A new fractional mathematical modelling of COVID-19 with the availability of vaccine

The most dangerous disease of this decade novel coronavirus or COVID-19 is yet not over. The whole world is facing this threat and trying to stand together to defeat this pandemic. Many countries have defeated this virus by their strong control strategies and many are still trying to do so. To date, some countries have prepared a vaccine against this virus but not in an enough amount. In this research article, we proposed a new SEIRS dynamical model by including the vaccine rate. First we formulate the model with integer order and after that we generalise it in Atangana-Baleanu derivative sense. The high motivation to apply Atangana-Baleanu fractional derivative on our model is to explore the dynamics of the model more clearly. We provide the analysis of the existence of solution for the given fractional SEIRS model. We use the famous Predictor-Corrector algorithm to derive the solution of the model. Also, the analysis for the stability of the given algorithm is established. We simulate number of graphs to see the role of vaccine on the dynamics of the population. For practical simulations, we use the parameter values which are based on real data of Spain. The main motivation or aim of this research study is to justify the role of vaccine in this tough time of COVID-19. A clear role of vaccine at this crucial time can be realised by this study.

[1]  E. Demirci,et al.  A Fractional Order SEIR Model with Density Dependent Death Rate ABSTRACT | FULL TEXT , 2011 .

[2]  N. Bairagi,et al.  Mathematical perspective of Covid-19 pandemic: Disease extinction criteria in deterministic and stochastic models , 2020, Chaos, Solitons & Fractals.

[3]  Delfim F. M. Torres,et al.  Fractional model of COVID-19 applied to Galicia, Spain and Portugal , 2021, Chaos, Solitons & Fractals.

[4]  Ndolane Sene,et al.  SIR epidemic model with Mittag–Leffler fractional derivative , 2020 .

[5]  K. N. Nabi,et al.  Forecasting of COVID-19 pandemic: From integer derivatives to fractional derivatives , 2020, Chaos, Solitons & Fractals.

[6]  Fanhai Zeng,et al.  The Finite Difference Methods for Fractional Ordinary Differential Equations , 2013 .

[7]  Delfim F. M. Torres,et al.  Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan , 2020, Chaos, Solitons & Fractals.

[8]  A. Guirao.  The Covid-19 outbreak in Spain. A simple dynamics model, some lessons, and a theoretical framework for control response , 2020, Infectious Disease Modelling.

[9]  D. G. Prakasha,et al.  A new study of unreported cases of 2019-nCOV epidemic outbreaks , 2020, Chaos, Solitons & Fractals.

[10]  A. Atangana,et al.  New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model , 2016, 1602.03408.

[11]  Dumitru Baleanu,et al.  Analysis and Dynamics of Fractional Order Mathematical Model of COVID-19 in Nigeria Using Atangana-Baleanu Operator , 2021, Computers, Materials & Continua.

[12]  Ivo Petráš,et al.  Simulation of Drug Uptake in a Two Compartmental Fractional Model for a Biological System. , 2011, Communications in nonlinear science & numerical simulation.

[13]  A. Rinaldo,et al.  Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures , 2020, Proceedings of the National Academy of Sciences.

[14]  A. Iomin,et al.  Toy model of fractional transport of cancer cells due to self-entrapping. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Hannah R. Meredith,et al.  The Incubation Period of Coronavirus Disease 2019 (COVID-19) From Publicly Reported Confirmed Cases: Estimation and Application , 2020, Annals of Internal Medicine.

[16]  Hamadjam Abboubakar,et al.  A malaria model with Caputo-Fabrizio and Atangana-Baleanu derivatives , 2020, Int. J. Model. Simul. Sci. Comput..

[17]  Muhammad Altaf Khan,et al.  Modeling the impact of non-pharmaceutical interventions on the dynamics of novel coronavirus with optimal control analysis with a case study , 2020, Chaos, Solitons & Fractals.

[18]  Pushpendra Kumar,et al.  A case study of Covid‐19 epidemic in India via new generalised Caputo type fractional derivatives , 2021, Mathematical methods in the applied sciences.

[19]  V. E. Tarasov.  Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media , 2011 .

[20]  D. Cucinotta,et al.  WHO Declares COVID-19 a Pandemic , 2020, Acta bio-medica : Atenei Parmensis.

[21]  K. N. Nabi,et al.  Projections and fractional dynamics of COVID-19 with optimal control strategies , 2021, Chaos, Solitons & Fractals.

[22]  D. Baleanu,et al.  On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag–Leffler kernel , 2018 .

[23]  Ritesh Gupta,et al.  Comorbidities in COVID-19: Outcomes in hypertensive cohort and controversies with renin angiotensin system blockers , 2020, Diabetes & Metabolic Syndrome: Clinical Research & Reviews.

[24]  Franco Blanchini,et al.  Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy , 2020, Nature Medicine.

[25]  Delfim F. M. Torres,et al.  Corrigendum to “Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan” [Chaos Solitons Fractals 135 (2020), 109846] , 2020, Chaos, Solitons & Fractals.

[26]  Vedat Suat Ertürk,et al.  A mathematical study of a tuberculosis model with fractional derivatives , 2021, Int. J. Model. Simul. Sci. Comput..

[27]  Vasily E. Tarasov,et al.  Fractional-order difference equations for physical lattices and some applications , 2015 .

[28]  V. S. Erturk,et al.  Prediction studies of the epidemic peak of coronavirus disease in Brazil via new generalised Caputo type fractional derivatives , 2021, Alexandria Engineering Journal.

[29]  Jiyuan Zhang,et al.  Pathological findings of COVID-19 associated with acute respiratory distress syndrome , 2020, The Lancet Respiratory Medicine.

[30]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[31]  P. Colaneri,et al.  Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy , 2020, Nature Medicine.

[32]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[33]  Nauman Ahmed,et al.  New applications related to Covid-19 , 2020, Results in Physics.

[34]  Pushpendra Kumar,et al.  The analysis of a time delay fractional COVID‐19 model via Caputo type fractional derivative , 2020, Mathematical methods in the applied sciences.

[35]  R. Verma,et al.  A numerical simulation of fractional order mathematical modeling of COVID-19 disease in case of Wuhan China , 2020, Chaos, Solitons & Fractals.

[36]  Khalid Hattaf,et al.  A fractional order SIR epidemic model with nonlinear incidence rate , 2018, Advances in Difference Equations.

[37]  Abdon Atangana,et al.  Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative , 2020, Alexandria Engineering Journal.

[38]  Vedat Suat Erturk,et al.  Environmental persistence influences infection dynamics for a butterfly pathogen via new generalised Caputo type fractional derivative , 2021 .

[39]  V. S. Erturk,et al.  Solution of a COVID-19 model via new generalized Caputo-type fractional derivatives , 2020, Chaos, Solitons & Fractals.

[40]  D. Baleanu,et al.  NEW GENERALIZATIONS IN THE SENSE OF THE WEIGHTED NON-SINGULAR FRACTIONAL INTEGRAL OPERATOR , 2020 .