On a Reaction-Diffusion Model of COVID-19

: 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 ( R 0 ), 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.


Introduction
The COVID-19 was declared a pandemic by the WHO on the 30 th 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. The droplets usually fall onto surfaces or to the ground rather than remaining in the air making it also possible for people to be infected by touching a contaminated surface or any contaminated object. According to the updated information available [27], incubation period ranges from 2 to 14 days and the main symptoms are fever, loss of appetite, shortness of breath, cough, fatigue, muscle aches and pain.
The majority of the infected individuals are asymptomatic and tend not to be tested thou they do play a role in the spreading of the disease. The recovery time which usually ranges from 2 to 6 weeks differs from person to person and it happens that even after that period some people still complain to not be fully recovered.
Depending on the main purpose, dynamic models usually try to encapsulate as much as possible important features of the disease in the simplest way [3][4][5][6][15][16][17][18][19] to provide a comprehensive view on the disease dynamic. That is why the majority of the current models on COVID-19 are very detailed in classes. For instance a quarantined class and/or an hospitalization class are often taken into account leading to models with 5 to 8 classes [20][21][22][23][24][25][26]. Knowing how challenging it is to find front traveling waves for R-D models of three or more than three equations, we have chosen to build a simpler model with only four classes (the susceptibles, the asymptomatic infected individuals, the symptomatic infected individuals and the removed). other approaches are regularly used to investigate on the existence existence of a traveling wave [5,7] but here we use the monotone iterative method by setting up a pair of ordered super-solutions. We consider that no major action is taken to stop the spreading, therefore we have no quarantine and people are still free to move. The interactions between the four classes are given in the Figure 1 waves for R-D models of three or more than three equations, we have chosen to build a simpler model that includes the maximum important features of the disease dynamics. We then consider only four classes: the susceptibles, the asymptomatic infected individuals, the symptomatic infected individuals and the removed. other approaches are regularly used to investigate on the existence existence of a traveling wave [5,7] but here we use the monotone iterative method by setting up a pair of ordered super-solutions. We consider that no major action is taken to stop the spreading, therefore we have no quarantine and people are still free to move. The interactions between the four classes are given in the Figure 1  (1) Every new-born is susceptible i.e there are only horizontal transmission; (2) An asymptomatic infected individual is an infectious person presenting no or very few symptoms; (3) A symptomatic infected individual is an infectious person presenting symptoms of COVID-19; (4) Every contact with an infectious person does not always lead to a transmission of SARS-Cov2; and the assumptions we make are the following: 1. Every new-born is susceptible i.e there are only horizontal transmission; 2. An asymptomatic infected individual is an infectious person presenting no or very few symptoms; 3. A symptomatic infected individual is an infectious person presenting symptoms of COVID-19; 4. Every contact with an infectious person does not always lead to a transmission of SARS-Cov2; 5. After an infectious contact there is always an incubating period but we do not take it into account here; 6. After a susceptible has been infected by either an asymptomatic infected individual or a symptomatic infected individual, he/she will go through an asymptomatic state he can remain into until he/she is totally healed or he/she will leave that state as soon as sufficient symptoms begin to appear; 7. A symptomatic infected individual can either die of COVID-19 or get healed; 8. We do not take into account reinfection by COVID-19. 9. The entire population has a per-capita death rate independent of COVID-19.

A Reaction Model of COVID-19
Consider a population with size N . We can divide it into sub-populations and denote their fractions by S, E, I, and R which respectively represent the fraction of susceptible, the fraction of asymptomatic infected individuals, the fraction of symptomatic infected individuals and the fraction of removed. Thus the sub-populations verify the identity: S + E + I + R = 1. Based on the assumptions made previously we can set a reaction model of COVID-19 as follows: The coefficients used into our model are explained in the following table: The last equation in (1) does not intervene into the transmission of the disease, we can simplify our system by reducing it into three equations as follows: To ensure the well posedness of the system we consider the proportion of the population in: 2.1. Equilibrium points and R 0 The equilibrium points areū for the disease free equilibrium (DFE) and for the endemic equilibrium (E.E).
To find R 0 we use the next generation operator [18,21]. Our DFE is given byū = ( Λ d , 0, 0) = (1, 0, 0) due to the fact that at this equilibrium point the entire population is susceptible i.e the fraction of the healthy people is 1. So we obtain the following sub-matrices Then Hence the basic reproduction number is: Proof. Let us suppose that R 0 ≤ 1 then we have the existence of the EE Thus u * = M N β+ηM , 0, 0 . for the proportion of the population being entirely in the first component we have Thus S * = Λ d and the unique equilibrium point in this situation is the disease free one.
≺ 0. Both components must be positive to ensure the existence of an endemic equilibrium, therefore there is none.
Let us now suppose that R 0 Thus there exists an endemic equilibrium u * as defined in (5).

