A reaction-diffusion system to better comprehend the unlockdown: Application of SEIR-type model with diffusion to the spatial spread of COVID-19 in France
AA reaction-diffusion system to better comprehend theunlockdown: Application of SEIR-type model withdiffusion to the spatial spread of COVID-19 in France
Youcef Mammeri a, ∗ a Laboratoire Ami´enois de Math´ematique Fondamentale et Appliqu´ee, CNRS UMR 7352,Universit´e de Picardie Jules Verne, 80069 Amiens, France
Abstract
A reaction-diffusion model was developed describing the spread of the COVID-19 virus considering the mean daily movement of susceptible, exposed andasymptomatic individuals. The model was calibrated using data on the con-firmed infection and death from France as well as their initial spatial distribu-tion. First, the system of partial differential equations is studied, then the basicreproduction number, R is derived. Second, numerical simulations, based ona combination of level-set and finite differences, shown the spatial spread ofCOVID-19 from March 16 to June 16. Finally, scenarios of unlockdown arecompared according to variation of distancing, or partially spatial lockdown. Keywords:
COVID-19, Reaction-diffusion, SE I s I a UR model, Reproductionnumber s, Unlockdown map
1. Introduction
In late 2019, a disease outbreak emerged in the city of Wuhan, China. Theculprit was a certain strain called Coronavirus Disease 2019 or COVID-19 inbrief (Wor, 2020a). This virus has been identified to cause fever, cough, short-ness of breath, muscle ache, confusion, headache, sore throat, rhinorrhoea, chestpain, and nausea (Hui et al., 2020; Chen et al., 2020). COVID-19 belongs tothe
Coronaviridae family. A family of coronaviruses that cause diseases in hu-mans and animals, ranging from the common cold to severe diseases. Althoughonly seven coronaviruses are known to cause disease in humans, three of these,COVID-19 included, can cause severe infection, and sometimes fatal to humans.COVID-19 spreads fast. According to WHO (Wor, 2020b), it only took 67days from the beginning of the outbreak in China last December 2019 for the ∗ Corresponding author: [email protected] a r X i v : . [ q - b i o . P E ] M a y irus to infect the first 100,000 people worldwide. As of the 5th of May 2020, acumulative total of 3,601,760 confirmed cases, while 251,910 deaths have beenrecorded for COVID-19 by World Health Organization (Wor, 2020c).Last 30th of January, WHO characterized COVID-19 as Public Health Emer-gency of International Concern (PHEI) and urge countries to put in place strongmeasures to detect disease early, isolate and treat cases, trace contacts, and pro-mote social distancing measures commensurate with the risk (Wor, 2020d). Inresponse, the world implemented its actions to reduce the spread of the virus.Limitations on mobility, social distancing, and self-quarantine have been ap-plied. More than 100 countries established full or partial lockdown. All theseefforts have been made to reduce the transmission rate of the virus. For thetime being, COVID-19 infection is still on the rise.Many mathematical models have been proposed to help governments as anearly warning device about the size of the outbreak, how quickly it will spread,and how effective control measures may be. Most of the model are (discrete orcontinuous) SIR-type and few are taken into account the spatial spread.Gardner (2020) implemented a metapopulation network model described bya discrete-time Susceptible-Exposed-Infected-Recovered (SEIR) compartmentalmodel. The model gives an estimate of the expected number of cases in mainlandChina at the end of January 2020, as well as the global distribution of infectedtravelers. Wu et al. (2020) fused an SEIR metapopulation model to simulateepidemic. Danon et al. (2020) incorporated daily movements in an SEIR model,while Giuliani et al. (2020) proposed a statistical model to handle the diffusionof covid-19 in Italy.Here, spatial propagation is translated by a diffusion and the reaction termsare deduced from an extension from the classical SEIR model by adding a com-partment of asymptomatic infected (Arcede et al., 2020). We aim at predictingthe spread of COVID-19 by giving maps the basic reproductive numbers R and its effective reproductive number R eff . Afterward, we also assess possiblescenarios of unlockdown.The rest of the paper is organized as follows. Section 2, outlines our method-ology. Here the model was explained, where the data was taken, and its param-eter estimates. Section 3 contained the qualitative analysis for the model. Here,we provide the reproductive number R , then compare strategies to handle un-lockdown. Finally, section 4 outlines our brief discussion on some measures tolimit the outbreak.
