### Analysis of Caputo fractional SEIR model for Covid-19 pandemic

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Mathematics, Statistics and Computer Science, Semnan University, Iran

Abstract

In this paper, we study the spread of COVID-19 and its effect on a population through mathematical models. We propose a Caputo time-fractional compartmental model (SEIR) comprising the susceptible, exposed, infected and recovered population for the dynamics of the COVID-19 pandemic. The proposed nonlinear fractional model is an extension of a formulated integer-order COVID-19 mathematical model. The existence of a unique solution for the proposed model was shown by using basic concepts such as continuity and Banach's fixed-point theorem. The uniqueness and boundedness of the solutions of the proposed model are investigated. We calculate a central quantity in epidemiology called the basic reproduction number, $R_{0}$ by the concept of the next-generation matrices approach. The equilibrium points of the model are calculated and the local asymptotic stability for the derived disease-free equilibrium point is discussed.

Keywords

[1] R.P. Agarwal, D. Baleanu, V. Hedayati, and S. Rezapour, Two fractional derivative inclusion problems via integral boundary conditions, Appl. Math. Comput. 257 (2015), 205—212.
[2] L.J. Allen, F. Brauer, P. van den Driessche, and J. Wu, Mathematical Epidemiology, Volume 1945. Springer, 2008.
[3] A. Al-Jabir, A. Kerwan, M. Nicola, Z. Alsafi, M. Khan, C. Sohrabi, N. O’Neill, C. Iosifidis, M. Griffin, G. Mathew, and R. Agha, Impact of the coronavirus (covid-19) pandemic on surgical practice-part 2 (surgical prioritisation), Int. J. Surgery 79 (2020), 233–248.
[4] A. Alsaedi, D. Baleanu, S. Etemad, and S. Rezapour, On coupled systems of time-fractional differential problems by using a new fractional derivative, J. Funct. Spaces 2016 (2016), Article ID 4626940.
[5] H. Al-Sulami, M. El-Shahed, and J.J. Nieto, On fractional order dengue epidemic model, Math. Probl. Eng. 4 (2014), 1–6.
[6] D. Baleanu, R.P. Agarwal, H. Mohammadi, and S. Rezapour, Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces, Bound. Value Probl. 2013 (2013), 112.
[7] E. Baleanu, S. Etemad, and S. Rezapour, On a fractional hybrid integro-differential equation with mixed hybrid integral boundary value conditions by using three operators, Alex. Eng. J. 59 (2020), no. 5, 3019–3027.
[8] D. Baleanu, V. Hedayati, S. Rezapour, and M.M. Al-Qurashi, On two fractional differential inclusions, Springer-Plus 5 (2016), no. 1, 882.
[9] D. Baleanu, H. Mohammadi, and S. Rezapour, A fractional differential equation model for the COVID-19 transmission by using the Caputo–Fabrizio derivative, Adv. Differ. Equ. 2020 (2020), 299.
[10] D. Baleanu, A. Mousalou and S. Rezapour, On the existence of solutions for some infinite coefficient-symmetric Caputo–Fabrizio fractional integro-differential equations, Bound. Value Probl. 2017 (2017), 145.
[11] F. Brauer, Mathematical epidemiology: Past, present, and future, Infect. Disease Model. 2 (2017), no. 2, 113–127.
[12] M. Caputo, Linear model of dissipation whose q is almost frequency independent, II, Geophys. J. Int. 13 (1967), no. 5, 529—539.
[13] L. Carenzo, E. Costantini, M. Greco, F. Barra, V. Rendiniello, M. Mainetti, R. Bui, A. Zanella, G. Grasselli, M. Lagioia, and A. and Protti, Hospital surge capacity in a tertiary emergency referral centre during the COVID-19 outbreak in Italy, Anaesthesia 75 (2020), no. 7, 928–934.
[14] Centers for Disease Control and Prevention, Trends in COVID-19 Incidence After Implementation of Mitigation Measures — Arizona, January 22-August 7, 2020. https://www.cdc.gov/ mmwr/volumes/69/wr/mm6940e3.htm, 2020. Accessed: 2021-03-20.
