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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Stochastic Dynamics of the COVID-19 Epidemic Via a New Mathematical Model

Discontinuity, Nonlinearity, and Complexity 12(3) (2023) 511--537 | DOI:10.5890/DNC.2023.09.004

I.M. Elbaz$^{1,2}$, M.A. Sohaly$^2$, H. El-Metwally$^2$

$^1$ Basic Sciences Department, Faculty of Engineering, The British University in Egypt, Cairo, Egypt

$^{2}$ Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

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Abstract

This work considers a new stochastic mathematical model for the transmission dynamics of the coronavirus COVID-19 by providing the healthy compartment together with the quarantine/isolation compartment. In the deterministic model, global stability conditions of the disease-free equilibrium $E_0$ and the endemic equilibrium $E_\star$ are derived in terms of the threshold quantity $R_0^d$. Based on the chaotic behavior, we develop and analyze a four-dimensional stochastic COVID-19 epidemic model. Uniqueness, boundedness, and positiveness of the proposed stochastic model are investigated in a biologically feasible region. In terms of the stochastic basic reproduction number $R_0^s$ of the stochastic model, extinction and persistence of the COVID-19 disease are derived. Our theoretical findings are supported by some numerical simulations. The sensitivity of the model with respect to the parameters involved in the system is studied to investigate the most sensitive parameter towards the highest number of infected individuals. We confirm the stability analysis by showing the elasticity of $R_0^s$ with respect to the variation of each parameter. We present real data of a case study with the first wave of the COVID-19 epidemic in the United Kingdom. We compare our numerical results with the real data.