2. Materials and methods
In this study, we used the publicly available dataset of COVID-19 pro-vided by the Sant´e Publique France. This dataset includes daily count of con-firmed infected cases, recovered cases, hospitalizations and deaths. Data canbe downloaded from . These data are collected by the National2ealth Agency and are directly reported public and unidentified patient data,so ethical approval is not required. The map of population density are fromG´eoportail ( ) established by the National GeographicInstitute. Data concerning transport are extracted from National Institute ofStatistics and Economic Studies ( https://statistiques-locales.insee.fr/ ). !" $ %&!" ' ( ' )* , !" + ( + )* ,( + ,- susceptible exposed asymptomatic infectedsymptomatic infected removedunder treatment ./$(1 − .)/$ 4( ' '
5- ,( ' Figure 1: Compartmental representation of the
SEI a I s UR − model. Blue arrows representthe infection flow. Purple arrows denote the death. Green compartments indicate movingindividuals. We focus our study on six components of the epidemic flow (Figure 1), i.e. the densities of Susceptible individual ( S ), Exposed individual ( E ), symp-tomatic Infected individual ( I s ), asymptomatic Infected individual ( I a ), Undertreatment individual ( U ) and Removed individual ( R ). To simplify the read-ings, treatments are not distinguished between quarantine, hospitalization ormedicine. To build the mathematical model, we followed the standard strategydeveloped in the literature concerning SIR model (Diekmann and Heesterbeek,2000; Brauer and Castillo-Chavez, 2012; Arcede et al., 2020). We assumed thatsusceptible can be infected by exposed and by infected individuals. We supposethat only susceptible, exposed and asymptotic individuals are moving. The dy-namics is governed by a system of three partial differential equations (PDE) andthree ordinary differential equations (ODE) as follows, for x = ( x, y ) ∈ Ω ⊂ R , t > ∂ t S − d ( t )∆ S = − ω ( t ) ( β e E + β s I s + β a I a ) SN∂ t E − d ( t )∆ E = ω ( t ) ( β e E + β s I s + β a I a ) SN − δE∂ t I a − d ( t )∆ I a = (1 − p ) δE − γI a I (cid:48) s = pδE − ( γ + µ + ν ) I s U (cid:48) = νI s − ( γ + µ ) UR (cid:48) = γ ( I a + I s + U ) . (1)3rontiers being now closed, homogeneous Neumann boundary conditions is im-posed. The total living population is N = S + E + I a + I s + U + R and thedeath is D = µ ( I s + U ) No new recruit is added. The parameters are describedin Table A of Figure 3. Latency period and infection period have been estimated as 5 days and 7days respectively (Lauer et al., 2020), and thus δ = 1 / , γ = 1 /
7. To accountfor the lockdown and unlockdown, the average number of contacts is updatedas follows (Liu et al., 2020) ω ( t ) = ω if t ≤ t bol ω e − ρ ( t − t bol ) if t bol ≤ t ≤ t eol (1 − η ) ω − η )e ρ ( t eol − t bol ) − − ρ ( t − t eol ) if t ≥ t eol , (2)while the diffusion coefficient is set up to d ( t ) = d if t ≤ t bol d e − ρ ( t − t bol ) if t bol ≤ t ≤ t eol d ρ ( t eol − t bol ) − − ρ ( t − t eol ) if t ≥ t eol . (3)Here bol denotes for beginning of lockdown and eol for end of lockdown. Unlock-down is assumed to be faster than lockdown. The parameter 0 ≤ η ≤ km , the value of d is fixed equal to (Okubo,1980; Shigesada and Kawasaki, 1997). Six parameters θ = ( ρ, β e , β s , β a , p, µ )remain to be determined. Given, for N days, the observations I s,obs ( t i ) and D obs ( t i ), the cost function consists of the nonlinear least square function J ( θ ) = N (cid:88) i =1 (cid:0) I s,obs ( t i ) − I s ( t i , θ ) (cid:1) + ( D obs ( t i ) − D ( t i , θ )) , with constraints θ ≥
0. Here I s ( t i , θ ) = (cid:82) Ω I s ( x , t i , θ ) d x and D ( t i , θ ) = (cid:82) Ω D ( x , t i , θ ) d x denote the output of the mathematical model at time t i computed with the pa-rameters θ . The optimization problem is solved using Approximate BayesianComputation combined with a quasi-Newton method (Csill´ery et al., 2010).4 i g u r e : A . M a p o f p o pu l a t i o nd e n s i t y f r o m G ´ e o p o r t a il. B . L e v e l - s e t a nd c o m pu t a t i o nd o m a i n c o n s i s t i n g i n t h ec a r t e s i a n g r i d . B . I n i t i a l p o pu l a t i o nd e n s i t y N ( x ) . C . C o nfi r m e d i n f ec t e d s w i t h r e s p ec tt o d e p a r t m e n t nu m b e r p r o v i d e db y S a n t ´ e P ub li q u e F r a n ce . C . I n i t i a li n f ec t i o n d e n s i t y I s , ( x ) . .4. Numerical discretization From the map of the country, the level-set (Osher and Fedkiw, 2002; Sethian,1999) is defined as the function φ such that the territoryΩ := (cid:8) x ∈ R ; φ ( x ) < (cid:9) , and its boundary is the zero level of φ The exterior normal is (cid:126)n = ∇ φ ||∇ φ || . Thecomputation domain consists in a cartesian grid given by the image pixels (Fig-ure 2). Then the map of population density allows to build initial populationdensity N ( x ). From data of confirmed infecteds, the spatial distribution ofinitial infection I s, ( x ) with respect to department number is created (Figure2). Finally, Runge-Kutta 4 is used for the time discretization and central finitedifference for the space discretization.
