, 178. [Google Scholar] [CrossRef] - Buonomo, B.; Della Marca, R. Oscillations and hysteresis in an epidemic model with information-dependent imperfect vaccination. Math. Comput. Simul.
**2019**, 162, 97–114. [Google Scholar] [CrossRef] - Ullah, S.; Khan, M.A.; Farooq, M.; Gul, T. Modeling and analysis of tuberculosis (TB) in Khyber Pakhtunkhwa, Pakistan. Math. Comput. Simul.
**2019**, 165, 181–199. [Google Scholar] [CrossRef] - Liu, Q.; Jiang, D.; Hayat, T.; Alsaedi, A. Dynamics of a stochastic multigroup SIQR epidemic model with standard incidence rates. J. Frankl. Inst.
**2019**, 356, 2960–2993. [Google Scholar] [CrossRef] - Nkamba, L.N.; Ntaganda, J.M.; Abboubakar, H.; Kamgang, J.C.; Castelli, L. Global stability of a SVEIR epidemic model: Application to poliomyelitis transmission dynamics. Open J. Model. Simul.
**2017**, 5, 98–112. [Google Scholar] [CrossRef] [Green Version] - Zaman, G.; Kang, Y.H.; Cho, G.; Jung, I.H. Optimal strategy of vaccination & treatment in an SIR epidemic model. Math. Comput. Simul.
**2017**, 136, 63–77. [Google Scholar] - Alonso-Quesada, S.; De la Sen, M.; Ibeas, A. On the discretization and control of an SEIR epidemic model with a periodic impulsive vaccination. Commun. Nonlinear Sci. Numer. Simul.
**2017**, 42, 247–274. [Google Scholar] [CrossRef] - Ling, L.; Jiang, G.; Long, T. The dynamics of an SIS epidemic model with fixed-time birth pulses and state feedback pulse treatments. Appl. Math. Model.
**2015**, 39, 5579–5591. [Google Scholar] [CrossRef] - Huang, H.; Chen, Y.; Yan, Z. Impacts of social distancing on the spread of infectious diseases with asymptomatic infection: A mathematical model. Appl. Math. Comput.
**2021**, 398, 1–13. [Google Scholar] [CrossRef] - Nadim, S.S.; Ghosh, I.; Chattopadhyay, J. Short-term predictions and prevention strategies for COVID-19: A model based study. Appl. Math. Comput.
**2021**, 404, 1–19. [Google Scholar] [CrossRef] - Han, D.; Shao, Q.; Li, D.; Sun, M. How the individuals’ risk aversion affect the epidemic spreading. Appl. Math. Comput.
**2020**, 369, 1–10. [Google Scholar] [CrossRef] - Tanimoto, J. Sociophysics Approach to Epidemics; Springer: Singapore, 2021. [Google Scholar]
- Kabir, K.M.A.; Tanimoto, J. Modelling and analysing the coexistence of dual dilemmas in the proactive vaccination game and retroactive treatment game in epidemic viral dynamics. Proc. R. Soc. A
**2019**, 475, 1–20. [Google Scholar] [CrossRef] [Green Version] - Moualeu, D.P.; Weiser, M.; Ehrig, R.; Deuflhard, P. Optimal control for a tuberculosis model with undetected cases in Cameroon. Commun. Nonlinear Sci. Numer. Simul.
**2015**, 20, 986–1003. [Google Scholar] [CrossRef] [Green Version] - Sharma, S.; Singh, F. Bifurcation and stability analysis of a cholera model with vaccination and saturated treatment. Chaos Solitons Fractals
**2021**, 146, 1–15. [Google Scholar] [CrossRef] - den Driessche, P.V.; Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci.
**2002**, 180, 29–48. [Google Scholar] [CrossRef] - Zhou, L.; Fan, M. Dynamics of an SIR epidemic model with limited medical resources revisited. Nonlinear Anal. Real World Appl.
**2012**, 13, 312–324. [Google Scholar] [CrossRef] - Dumont, Y.; Chiroleu, F.; Domerg, C. On a temporal model for the Chikungunea disease: Modeling, theory and numerics. Math. Biosci.
**2008**, 213, 80–91. [Google Scholar] [CrossRef] [PubMed] - Mwasunda, J.A.; Irunde, J.I.; Kajunguri, D.; Kuznetsov, D. Modeling and analysis of taeniasis and cysticercosis transmission dynamics in human, pigs and cattle. Adv. Differ. Equ.
**2021**, 2021, 176. [Google Scholar] [CrossRef] - Vidyasagar, M. Nonlinear Systems Analysis, 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, USA, 1993. [Google Scholar]

**Figure 4.**Time evolution of the whole population in the SIRS model with vaccination and treatment within the 100 first days.