[15] Centers for Disease Control and Prevention, About COVID-19. https://www.cdc.gov/ coronavirus/2019-ncov/cdcresponse/about-COVID-19.html, 2020. Accessed: 2021-03-20.
[16] Centers for Disease Control and Prevention, Interim Clinical Guidance for Management of Patients with Confirmed Coronavirus Disease (COVID-19). https://www.cdc.gov/ coronavirus/2019-ncov/hcp/clinical-guidancemanagement-patients.html, 2021. Accessed: 2021-03-20.
[17] Centers for Disease Control and Prevention, https://www.cdc.gov/coronavirus/2019- ncov/symptomstesting/symptoms.html. https://www.cdc.gov/coronavirus/2019-ncov/symptoms-testing/symptoms.html, 2021. Accessed: 2021-03-20.
[18] T. Chen, J. Rui, Q. Wang, Z. Zhao, J.A. Cui, and L. Yin, A mathematical model for simulating the transmission of Wuhan novel coronavirus, Infect. Dis. Poverty 9 (2020), 24.
[19] D. Cucinotta and M. Vanelli, Who declares covid-19 a pandemic, Acta Bio Medica: Atenei Parmensis 91 (2020), no. 1, 157.
[20] O. Diekmann, J. Heesterbeek, and J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28 (1990), 365-–382.
[21] O. Diekmann, J.A.P. Heesterbeek, and M.G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc. Interface 7 (2010), no. 47, 873–885.
[22] A. Dighe, T. Jombart, M.V. Kerkhove, and N. Ferguson, A mathematical model of the transmission of Middle East respiratory syndrome coronavirus in dromedary camels (Camelus dromedarius), Int. J. Infect. Dis. 79 (2019), no. S1, 1-–150.
[23] M. Du, Z. Wang, and H. Hu, Measuring memory with the order of fractional derivative, Sci. Rep. 3 (2013), 3431.
[24] M. Egger, L. Johnson, C. Althaus, A. Sch¨oni, G. Salanti, N. Low, and S.L. Norris, Developing who guidelines: Time to formally include evidence from mathematical modelling studies, F1000Research 6 (2017), 1584.
[25] N. Ferguson, D. Laydon, G. Nedjati Gilani, N. Imai, K. Ainslie, M. Baguelin, S. Bhatia, A. Boonyasiri, Z.U.L.M.A. Cucunuba Perez, G. Cuomo-Dannenburg, and A. Dighe, Report 9: Impact of non-pharmaceutical interventions (npis) to reduce covid19 mortality and healthcare demand, Imperial College COVID-19 Response Team, 16 March 2020.
[26] S.M. Garba, J.M.-S. Lubuma, and B. Tsanou, Modeling the transmission dynamics of the covid-19 pandemic in South Africa, Mathematical Biosci. 328 (2020), 108441.
[27] F. Haq, K. Shah, G. Rahman, and M. Shahzad, Numerical analysis of fractional order model of HIV-1 infection of CD4+ T-cells, Comput. Meth. Differ. Equ. 5 (2017), no. 1, 1—11.
[28] V. Hedayati and S. Rezapour, The existence of solution for a k-dimensional system of fractional differential inclusions with anti-periodic boundary value problems, Filomat 30 (2016), no. 6, 1601-–1613.
[29] M. Higazy, Novel fractional order SIDARTHE mathematical model of COVID-19 pandemic, Chaos Solitons Fractals 138 (2020), 110007.
[30] E.A. Iboi, C.N. Ngonghala, and A.B. Gumel, Will an imperfect vaccine curtail the covid-19 pandemic in the us?, Infect. Disease Model. 5 (2020), 510–524.
[31] S. Khajanchi, D.K. Das, and T.K. Kar, Dynamics of tuberculosis transmission with exogenous reinfections and endogenous reactivation, Phys. A: Statist. Mech. Appl. 497 (2018), 52—71.
[32] I. Koca, Analysis of rubella disease model with non-local and non-singular fractional derivatives, Int. J. Optim. Control Theor. Appl. 8 (2018), no. 1, 17-–25.
[33] W. Lin, Global existence theory and chaos control of fractional differential equations, J. Math. Anal. Appl. 332 (2007), no. 1, 709—726.
[34] F. Ndaırou, I. Area, J.J. Nieto, C.J. Silva, and D.F.M. Torres, Fractional model of COVID-19 applied to Galicia, Spain and Portugal, Chaos Solitons Fractals 144 (2021), 110652.