3. Results
Let us suppose that for all time, d ≤ d ( t ) ≤ d . We first prove that themodel is globally well posed. Theorem 3.1.
Let ≤ S , E , I a, , I s, , U , R ≤ N be the initial datum.Then there exists a unique global in time weak solution ( S, E, I a , I s , U, R ) ∈ L ∞ ( R + , L ∞ (Ω)) , of the initial boundary value problem. Moreover, the solutionis nonnegative and S + E + I a + I s + U + R ≤ N . Proof.
Let Y = ( S, E, I a , I s , U, R ). Thanks to the comparison principle andaccording to the Duhamel formulation, we look for a time T > Y ) := (cid:18) G d ∗ S − (cid:90) t G d ∗ (cid:18) ω ( β e E + β s I s + β a I a ) SN (cid:19) ds, K e ∗ E + (cid:90) t K e ∗ (cid:18) ω ( β e E + β s I s + β a I a ) SN (cid:19) ds, K s ∗ I a, + (1 − p ) δ (cid:90) t K s ∗ Eds, e − ( γ + µ + ν ) t I + pδ (cid:90) t e − ( γ + µ )( t − s ) Eds, e − ( γ + µ ) t U + ν (cid:90) t e − ( γ + µ )( t − s ) I s ds,R + γ (cid:90) t I s + I a + U ds (cid:19)
6s a contraction mapping from the closed ball (cid:40) Y = ( S, E, I a , I s , U, R ) ∈ L ∞ ( R + , L ∞ (Ω)) ; sup t ∈ [0 ,T ] || Y ( t, . ) − Y || L ∞ (Ω) < + ∞ (cid:41) onto itself. Here G d , K e , and K s are the kernels of the respective operators ∂ t − d ∆, ∂ t − d ∆ + δ , and ∂ t − d ∆ + γ for d = d or d . According to Ouhabaz(2005), there exists a constant C Ω > ||G d ( t, . ) || L (Ω) ≤ C Ω , ||K e ( t, . ) || L (Ω) ≤ C Ω , ||K s ( t, . ) || L (Ω) ≤ C Ω . Combining with the fact that the integral terms of the right-hand-side are lo-cally Lipschitz, choosing T (cid:28) C Ω allows to apply Picard’s fixed point theorem(Henry, 1981).If f = ( f , . . . , f ) denotes the right-hand-side of the system (1), and Y =( S, E, I a , I s , U, R ), since f i ( Y i = 0) ≥
0, we deduce that the solution is nonneg-ative if the initial datum is nonnegative. Finally, maximum principle providesboundedness of solution. (cid:3)
We give a condition on parameters such that the disease has an exponentialinitial growth.
Theorem 3.2.
Let ( S , E , I a, , I s, , , be a nonnegative initial datum. If thebasic reproduction number R := ω (cid:18) β e δ + (1 − p ) β a γ + pβ s γ + µ + ν (cid:19) S N > , then ( E, I a , I s ) exponentially grows. This number has an epidemiological meaning. The term β e δ represents thetransmission rate by exposed during the average latency period 1 /δ . The term (1 − p ) β a γ is the transmission rate by asymptomatic during the average infectionperiod 1 /γ , and the last one is the part of symptomatic. Proof.