References

  1. [1]  World Health Organization and others. (2019), Measles.
  2. [2]  World Health Organization. (2019), World malaria report.
  3. [3]  World Health Organization and others. (2013), Weekly Epidemiological Record, Weekly Epidemiological Record= Relev{e {e}pid{e}miologique hebdomadaire} , 88, 365-380.
  4. [4]  Spreeuwenberg, P., Kroneman, M., and Paget, J. (2018), Reassessing the global mortality burden of the 1918 influenza pandemic, American journal of epidemiology, 187, 2561-2567.
  5. [5]  World Health Organization. (2021), Global HIV and AIDS statistics-fact sheet.
  6. [6]  ArcGIS. Johns Hopkins University. (2021), COVID-19 Dashboard by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University (JHU).
  7. [7]  World Health Organization. (2020), WHO Director-General's opening remarks at the media briefing on COVID-19.
  8. [8]  Vaccine Centre, London School of Hygiene and Tropical Medicine. (2021), COVID-19 vaccine development pipeline (Refresh URL to update).
  9. [9]  So, A.D. and Woo, J. (2020), Reserving coronavirus disease 2019 vaccines for global access: cross sectional analysis, bmj, 371, 1-8.
  10. [10]  Lau, H., Khosrawipour, V., Kocbach, P., Mikolajczyk, A., Schubert, J., Bania, J., and Khosrawipour, T. (2020), The positive impact of lockdown in Wuhan on containing the COVID-19 outbreak in China, Journal of Travel Medicine, 27, 1-7.
  11. [11]  Prem, K., Liu, Y., Russell, T.W., Kucharski, A.J., Eggo, R.M., Davies, N., Flasche, S., Clifford, S., Pearson, C.A.B., and Munday, J.D. (2020), The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China: a modelling study, The Lancet Public Health, 5, 261-270.
  12. [12]  Guerra, F.M., Bolotin, S., Lim, G., Heffernan, J., Deeks, S.L., Li, Y., and Crowcroft, N.S. (2017), The basic reproduction number (R0) of measles: a systematic review, The Lancet Infectious Diseases, 17, 420-428.
  13. [13]  Liu, Y., Gayle, A.A., Wilder-Smith, A., and Rockl{\"o}v, J. (2020), The reproductive number of COVID-19 is higher compared to SARS coronavirus, Journal of travel medicine, 1-4.
  14. [14]  Roda, W.C., Varughese, M.B., Han, D., and Li, M.Y. (2017), Why is it difficult to accurately predict the COVID-19 epidemic?, Infectious Disease Modelling, 5, 271-281.
  15. [15]  Roy, M., Son, A., and May, R.M. (1982), Directly transmitted infectious diseases: control by vaccination, Science, 215, 492.
  16. [16]  World Health Organization and others. (2003), Consensus document on the epidemiology of severe acute respiratory syndrome (SARS).
  17. [17]  Choi, S., Jung, E., Choi, B.Y., Hur, Y.J., and Ki, M. (2018), High reproduction number of middle east respiratory syndrome coronavirus in nosocomial outbreaks: mathematical modelling in Saudi Arabia and South Korea, Journal of Hospital Infection, 99, 162-168.
  18. [18]  Chowell, G., Ammon, C.E., Hengartner, N.W., and Hyman, J.M. (2006), Estimation of the reproductive number of the Spanish flu epidemic in Geneva, Switzerland, Vaccine, 24, 44-46.
  19. [19]  Read, J.M., Bridgen, J.R.E., Cummings, D.A.T., Ho, A., and Jewell, C.P. (2020), Novel coronavirus 2019-nCoV: early estimation of epidemiological parameters and epidemic predictions, MedRxiv.
  20. [20]  Wyper, G.M.A., Assun{\c{c}}{\~a}o, R., Cuschieri, S., Devleesschauwer, B., Fletcher, E., Haagsma, J.A., Hilderink, H.B.M.,Idavain, J., Lesnik, T., Von der Lippe, E., and others (2020), Population vulnerability to COVID-19 in Europe: a burden of disease analysis, Archives of Public Health, 78, 1-8.
  21. [21]  Adam, D. (2020), Special report: The simulations driving the world's response to COVID-19., Nature, 580, 316.
  22. [22]  Van den Driessche, P. (2017), Reproduction numbers of infectious disease models, Infectious Disease Modelling, 2, 288-303.
  23. [23]  Liu, X., Takeuchi, Y., and Iwami, S. (2017), SVIR epidemic models with vaccination strategies, Journal of Theoretical Biology, 253, 1-11.
  24. [24]  Din, R.U., Shah, K., Ahmad, I., and Abdeljawad, T. (2020), Study of transmission dynamics of novel COVID-19 by using mathematical model, Advances in Difference Equations, 2020, 1-13.
