Optimal Control Studies on Age Structured Modeling of COVID-19 in Presence of Saturated Medical Treatment of Holling Type III

doi:10.1007/s12591-022-00593-z

. 2022 Feb 18:1-40.

doi: 10.1007/s12591-022-00593-z. Online ahead of print.

Optimal Control Studies on Age Structured Modeling of COVID-19 in Presence of Saturated Medical Treatment of Holling Type III

Bishal Chhetri¹, D K K Vamsi¹, Carani B Sanjeevi^{2

3}

Affiliations

¹ Department of Mathematics and Computer Science, Sri Sathya Sai Institute of Higher Learning-SSSIHL, Anantapur, India.
² Department of Medicine, Karolinska Institute, Stockholm, Sweden.
³ Sri Sathya Sai Institute of Higher Learning-SSSIHL, Anantapur, India.

PMID: 35194346
PMCID: PMC8855658
DOI: 10.1007/s12591-022-00593-z

Optimal Control Studies on Age Structured Modeling of COVID-19 in Presence of Saturated Medical Treatment of Holling Type III

Bishal Chhetri et al. Differ Equ Dyn Syst. 2022.

. 2022 Feb 18:1-40.

doi: 10.1007/s12591-022-00593-z. Online ahead of print.

Authors

Bishal Chhetri¹, D K K Vamsi¹, Carani B Sanjeevi^{2

3}

Affiliations

¹ Department of Mathematics and Computer Science, Sri Sathya Sai Institute of Higher Learning-SSSIHL, Anantapur, India.
² Department of Medicine, Karolinska Institute, Stockholm, Sweden.
³ Sri Sathya Sai Institute of Higher Learning-SSSIHL, Anantapur, India.

PMID: 35194346
PMCID: PMC8855658
DOI: 10.1007/s12591-022-00593-z

Abstract

COVID-19 pandemic has caused the most severe health problems to adults over 60 years of age, with particularly fatal consequences for those over 80. In this case, age-structured mathematical modeling could be useful to determine the spread of the disease and to develop a better control strategy for different age groups. In this study, we first propose an age-structured model considering two different age groups, the first group with population age below 30 years and the second with population age above 30 years, and discuss the stability of the equilibrium points and the sensitivity of the model parameters. In the second part of the study, we propose an optimal control problem to understand the age-specific role of treatment in controlling the spread of COVID -19 infection. From the stability analysis of the equilibrium points, it was found that the infection-free equilibrium point remains locally asymptotically stable when $R_{0} < 1$ , and when $R_{0}$ is greater than one, the infected equilibrium point remains locally asymptotically stable. The results of the optimal control study show that infection decreases with the implementation of an optimal treatment strategy, and that a combined treatment strategy considering treatment for both age groups is effective in keeping cumulative infection low in severe epidemics. Cumulative infection was found to increase with increasing saturation in medical treatment.

Keywords: Age structured modeling; Basic Reproduction number; COVID-19; Models; Optimal control problem; Type III recovery rate.

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Figures

**Fig. 1**
Figure depicting local asymptotic stability of $E_{0}$ whenever $R_{0} < 1$

**Fig. 2**
Figure depicting local asymptotic stability of $E_{1}$ whenever $R_{0} > 1$

**Fig. 3**
Figure depicting the sensitivity Analysis of $μ_{12}$ varied in three intervals in Table 5. The plots depict the infected population for each varied value of the parameter $μ_{12}$ per interval along with the mean infected population and the mean square error in the same interval

**Fig. 4**
Figure depicting the sensitivity analysis of $b_{1}$ varied in three intervals in Table 5. The plots depict the infected population for each varied value of the parameter $b_{1}$ per interval along with the mean infected population and the mean square error in the same interval

**Fig. 5**
Figure depicting the sensitivity analysis of $μ$ varied in three intervals in Table 5. The plots depict the infected population for each varied value of the parameter $μ$ per interval along with the mean infected population and the mean square error in the same interval

**Fig. 6**
Figure depicting the sensitivity analysis of $β_{1}$ varied in two intervals in Table 5. The plots depict the infected population for each varied value of the parameter $β_{1}$ per interval along with the mean infected population and the mean square error in the same interval