A linearization around ( S , E , I a, , I s, , ,
0) is written as the linearsystem of differential equations EI a I s (cid:48) ( t ) = d ∆ + ω β e S N − δ ωβ a S N ωβ s S N (1 − p ) δ d ∆ − γ pδ − ( γ + µ + ν ) EI a I s . Let ( v k ) k ≥ be an orthonormal basis of eigenfunctions of the Laplace opera-tor with the homogeneous Neumann boundary condition, i.e. − ∆ v k = k v k .Therefore, the characteristic polynomial of the matrix − d k + ω β e S N − δ ωβ a S N ωβ s S N (1 − p ) δ − d k − γ pδ − ( γ + µ + ν ) , P ( x ) = x + a x + a x + a , with a = ( γ + µ )( d k + γ )( d k + δ ) (1 − R k )and R k := ω (cid:18) β e d k + δ + (1 − p ) β a d k + γ + pβ s γ + µ + ν (cid:19) S N . If R k >
1, there is at least one positive eigenvalue that coincides with an initialexponential growth rate of solutions. (cid:3)
To reflect the spatiotemporal dynamic of the disease, we consider the effectivereproduction number R eff ( x , t ) := ω ( t ) (cid:18) β e δ + (1 − p ) β a γ + pβ s γ + µ + ν (cid:19) S ( x , t ) N ( x , t ) , and its mean with respect to the domain Ω R eff ( t ) := 1 A (Ω) (cid:90) Ω R eff ( x , t ) d x . The value of R is computed in Table A of Figure 3.We now establish the asymptotic behavior of solution. Theorem 3.3.
With the same assumptions as Theorem 3.1. Suppose moreover ω β e ≤ δ . Then the solution converges almost everywhere to the Disease FreeEquilibrium ( S ∗ , , , , R ∗ ) with S ∗ + R ∗ = N ∗ . Proof.
From the last differential equation in system (1), we deduce that R isan increasing function bounded by N (0). Thus R ( t ) converges to R ∗ a.e. as t goes to + ∞ . Then integrating over time this equation provides R ( x , t ) − R ( x ,
0) = γ (cid:90) t I a ( x , s ) + I s ( x , s ) + U ( x , s ) ds and R ∗ ( x ) − R ( x ) = γ (cid:90) + ∞ I a ( x , s ) + I s ( x , s ) + U ( x , s ) ds, which is finite. Furthermore, I s , I a , U also go to 0 a.e. as t → + ∞ thanks to thepositivity of the solution. Multiplying the second equation by E and integratingover Ω give12 ddt (cid:90) Ω E ( x , t ) d x + d (cid:90) Ω ( ∇ E ) ( x , t ) d x = (cid:90) Ω ω ( β s I s + β a I a ) SN E +( ωβ e SN − δ ) E . Since 0 ≤ SN ≤ ω ≤ ω , Young’s inequality followed by Poincar´e inequalityprovide12 ddt (cid:90) Ω E ( x , t ) d x + dC Ω (cid:90) Ω E ( x , t ) d x ≤ (cid:90) Ω ω β s ε I s + ω β a ε I a +( ε ω β e − δ ) E . Since I s and I a go to 0 when t → + ∞ , it is enough to choose ε > dC Ω + δ − ω β e − ε >
0, to conclude that E → (cid:3) emark 3.4. The basic reproduction number R can be computed thanks to thenext generation matrix of the model without diffusion as in van den Driesscheand Watmough (2000). Since the infected individuals are in E, I a and I s , newinfections ( F ) and transitions between compartments ( V ) can be rewritten as F = ω ( β e E + β s I s + β a I a ) SN , V = δ e EγI a − (1 − p ) δE ( γ + µ + ν ) I s − pδE . Thus, R = ρ ( F V − ) of the next generation matrix F V − = ωβ e S δN + ω (1 − p ) β a S γN + ωpβ s S ( γ + µ + ν ) N ωβ a S γN ωβ s S ( γ + µ + ν ) N . No treatment is applied, then ν = 0 Calibration of the model is done fromJanuary 24, 2020, the day of first confirmed infection, to April 30, i.e.
