Global investigation for an "SIS" model for COVID-19 epidemic with asymptomatic infection

Mohammed H. Alharbi; Mohammed H. Alharbi

doi:10.3934/mbe.2023245

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 3: 5298-5315. doi: 10.3934/mbe.2023245

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Global investigation for an "SIS" model for COVID-19 epidemic with asymptomatic infection

Mohammed H. Alharbi ^,

Department of Mathematics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia

Academic Editor: Pierre Magal

Received: 08 November 2022 Revised: 21 December 2022 Accepted: 25 December 2022 Published: 11 January 2023

In this paper, we analyse a dynamical system taking into account the asymptomatic infection and we consider optimal control strategies based on a regular network. We obtain basic mathematical results for the model without control. We compute the basic reproduction number ($ \mathcal{R} $) by using the method of the next generation matrix then we analyse the local stability and global stability of the equilibria (disease-free equilibrium (DFE) and endemic equilibrium (EE)). We prove that DFE is LAS (locally asymptotically stable) when $ \mathcal{R} < 1 $ and it is unstable when $ \mathcal{R} > 1 $. Further, the existence, the uniqueness and the stability of EE is carried out. We deduce that when $ \mathcal{R} > 1 $, EE exists and is unique and it is LAS. By using generalized Bendixson-Dulac theorem, we prove that DFE is GAS (globally asymptotically stable) if $ \mathcal{R} < 1 $ and that the unique endemic equilibrium is globally asymptotically stable when $ \mathcal{R} > 1 $. Later, by using Pontryagin's maximum principle, we propose several reasonable optimal control strategies to the control and the prevention of the disease. We mathematically formulate these strategies. The unique optimal solution was expressed using adjoint variables. A particular numerical scheme was applied to solve the control problem. Finally, several numerical simulations that validate the obtained results were presented.
- COVID-19,
- SIS epidemic model,
- asymptomatic and symptomatic individuals,
- nonlinear incidence rates,
- Bendixson-Dulac theorem,
- optimal strategies,
- sensitivity analysis
Citation: Mohammed H. Alharbi. Global investigation for an 'SIS' model for COVID-19 epidemic with asymptomatic infection[J]. Mathematical Biosciences and Engineering, 2023, 20(3): 5298-5315. doi: 10.3934/mbe.2023245

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Abstract

In this paper, we analyse a dynamical system taking into account the asymptomatic infection and we consider optimal control strategies based on a regular network. We obtain basic mathematical results for the model without control. We compute the basic reproduction number ($ \mathcal{R} $) by using the method of the next generation matrix then we analyse the local stability and global stability of the equilibria (disease-free equilibrium (DFE) and endemic equilibrium (EE)). We prove that DFE is LAS (locally asymptotically stable) when $ \mathcal{R} < 1 $ and it is unstable when $ \mathcal{R} > 1 $. Further, the existence, the uniqueness and the stability of EE is carried out. We deduce that when $ \mathcal{R} > 1 $, EE exists and is unique and it is LAS. By using generalized Bendixson-Dulac theorem, we prove that DFE is GAS (globally asymptotically stable) if $ \mathcal{R} < 1 $ and that the unique endemic equilibrium is globally asymptotically stable when $ \mathcal{R} > 1 $. Later, by using Pontryagin's maximum principle, we propose several reasonable optimal control strategies to the control and the prevention of the disease. We mathematically formulate these strategies. The unique optimal solution was expressed using adjoint variables. A particular numerical scheme was applied to solve the control problem. Finally, several numerical simulations that validate the obtained results were presented.