**Figure 5.**Time evolution of the whole population in the SIRS model with vaccination and treatment along 100,000 days.

**Figure 8.**Time evolution of the subpopulations in the SIRS model under an insufficient vaccination and treatment.

**Figure 10.**Time evolution of the infectious subpopulation for the four considered values of the control parameter ${c}_{2}$.

**Figure 11.**Time evolution of the vaccination effort for the four considered values of the control parameter ${c}_{2}$.

**Figure 12.**Time evolution of the treatment effort for the four considered values of the control parameter ${c}_{2}$.

**Figure 13.**Number of applied vaccines each day for the four considered values of the control parameter ${c}_{2}$.

**Figure 14.**Number of applied medicaments each day for the four considered values of the control parameter ${c}_{2}$.

**Figure 15.**Time evolution of the infectious subpopulation for the four considered values of the control parameter ${c}_{6}$.

**Table 1.**Specifications of the model behaviour for the four considered values of the parameter ${c}_{2}$.

${\mathit{c}}_{2}=1$ ${\mathit{R}}_{\mathit{c}}=2.4687$ | ${\mathit{c}}_{2}=0.9$ ${\mathit{R}}_{\mathit{c}}=2.6912$ | ${\mathit{c}}_{2}=0.8$ ${\mathit{R}}_{\mathit{c}}=2.9579$ | ${\mathit{c}}_{2}=0.7$ ${\mathit{R}}_{\mathit{c}}=3.2832$ | |
---|---|---|---|---|

Infectious peak | 215 | 229 | 245 | 261 |

Transient duration (days) | 64 | 66 | 69 | 72 |

Vaccination peak | 39 | 35 | 31 | 27 |

Treatment peak | 29 | 31 | 33 | 35 |

S_{DFE} | 182 | 198 | 218 | 242 |

R_{DFE} | 868 | 852 | 832 | 808 |

${v}_{DFE}$ | 7 | 7 | 7 | 7 |

Vaccines | 578 | 537 | 502 | 467 |

Medicaments | 483 | 515 | 542 | 570 |

**Table 2.**Specifications of the model behaviour for the four considered values of the parameter ${c}_{6}$.

${\mathit{c}}_{6}=0.24$ ${\overline{\mathit{R}}}_{\mathit{c}}=3.635$ | ${\mathit{c}}_{6}=0.27$ ${\overline{\mathit{R}}}_{\mathit{c}}=3.9644$ | ${\mathit{c}}_{6}=0.3$ ${\overline{\mathit{R}}}_{\mathit{c}}=4.2937$ | ${\mathit{c}}_{6}=0.33$ ${\overline{\mathit{R}}}_{\mathit{c}}=4.6231$ | |
---|---|---|---|---|

Infectious peak | 286 | 261 | 237 | 215 |

Transient duration (days) | 87 | 72 | 63 | 58 |

Vaccination peak | 27 | 27 | 27 | 27 |

Treatment peak | 34 | 35 | 35 | 35 |

Vaccines | 525 | 467 | 439 | 428 |

Medicaments | 573 | 570 | 565 | 554 |

**Table 3.**Specifications of the model behaviour for seven considered pairs of values of the control parameters ${c}_{1}$ and ${c}_{2}$ when there is not treatment to control the disease propagation.

Infectious Peak | Transient Duration (Days) | Vaccines | DFE Point | |
---|---|---|---|---|

Case 1 ${c}_{1}=25;{c}_{2}=10$ ${R}_{c}=0.2924$ | 33 | 120 | 1869 | ${S}_{DFE}=22$ ${R}_{DFE}=1028$ ${v}_{DFE}=9$ |

Case 2 ${c}_{1}=22;{c}_{2}=10$ ${R}_{c}=0.258$ | 29 | 110 | 1800 | ${S}_{DFE}=19$ ${R}_{DFE}=1031$ ${v}_{DFE}=9$ |

Case 3 ${c}_{1}=19;{c}_{2}=10$ ${R}_{c}=0.2233$ | 25 | 101 | 1740 | ${S}_{DFE}=16$ ${R}_{DFE}=1034$ ${v}_{DFE}=9$ |