[35] F. Ndaırou, I. Area, J.J. Nieto, and D.F. Torres, Mathematical modeling of covid-19 transmission dynamics with a case study of Wuhan, Chaos Solitons Fractals 135 (2010), 109846.
[36] M. Naveed, M. Rafiq, A. Raza, N. Ahmed, I. Khan, K.S. Nisar, and A.H. Soori, Mathematical analysis of novel Coronavirus (2019-nCov) delay pandemic model, Comput. Mater. Continua 64 (2020), no. 3, 1401—1414.
[37] N. Ngonghala, E. Iboi, S. Eikenberry, M. Scotch, C.R. MacIntyre, M.H. Bonds, and A.B. Gumel, Mathematical assessment of the impact of non-pharmaceutical interventions on curtailing the 2019 novel coronavirus, Math. Biosci. 325 (2020), 108364.
[38] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198, Academic Press, 1999.
[39] S.Z. Rida, A.A.M. Arafa, and Y.A. Gaber, Solution of the fractional epidemic model by l-adm, J. Fract. Calc. Appl. 7 (2016), no. 1, 189—195.
[40] S. Rosa and D.F.M. Torres, Optimal control and sensitivity analysis of a fractional order TB model, Stat. Optim. Inf. Comput. 7 (2019), 617—625.
[41] M.E. Samei and S. Rezapour, On a system of fractional q-differential inclusions via sum of two multi-term functions on a time scale, Bound. Value Probl. 2020 (2020), 135.
[42] P. Samui, J. Mondal, and S. Khajanchi, A mathematical model for covid-19 transmission dynamics with a case study of India, Chaos Solitons Fractals 140 (2020), 110173.
[43] A.S. Shaikh, V.S. Jadhav, M.G. Timol, K.S. Nisar, and I. Khan, Analysis of the covid-19 pandemic spreading in India by an epidemiological model and fractional differential operator, Preprints 2020050266 (2020). https://doi.org/10.20944/preprints202005.0266.v1
[44] J.E. Tonna, H.A. Hanson, J.N. Cohan, M.L. McCrum, J.J. Horns, B.S. Brooke, R. Das, B.C. Kelly, A.J. Campbell, and J. Hotaling, Balancing revenue generation with capacity generation: Case distribution, financial impact and hospital capacity changes from cancelling or resuming elective surgeries in the us during covid-19, BMC Health Services Res. 20 (2020), no. 1, 1–7.
[45] J. Turner-Musa, O. Ajayi, and L. Kemp, Examining social determinants of health, stigma, and covid-19 disparities, Healthcare 8 (2020), 168.
[46] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), no. 1, 29—48.
[47] K. Verma, S.C. Sivastava, K. Attur, and A. Kubavat, Rationale for infection control and its applicability in dental clinics-an overview, Nat. J. Integ. Res. Med. 11 (2020), no. 4.
[48] C. Wang, P.W. Horby, F.G. Hayden, and E.F. Gao, A novel coronavirus outbreak of global health concern, Lancet, 395 (2020), no. 10223, 470—473.
[49] A. Wilder-Smith, C.J. Chiew, and V.J. Lee, Can we contain the covid-19 outbreak with the same measures as for sars?, Lancet Infect. Diseases 20 (2020), no. 5, 102–107.
[50] Y. Zhang, B. Jiang, J. Yuan, and Y. Tao, The impact of social distancing and epicenter lockdown on the covid-19 epidemic in mainland China: A data-driven SEIQR model study, MedRxiv, 2020, Doi:
10.1101/2020.03.04.20031187.
[51] Y. Zheng, E. Goh, and J. Wen, The effects of misleading media reports about covid-19 on Chinese tourists’ mental health: A perspective article, Anatolia 31 (2020), no. 2, 337-–340.
[52] Y. Zhou, Z. Ma, and F. Brauer, A discrete epidemic model for SARS transmission and control in China, Math. Comput. Model. 40 (2004), no. 13, 1491—1506.
[53] L. Zou, F. Ruan, M. Huang, L. Liang, H. Huang, Z. Hong, J. Yu, M. Kang, Y. Song, J. Xia, and Q. Guo, Sars-cov-2 viral load in upper respiratory specimens of infected patients, New England J. Med. 382 (2020), no. 2, 1177—1179.
###### Volume 14, Issue 12December 2023Pages 305-314
• Receive Date: 22 March 2022
• Revise Date: 28 April 2022
• Accept Date: 31 May 2022