  25. [25]  Higazy, M. (2020), Novel fractional order SIDARTHE mathematical model of COVID-19 pandemic, Chaos, Solitons \& Fractals, 138, 110007.
  26. [26]  Fredj, H.B. and Ch{e}rif, F. (2020), Novel Corona virus Disease infection in Tunisia: Mathematical model and the impact of the quarantine strategy, Chaos, Solitons \& Fractals, 138, 109969.
  27. [27]  Zhang, Z., Zeb, A., Hussain, S., and Alzahrani, E. (2020), Dynamics of COVID-19 mathematical model with stochastic perturbation, Advances in Difference Equations, 2020, 1-12.
  28. [28]  Yanev, N.M., Stoimenova, V.K., and Atanasov, D.V. (2020), Stochastic modeling and estimation of COVID-19 population dynamics, arXiv preprint arXiv.
  29. [29]  Boukanjime, B., Caraballo, T., El Fatini, M., and El Khalifi, M. (2020), Dynamics of a stochastic coronavirus (COVID-19) epidemic model with Markovian switching, Chaos, Solitons \& Fractals, 141, 110361.
  30. [30]  El-Metwally, H., Sohaly, M.A., and Elbaz, I.M. (2020), Stochastic global exponential stability of disease-free equilibrium of HIV/AIDS model, The European Physical Journal Plus, 135, 1-14.
  31. [31]  Khan, T., Khan, A., and Zaman, G. (2018), The extinction and persistence of the stochastic hepatitis B epidemic model, Chaos, Solitons \& Fractals, 108, 123-128.
  32. [32]  Li, R., Pei, S., Chen, B., Song, Y., Zhang, T., Yang, W., and Shaman, J. (2020), Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV-2), Science, 368, v.
  33. [33]  Lauer, S.A., Grantz, K.H., Bi, Q., Jones, F.K., Zheng, Q., Meredith, H.R., Azman, A.S., Reich, N.G., and Lessler, J. (2020), The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: estimation and application, Annals of internal medicine, 172, 577-582.
  34. [34]  Bai, Y., Yao, L., Wei, T., Tian, F., Jin, D.Y., Chen, L., and Wang, M. (2020), Presumed asymptomatic carrier transmission of COVID-19, Jama, 323, 1406-1407.
  35. [35]  Evans, L.C. (2012), An Introduction to Stochastic Differential Equations, (82 vol.)(American Mathematical Soc.).
  36. [36]  Mao, X. (2007), Stochastic Differential Equations and Applications, (Stochastic differential equations and applications).
  37. [37]  Yassen, M.T., Sohaly, M.A., and Elbaz, I.M. (2016), Random crank-nicolson scheme for random heat equation in mean square sense, American Journal of Computational Mathematics, 6, 66-73.
  38. [38]  Yassen, M.T., Sohaly, M.A., and Elbaz, I.M. (2017), Stochastic solution for Cauchy one-dimensional advection model in mean square calculus, Journal of the Association of Arab Universities for Basic and Applied Sciences, 24, 263-270.
  39. [39]  Sohaly, M.A., Yassen, M.T., and Elbaz, I.M. (2018), Stochastic consistency and stochastic stability in mean square sense for Cauchy advection problem, Journal of Difference Equations and Applications, 24, v.
  40. [40]  Almutairi, A., El-Metwally, H., Sohaly, M.A., and Elbaz, I.M. (2021), Lyapunov stability analysis for nonlinear delay systems under random effects and stochastic perturbations with applications in finance and ecology, Advances in Difference Equations , 2021, 1-32.
  41. [41]  Sohaly, M.A., Yassen, M.T., and Elbaz, I.M. (2017), The Variational Methods for Solving Random Models, IJIRCST , 5.
  42. [42]  Fares, M.E., Sohaly, M.A., Yassen, M.T., and Elbaz, I.M. (2017), Mean square calculus for cauchy problems stochastic heat and stochastic advection models, International Journal of Contemporary Mathematical Sciences, 15, 177-186.
  43. [43]  Yang, X., Chen, L., and Chen, J. (1996), Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models, Computers \& Mathematics with Applications , 32, 109-116.
  44. [44]  La Salle, J.P. (1976), The Stability of Dynamical Systems, (SIAM).
  45. [45]  Higham, D.J., Mao, X., and Stuart, A.M. (2002), Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM Journal on Numerical Analysis, 40, 1041-1063.
  46. [46]  N'i, M. and Yattara, I. (2018), The dynamics of a delayed deterministic and stochastic epidemic SIR models with a saturated incidence rate, Random Operators and Stochastic Equations, 26, 235-245.
