Abstract
Nowadays mathematical models play a major role in epidemiology since they can help in predicting the spreading and the evolution of diseases. Many of them are based on ODEs on the assumption that the populations being studied are homogenous sets of fixed points (individuals) but actually populations are far from being homogenous and people are constantly moving. In fact, thanks to science progresses, distances are no longer what they used to be in the past and a disease can travel and reach out even the most remote places on the globe in a matter of hours. HIV and Covid-19 outbreaks are perfect illustrations of how far and fast a disease can now spread. When it comes to studying the spatio-temporal spreading of a disease, instead of ODEs dynamic models the Reaction-Diffusion ones are best suited. They are inspired by the second Fick’s law in physics and are getting more and more used. In this article we make a study of the spatio-temporal spreading of the COVID-19. We first present our SEIR dynamic model, we find the two equilibrium points and an expression for the basic reproduction number (R0), we use the additive compound matrices and show that only one condition is necessary to show the local stability of the two equilibrium points instead of two like it is traditionally done, and we study the conditions for the DFE (Disease Free Equilibrium point) and the EE (Endemic Equilibrium point) to be globally asymptotically stable. Then we construct a diffusive model from our previous SEIR model, we investigate on the existence of a traveling wave connecting the two equilibrium thanks to the monotone iterative method and we give an expression for the minimal wave speed. Then in the last section we use the additive compound matrices to show that the DFE remains stable when diffusion is added whereas there will be appearance of Turing instability for the EE once diffusion is added. The conclusion of our article emphasizes the importance of barrier gestures and the fact that the more people are getting tested the better governments will be able to handle and tackle the spreading of the disease.
Highlights
The COVID-19 was declared a pandemic by the WHO on the 30th Juanary 2020
When diffusion is added to a dynamic model it can radically change the nature of the equilibrium points and generate diffusion-driven (Turing) instabilities [12,13,14]
If the disease free equilibrium (DFE) is stable the dynamic model remains stable under the condition R0 ≺ 1 even if a diffusion term is introduced and there will not be any appearance of a traveling wave solution but some conditions must be fulfilled:
Summary
The COVID-19 was declared a pandemic by the WHO on the 30th Juanary 2020. The responsible agent is a coronavirus (SARS-Cov2) that spreads between people thanks to close contacts, usually via droplets produced by coughing, sneezing or talking. We consider only four classes: the susceptibles, the asymptomatic infected indIinvteirdnuataiolnsa,l Jtohuernaslyomf SpytstoemmsaStciicenicne faencdtAepdpliienddMivatihdeumaatliscsa2n02d1;t6h(1e):r2e2m-34oved. Other approaches are regularly used to investigate on the existence existence of a traveling amnohdoeslspiwtailtwoihzraa5dtviteooern8e[cd5cla,lass7ssus]aeprsbee[uor2t-f0ts-eho2n6leut]ra.tekKieownnnoeiswn.tuionsgaWechcteoohwuecncothmnlaeslaolidendinnoeggtrinotgotnheaituwtseanervdoaett[omi5v,iean7vjm]oesrbetuitgathachtoteedirooennbwyteihsesuesteteaxtikitsnheteegnnmcuteoopnexoasittsotopnepeanciirettehrooaefftiavetrmaveetlhinogd it is to finsdpfrreoantdtirnagve,litnhgerweafvoersefowreRh-Davmeondeolsqoufatrharneetinebyansedttipngeoupplea paarier osftiolrldefrreede stuopemr-soovluet.ionTs.heWe consider or more thinantetrharecetioeqnusatbioentsw, eween htahve fcohuosrenclatossbeusildarae givtheant nion mthajeorFaicgtiuonreis1taken to stop the spreading, simpler model with only four classes An asymptomatic infeFctiegduirndeiv1id.uaDl iysnaanminicfecgtrioaupsh o2f .COAVRIDe-a1c9ttiorannsMmiossdioenl of COVID-19 person presenting no or very few symptoms; Asymptomatic infected individual is an infectious person parensdenttihnge saysmsputmompstoiof nCsOVwIeD-m19a;ke are the foslluobCw-opinonspigud:leartiaonpsoapnudlatdieonnowteitthhesiirzefraNct.ioWnsebcyanS,dEiv,idIe, it into and R. Asymptomatic tionmfecsteodf iCndOivVidIuDal-1o9r; a symptomatic reaction model of COVID-19 as follows: infected (i4n)divEidvuearl,y choe/nsthaectwwillithgoanthirnoufegchtioauns asymptomatic mstaitsesihoencaonf rSeAmaRinS-inCtoovu2n;til he/she is totally healed or he/she will leave that state as soon as person does. We do not take into account reinfection by COVID-19
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