97 days.Since R = 3 . − R = 0 . J is computed to provide a relative error of order less than 10 − . In Figure 3,Table A shows estimated parameters. Remark that ω β e ≤ δ . The rest of theFigure compares the data and the fitted total symptomatic infected and deathof the posterior distribution. B. w b e w b s w b e p µ r C. D.A. symbol description value J relative cost function 9.61E-03 w b e transmission rate from S to E from contact with E w b s transmission rate from S to E from contact with I s w b a transmission rate from S to E from contact with I a r lockdown decay 0.043198 d latency rate 1/5 p probability of be symptomatic 0.503939 probability of be asymptomatic 0.496061 g recovery rates 1/7 µ death rate 0.010381 R initial effective reproduction number 3.425257median R median effective reproduction number 3.500671mean R mean effective reproduction number 3.454132 Figure 3: A. Parameters calibrated according to data from Sant´e Publique France. B. Boxplotof the posterior distribution computed from these data. C. Fitted total symptomatic infectedof the posterior distribution in grey , median in red straight line, mean is dotted line. D.Fitted total death of the posterior distribution in grey, median in black straight line, mean isdotted line. .3. Spatial spread of covid-19 Simulations are performed from January 24, 2020, the day of first confirmedinfection, to June 16. Images are 957 ×
984 pixels, time step is chosen to verifythe stability condition preserving positivity, and η = 0 . η = 0 . a r c h M a y J un e A . B . B . C . C . w i t h l o c k d o w n w i t h o u t l o c k d o w n F i g u r e : A . Sp a t i a l d i s t r i bu t i o n o f s y m p t o m a t i c i n f ec t e dd e n s i t y o n M a r c h , o n e d a y b e f o r e l o c k d o w n . B . Sp a t i a l d i s t r i bu t i o n o f s y m p t o m a t i c i n f ec t e dd e n s i t y o n M a y ( B ) , o n e d a y b e f o r e un l o c k d o w n , a nd o n J un e ( B ) . C . Sp a t i a l d i s t r i bu t i o n o f s y m p t o m a t i c i n f ec t e dd e n s i t y o n M a y ( B ) a nd o n J un e ( B ) a ss u m i n g t h a t n o l o c k d o w nh a s b ee nd o n e . .4. Strategies for unlockdown A naive strategy is to unlock the regions where R eff ( x , . ) is the lowest. Look-ing at the maps in Figures 5, we see that the effective reproduction numberbefore the lockdown is between 3 .
34 and 3 .
42 (Figures 5-A). On May 10, itgoes down and is between 0 .
38 and 0 .
42 (Figures 5-B). After May 10, with adistancing mostly respected ( η = 0 . R eff ( x , . ) increases again. It exceeds 1with lower value (1 .
04) in the regions that have been most affected upstream(Figures 5-C).
March 16 May 10 June 16 h = 0.6 h = 0.7 h = 0.8 h = 0.6 h = 0.7 A. B.A. B. C. h = 0.8 h = 0.63 Figure 5: Spatial distribution of effective reproduction number one day before lockdown March16 (A), one day before unlockdown May 10 (B), and on June 16 (C).
A less forced strategy is to believe in the respect of distancing after lockdown.With this set of parameters, R eff goes to 1 when η = 0 .
63. As shown in Figure6-A, provided that at least 63% of the individuals respect the distancing rules,the number of symptomatic infected individuals is controlled. Figure 6-B showsthat in this case, the effective reproduction number R eff remains less than 1.Below 63% regarding distancing, the number of symptomatic infected increasesagain and R eff becomes greater than 1.A third strategy, more constraining for some, is to continue the lockdown inthe most affected areas. Here, lockdown is continued on the eastern half, whilethe western half is free. Then the number of symptomatic infected and theeffective reproduction number pursue their decay as shown in Figure 6 (dottedlines). It is important to note that in the simulation no individuals move froma confined to an unconfined region. A homogeneous Neumann condition, equalto 0, is imposed along this fictitious frontier.12 arch 16 May 10 June 1 h = 0.6 h = 0.7 h = 0.8 h = 0.6 h = 0.7 A. B.A. B. C. h = 0.8 h = 0.63 Figure 6: A. Total number of symptomatic infected with η = 0 . , . , . η = 0 . , . , .
4. Discussion
Lockdown has reduced both the number of infections and the spread. Thepropagation focused in the Eastern half and kept the West intact. Withoutlockdown, the whole country would be affected and more severely. Lockdowncaused a significant reduction of the reproduction number, from 3 .
42 to 0 . R eff grows and can exceed 1.With a lack of treatment, social distancing remains the most effective means.We notice that it has to be highly respected (here at over 63%). As shown in Fig-ure 6-A, the number of symptomatic infected individuals, therefore potentiallyhospitalized, is restrained as soon as at least 63% of the individuals respect thedistancing, Nevertheless, this constraint can be relaxed since it can be imposedonly in the most infectious areas.In summary, to obtain a possible unlockdown map, the local value of the ef-fective reproduction number should be taken into account, as well as the numberof infected individuals and the direction of the spread of the disease.Of course, this is a simplified model based only the population density andmean daily commuting. For example, the model could be improved by consid-ering a larger diffusion along major axes of travel ( i.e. d = d ( x , t )), by takinglocal effects of distancing ( e.g η = η ( x , t ) where η ( x , t ) = 0 in closed schools),or by opening of the frontiers (by changing the boundary conditions and addingnew recruits). Supplementary video
Spatiotemporal propagation of COVID-19 from March16 to June 16 with lockdown occurring from March 17 to May 11 (left) andwithout intervention (right).
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