References

[1]	L. Brunese, F. Mercaldo, A. Reginelli, A. Santone, Explainable deep learning for pulmonary 110 disease and coronavirus covid-19 detection from x-rays, Comput. Methods Programs Biomed., 196 (2020), 105608. https://doi.org/10.1016/j.cmpb.2020.105608 doi: 10.1016/j.cmpb.2020.105608
[2]	D. Kalajdzievska, M. Y. Li, Modeling the effects of carriers on transmission dynamics of infectious diseases, Math. Biosc. Eng., 8 (2011) 711–722. https://doi.org/10.3934/mbe.2011.8.711 doi: 10.3934/mbe.2011.8.711
[3]	B. Batista, D. Dickenson, K. Gurski, M. Kebe, N. Rankin, Minimizing disease spread on a 130 quarantined cruise ship: A model of covid-19 with asymptomatic infections, Math. Biosci., 329 (2020), 108442. https://doi.org/10.1016/j.mbs.2020.108442 doi: 10.1016/j.mbs.2020.108442
[4]	J. L. Gevertz, J. M. Greene, C. H. Sanchez-Tapia, E. D. Sontag, A novel covid-19 epidemiological model with explicit susceptible and asymptomatic isolation compartments reveals unexpected consequences of timing social distancing, J. Theor. Biol., 510 (2021), 110539. https://doi.org/10.1016/j.jtbi.2020.110539 doi: 10.1016/j.jtbi.2020.110539
[5]	O. Khyar, K. Allali, Global dynamics of a multi-strain SEIR epidemic model with general incidence rates: Application to COVID-19 pandemic, Non-Linear Dynamics, (2020), 1–21. https://doi.org/10.1007/s11071-020-05929-4 doi: 10.1007/s11071-020-05929-4
[6]	A. Alshehri, M. El Hajji, Mathematical study for Zika virus transmission with general incidence rate, AIMS Math., 7 (2022), 2853–2875. http://doi.org/2010.3934/math.2022397
[7]	M. H. Alharbi, C. M. Kribs, A mathematical modeling study: Assessing impact of mismatch between influenza vaccine strains and circulating strains in Hajj, Bull. Math. Biol., 83 (2021). https://doi.org/10.1007/s11538-020-00836-6 doi: 10.1007/s11538-020-00836-6
[8]	M. El Hajji, A. H. Albargi, A mathematical investigation of an "SVEIR" epidemic model for the Measles transmission, Math. Biosci. Eng., 19 (2022), 2853–2875. https://doi.org/10.3934/mbe.2022131 doi: 10.3934/mbe.2022131
[9]	M. De la Sen, A. Ibeas, S. Alonso-Quesada, R. Nistal, On a new epidemic model with asymptomatic and dead-infective subpopulations with feedback controls useful for Ebola disease, Discrete Dyn. Nat. Soc., 2017 (2017), 22. https://doi.org/10.1155/2017/4232971 doi: 10.1155/2017/4232971
[10]	M. De la Sen, S. Alonso-Quesada, A. Ibeas, On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules, Appl. Math. Comput., 270 (2015), 953–976. https://doi.org/10.1016/j.amc.2015.08.099 doi: 10.1016/j.amc.2015.08.099
[11]	S. Saha, P. Dutta, G. Samanta, Dynamical behavior of SIRS model incorporating government action and public response in presence of deterministic and fluctuating environments, Chaos Solitons Fractals, 164 (2022), 112643. https://doi.org/10.1016/j.chaos.2022.112643 doi: 10.1016/j.chaos.2022.112643
[12]	X. Chen, Infectious disease modeling and epidemic response measures analysis considering asymptomatic infection, IEEE Access, 8 (2020), 149652–149660. https://doi.org/10.1109/ACCESS.2020.3016681 doi: 10.1109/ACCESS.2020.3016681
[13]	J.P. Vandenbroucke, N. Pearce, Incidence rates in dynamic populations, Int. J. Epidemiol., 41 (2012), 1472–1479. https://doi.org/10.1093/ije/dys142 doi: 10.1093/ije/dys142
[14]	X. Z. Li, W. S. Li, M. Ghosh, Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment, Appl. Math. Comput., 210 (2009), 141–150. https://doi.org/10.1016/j.amc.2008.12.085 doi: 10.1016/j.amc.2008.12.085
[15]	O. Diekmann, J. Heesterbeek, On the definition and the computation of the basic reproduction ratio $\mathcal{R}_0$ in models for infectious diseases in heterogeneous populations, J. Math. Bio., 28 (1990), 365–382. https://doi.org/10.1007/BF00178324 doi: 10.1007/BF00178324
[16]	P. Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
[17]	H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755–763. https://doi.org/10.1007/BF00173267 doi: 10.1007/BF00173267
[18]	H. L. Smith, P. Waltman, The theory of the chemostat, Dynamics of microbial competition, in Cambridge Studies in Mathematical Biology, Cambridge University Press. 1995.
[19]	W. H. Fleming, R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer verlag, New York. 1975.
[20]	S. Lenhart, J. T. Workman, Optimal Control Applied to Biological Models, Chapman and Hall, 2007.
[21]	M. El Hajji, Modelling and optimal control for Chikungunya disease, Theory Biosci., 140 (2021), 27–44. https://doi.org/10.1007/s12064-020-00324-4 doi: 10.1007/s12064-020-00324-4
[22]	M. El Hajji, A. Zaghdani, S. Sayari, Mathematical analysis and optimal control for Chikungunya virus with two routes of infection with nonlinear incidence rate, Int. J. Biomath., 15 (2021), 2150088. https://doi.org/10.1142/S1793524521500881 doi: 10.1142/S1793524521500881
[23]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The mathematical theory of optimal processes, Wiley, New York, 1962.
[24]	N. Chitnis, J. M. Hyman, J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272–1296. https://doi.org/10.1007/s11538-008-9299-0 doi: 10.1007/s11538-008-9299-0
[25]	C. J. Silva, D. F. M. Torres, Optimal control for a tuberculosis model with reinfection and post-exposure interventions, Math. Biosci., 244 (2013), 154–164. https://doi.org/10.1016/j.mbs.2013.05.005 doi: 10.1016/j.mbs.2013.05.005

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