Case 4 ${c}_{1}=25;{c}_{2}=15$ ${R}_{c}=0.1963$ | 22 | 94 | 1693 | ${S}_{DFE}=14$ ${R}_{DFE}=1036$ ${v}_{DFE}=9$ |

Case 5 ${c}_{1}=25;{c}_{2}=17$ ${R}_{c}=0.1735$ | 20 | 88 | 1651 | ${S}_{DFE}=13$ ${R}_{DFE}=1037$ ${v}_{DFE}=9$ |

Case 6 ${c}_{1}=25;{c}_{2}=20$ ${R}_{c}=0.1477$ | 18 | 82 | 1611 | ${S}_{DFE}=11$ ${R}_{DFE}=1039$ ${v}_{DFE}=9$ |

Case 7 ${c}_{1}=25;{c}_{2}=25$ ${R}_{c}=0.1184$ | 16 | 76 | 1569 | ${S}_{DFE}=9$ ${R}_{DFE}=1041$ ${v}_{DFE}=9$ |

**Table 4.**Specifications of the model behaviour for the four considered pairs of values of the control parameters ${c}_{5}$ and ${c}_{6}$ governing the treatment effort.

${\mathit{c}}_{5}=2$ ${\mathit{c}}_{6}=0.24$ ${\overline{\mathit{R}}}_{\mathit{c}}=3.635$ | ${\mathit{c}}_{5}=2$ ${\mathit{c}}_{6}=0.27$ ${\overline{\mathit{R}}}_{\mathit{c}}=3.9644$ | ${\mathit{c}}_{5}=2$ ${\mathit{c}}_{6}=0.3$ ${\overline{\mathit{R}}}_{\mathit{c}}=4.2937$ | ${\mathit{c}}_{5}=2$ ${\mathit{c}}_{6}=0.33$ ${\overline{\mathit{R}}}_{\mathit{c}}=4.6231$ | |
---|---|---|---|---|

Infectious peak | 424 | 395 | 369 | 342 |

Treatment peak | 51 | 53 | 55 | 56 |

${S}_{EE}$ | 255 | 280 | 304 | 329 |

${I}_{EE}$ | 36 | 32 | 29 | 26 |

${R}_{EE}$ | 710 | 694 | 677 | 659 |

${N}_{EE}$ | 1001 | 1006 | 1010 | 1014 |

$\frac{{I}_{EE}}{{N}_{EE}}\times 100$ | 3.59% | 3.2% | 2.87% | 2.59% |

${t}_{r}{}_{EE}$ | 4 | 4 | 4 | 4 |

**Table 5.**Specifications of the model behaviour for the four considered values of the parameter $\rho $.

. | $\mathit{\rho}=\frac{1}{100}$ | $\mathit{\rho}=\frac{1}{120}$ | $\mathit{\rho}=\frac{1}{150}$ | $\mathit{\rho}=0$ |
---|---|---|---|---|

Infectious peak | 262 | 261 | 259 | 253 |

Transient duration (days) | 91 | 72 | 64 | 49 |

Vaccination peak | 27 | 27 | 27 | 27 |

Treatment peak | 35 | 35 | 35 | 34 |

S_{DFE} | 277 | 242 | 203 | 1 |

R_{DFE} | 773 | 808 | 847 | 1049 |

${v}_{DFE}$ | 8 | 7 | 6 | 0 |

Vaccines | 615 | 467 | 400 | 290 |

Medicaments | 589 | 570 | 556 | 512 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Alonso-Quesada, S.; De la Sen, M.; Nistal, R.
An SIRS Epidemic Model Supervised by a Control System for Vaccination and Treatment Actions Which Involve First-Order Dynamics and Vaccination of Newborns. *Mathematics* **2022**, *10*, 36.
https://doi.org/10.3390/math10010036

**AMA Style**

Alonso-Quesada S, De la Sen M, Nistal R.
An SIRS Epidemic Model Supervised by a Control System for Vaccination and Treatment Actions Which Involve First-Order Dynamics and Vaccination of Newborns. *Mathematics*. 2022; 10(1):36.
https://doi.org/10.3390/math10010036

**Chicago/Turabian Style**

Alonso-Quesada, Santiago, Manuel De la Sen, and Raúl Nistal.
2022. "An SIRS Epidemic Model Supervised by a Control System for Vaccination and Treatment Actions Which Involve First-Order Dynamics and Vaccination of Newborns" *Mathematics* 10, no. 1: 36.
https://doi.org/10.3390/math10010036