  47. [47]  Mohebbi, R., Delouei, A.A., Jamali, A., Izadi, M., and Mohamad, A.A. (2019), Pore-scale simulation of non-Newtonian power-law fluid flow and forced convection in partially porous media: thermal lattice Boltzmann method, Physica A: Statistical Mechanics and Its Applications , 525, 642-656.
  48. [48]  Delouei, A.A., Nazari, M., Kayhani, M.H., and Ahmadi, G. (2017), Direct-forcing immersed boundary--non-Newtonian lattice Boltzmann method for transient non-isothermal sedimentation, Journal of Aerosol Science, 104, 106-122.
  49. [49]  Karimnejad, S., Delouei, A.A., Nazari, M., Shahmardan, M.M., Rashidi, M.M., and Wongwises, S. (2019), Immersed boundary---thermal lattice Boltzmann method for the moving simulation of non-isothermal elliptical particles, Journal of Thermal Analysis and Calorimetry, 138, 4003-4017.
  50. [50]  Jalali, A., Delouei, A.A., Khorashadizadeh, M., Golmohamadi, A.M., and Karimnejad, S. (2020), Mesoscopic simulation of forced convective heat transfer of Carreau-Yasuda fluid flow over an inclined square: temperature-dependent viscosity, Journal of Applied and Computational Mechanics, 6, 307-319.
  51. [51]  Afra, B., Delouei, A.A., Mostafavi, M., and Tarokh, A. (2021), Fluid-structure interaction for the flexible filament's propulsion hanging in the free stream, Journal of Thermal Analysis and Calorimetry, 323, 114941.
  52. [52]  Sajjadi, H., Atashafrooz, M., Delouei, A.A., and Wang, Y. (2021), The effect of indoor heating system location on particle deposition and convection heat transfer: DMRT-LBM, Computers \& Mathematics with Applications, 86, 90-105.
  53. [53]  Lepingle, D. (1995), Euler scheme for reflected stochastic differential equations, Journal of Thermal Analysis and Calorimetry, 38, 119-126.
  54. [54]  Mao, X. (2016), Convergence rates of the truncated Euler--Maruyama method for stochastic differential equations, Journal of Computational and Applied Mathematics, 296, 362-375.
  55. [55]  Sajjadi, H., Delouei, A.A., Mohebbi, R., Izadi, M., and Succi, S. (2021), Natural convection heat transfer in a porous cavity with sinusoidal temperature distribution using Cu/water nanofluid: Double MRT lattice Boltzmann method, Journal of Thermal Analysis and Calorimetry, 29, 292-318.
  56. [56]  Heffernan, J.M., Smith, R.J., and Wahl, L.M. (2005), Perspectives on the basic reproductive ratio, Journal of the Royal Society Interface, 2, 281-293.
  57. [57]  worldometers. (2020), Coronavirus pandemic.
  58. [58]  Wyper, G.M.A., Assun{\c{c}}{\~a}o, R., Cuschieri, S., Devleesschauwer, B., Fletcher, E., Haagsma, J.A., Hilderink, H.B.M., Idavain, J., Lesnik, T., Von der Lippe, E., and others (2020), Population vulnerability to COVID-19 in Europe: a burden of disease analysis, Archives of Public Health, 78, 1-8.
  59. [59]  Ahmad, S., Owyed, S., Abdel-Aty, A., Mahmoud, E.E., Shah, K., Alrabaiah, H., and others (2021), Mathematical analysis of COVID-19 via new mathematical model, Chaos, Solitons \& Fractals, 143, 110585.
  60. [60]  Qureshi, S. (2020), Periodic dynamics of rubella epidemic under standard and fractional Caputo operator with real data from Pakistan, Mathematics and Computers in Simulation, 178, 151-165.
  61. [61]  Jajarmi, A., Hajipour, M., Sajjadi, S.S., and Baleanu, D. (2019), A robust and accurate disturbance damping control design for nonlinear dynamical systems, Optimal Control Applications and Methods, 40, 375-393.
  62. [62]  Rezapour, S., Mohammadi, H., and Jajarmi, A. (2020), A new mathematical model for Zika virus transmission, Advances in Difference Equations, 2020, 1-15.
  63. [63]  Musa, S.S., Qureshi, S., Zhao, S., Yusuf, A., Mustapha, U.T., and He, D. (2021), Mathematical modeling of COVID-19 epidemic with effect of awareness programs, Infectious Disease Modelling, 6, 448-460.
  64. [64]  Mohammadi, H., Rezapour, S. and Jajarmi, A. (2021), On the fractional SIRD mathematical model and control for the transmission of COVID-19: the first and the second waves of the disease in Iran and Japan, ISA transactions.
  65. [65]  El-Metwally, H., Sohaly, M.A., and Elbaz, I.M. (2021), Mean-square stability of the zero equilibrium of the nonlinear delay differential equation: Nicholson's blowflies application, Nonlinear Dynamics, 105, 1713-1722.

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