**Fig. 7**
$I_{2}$ under optimal controls $μ_{11}^{*}$ , $μ_{12}^{*}$

**Fig. 8**
$I_{1}$ under optimal controls $μ_{11}^{*}$ , $μ_{12}^{*}$

**Fig. 9**
$R_{1}$ under optimal controls $μ_{11}^{*}$ , $μ_{12}^{*}$

**Fig. 10**
$R_{2}$ under optimal controls $μ_{11}^{*}$ , $μ_{12}^{*}$

**Fig. 11**
Cumulative infection ( $I_{1} + I_{2}$ ) under different controls

**Fig. 12**
Effect of $R_{0}$ on $I_{1}$ under different controls

**Fig. 13**
Effect of $R_{0}$ on $I_{2}$ under different controls

**Fig. 14**
Effect of $R_{0}$ on cumulative infected population under different controls

**Fig. 15**
Effect of $R_{0}$ on cumulative recovered population under different controls

**Fig. 16**
Effect of $α$ on the cumulative disease burden with $μ_{11}^{*}$ and $μ_{12}^{*}$

**Fig. 17**
Effect of $α$ on the cumulative disease burden with $μ_{12}^{*}$

**Fig. 18**
Sensitivity analysis of $u_{11}$

**Fig. 19**
Sensitivity analysis of $β_{2}$

**Fig. 20**
Sensitivity analysis of $β_{3}$

**Fig. 21**
Sensitivity analysis of $β_{4}$

**Fig. 22**
Sensitivity analysis of $d_{1}$

**Fig. 23**
Sensitivity analysis of $d_{2}$

**Fig. 26**
Sensitivity analysis of $δ_{1}$

**Fig. 27**
Sensitivity analysis of $δ_{2}$

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References

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[1] Abdo MS, Shah K, Wahash HA, Panchal SK. On a comprehensive model of the novel coronavirus (covid-19) under Mittag–Leffler derivative. Chaos Solitons Fractals. 2020;135:109867. doi: 10.1016/j.chaos.2020.109867. - DOI - PMC - PubMed

[2] Abdo MS, Shah K, Wahash HA, Panchal SK. On a comprehensive model of the novel coronavirus (covid-19) under Mittag–Leffler derivative. Chaos Solitons Fractals. 2020;135:109867. doi: 10.1016/j.chaos.2020.109867. - DOI - PMC - PubMed

[3] Abdulwasaa MA, Abdo MS, Shah K, Nofal TA, Panchal SK, Kawale SV, Abdel-Aty A-H. Fractal-fractional mathematical modeling and forecasting of new cases and deaths of covid-19 epidemic outbreaks in India. Results Phys. 2021;20:103702. doi: 10.1016/j.rinp.2020.103702. - DOI - PMC - PubMed

[4] Abdulwasaa MA, Abdo MS, Shah K, Nofal TA, Panchal SK, Kawale SV, Abdel-Aty A-H. Fractal-fractional mathematical modeling and forecasting of new cases and deaths of covid-19 epidemic outbreaks in India. Results Phys. 2021;20:103702. doi: 10.1016/j.rinp.2020.103702. - DOI - PMC - PubMed

[5] Ahmad S, Owyed S, Abdel-Aty A-H, Mahmoud EE, Shah K, Alrabaiah H, et al. Mathematical analysis of covid-19 via new mathematical model. Chaos Solitons Fractals. 2021;143:110585. doi: 10.1016/j.chaos.2020.110585. - DOI - PMC - PubMed

[6] Ahmad S, Owyed S, Abdel-Aty A-H, Mahmoud EE, Shah K, Alrabaiah H, et al. Mathematical analysis of covid-19 via new mathematical model. Chaos Solitons Fractals. 2021;143:110585. doi: 10.1016/j.chaos.2020.110585. - DOI - PMC - PubMed

[7] Ahmad S, Ullah A, Al-Mdallal QM, Khan H, Shah K, Khan A. Fractional order mathematical modeling of covid-19 transmission. Chaos Solitons Fractals. 2020;139:110256. doi: 10.1016/j.chaos.2020.110256. - DOI - PMC - PubMed

[8] Ahmad S, Ullah A, Al-Mdallal QM, Khan H, Shah K, Khan A. Fractional order mathematical modeling of covid-19 transmission. Chaos Solitons Fractals. 2020;139:110256. doi: 10.1016/j.chaos.2020.110256. - DOI - PMC - PubMed

[9] Ahmed HM, Elbarkouky RA, Omar OAM, Ragusa MA. Models for covid-19 daily confirmed cases in different countries. Mathematics. 2021;9(6):659. doi: 10.3390/math9060659. - DOI

[10] Ahmed HM, Elbarkouky RA, Omar OAM, Ragusa MA. Models for covid-19 daily confirmed cases in different countries. Mathematics. 2021;9(6):659. doi: 10.3390/math9060659. - DOI

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Optimal Control Studies on Age Structured Modeling of COVID-19 in Presence of Saturated Medical Treatment of Holling Type III

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Optimal Control Studies on Age Structured Modeling of COVID-19 in Presence of Saturated Medical Treatment of Holling Type III

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