Stability of the Equilibria
then the DFE given in (4) is locally asymptotically stable in G.
then the E.E given in (5) is locally asymptotically stable in G.
Proof: It suffices to show that the eigenvalues of the two Jacobian matrices at the two equilibria have real negative part. Next we use a property of the additive compound matrices to state: Theorem 2.3. Let Jū and J u * be the Jacobian matrices at the DFE and the EE. If − |Jū| 0 then the DFE given in (4) is locally asymptotically stable in G.
then the E.E given in (5) is locally asymptotically stable in G.
Proof: To show thatū is stable we must prove under which conditions − |Jū| 0 and µ(J The second compound matrix of Jū is given by: Let us use µ 1 as our Lozinskǔ mesure with From the first column we have: We proceed the same way for the second and the third columns and find respectively: Two conditions are necessary to the stability of the equilibriumū. The first one is the sign of the Jacobian Jū.
0 then R 0 ≺ 1 and this condition is necessary for the local stability ofū. If µ J we get a condition on the parameters and this has no much meaning and impact for our model. Theorem 2.4. When η ≤ β and R 0 ≤ 1, then disease-free equilibriumū for (2) is globally asymptotically stable. Proof We use an approach given by Zhisheng Shuai and P. Van Den Driessche to construct our Lyapounov function [22,29] . Let F, V and V −1 defined like in (7) and (8).
If w T = (x y) denotes the left eigenvector of V −1 F then we have and w T = (1, 1).
Algorithms on the calculation of µ(J [2] u ) are given in [8][9][10] and the Lyapounov function is given by: We have Since From the hypothesis, R 0 ≤ 1 and from (13)  The endemic equilibrium u * for (2) is globally asymptotically stable when R 0 1 Proof. Let Our Lyapounov function is given by: The associated weighted diagram given in Figure 2 has three vertices and two cycles. Along each cycle, G 21 +G 32 +G 13 = 0 and G 21 + G 12 = 0. Then there exist

A Reaction-Diffusion Model on COVID-19
Assume now that the individual in the population can move (diffuse) with the same diffusion coefficient. If the susceptibles and the asymptomatic infectious are free to move the same way, we suppose that the symptomatic infectious still have contacts with people able to diffuse. Then we can formulate our R-D model like: Using wave coordinates ξ = x + ct in (18) yields: Asymptotically the system (19) satisfies the following boundary conditions: Linearizing (19) about ( Λ d , 0, 0) = (1, 0, 0)we obtain: The second equation in (22) provides the speed of the wave. In fact its characteristic equation is: To ensure the existence of real solutions we must have Hence the minimal speed is 23) and the roots to the characteristic equation are: The solutions of the first equation are: and those of the third: Hence the profile of the traveling wave solution to (22) is

Existence of a Traveling Wave
To prove the existence of a front traveling wave solution to (22) we shall use the monotone iterative method which relies on the following principle: Principle 4.1.
[Monotone Iterative Method] Consider the general second order ODE with Dirichlet boundary conditions value given by : with f : I × R 2 −→ R a continuous function and A, B ∈ R. If in C 2 (I) there exist U (t) a lower solution to (25) and U (t) an upper solution to (25) such that U (t) ≤ U (t) on I. Then the existence of a solution to the problem (25) lying between U (t) and U (t) is proved. Lemma 4.1. Let X(ξ) = (S(ξ), E(ξ), I(ξ)) = (0, 0, 0), then X is a lower solution to (19).
Proof It is obvious the last two equations of (19) vanish and for the first one we have Λ ≥ 0.
For the first equation of (19) we have: For the second equation of (19) we have: by their definitions. For the third equation of (19) we have: Now if ξ 0, then: International Journal of Systems Science and Applied Mathematics 2021; 6(1): 22-34 29 For the first equation of (19) we have: For the second equation of (19) we have: For the third equation of (19) we have: (4.8) and (4.9) are obtained due to the fact that (S * , E * , I * ) is an equilibrium to (2). For the boundary conditions we have: Hence, for both values ξ 0 and ξ ≤ 0, X(ξ) is an uppersolution to (19).
Theorem 4.1. If R 0 1 then there exists a traveling wave solution to (19) with a minimal speed c * = 2 √ η − N . If R 0 ≺ 1 then there does not exist any traveling wave solution to (19).

Turing Instability
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]. In this section we also use the additive compound matrices to investigate whether there will be appearance of Turing instability.
Theorem 5.1. Suppose 0 ≺ β N −η ≺ M , the DFE will be locally asymptotically stable for all diffusion matrix D 0.

Proof
The principal minor matrices to Jū are From (a) and (b) we know that Jū satisfy the minor conditions if 0 ≺ β N −η ≺ M i.e under that condition the DFE will remain locally asymptotically stable even if diffusion is introduced in the Reaction model (2).
Theorem 5.2. There will always be a Turing instability at the EE defined in (5) for all diffusion matrix D 0.

Proof
The principal minor matrices of J u * are |K 3 | = 0 implies that the minor conditions will never be satisfied on J u * . Then if diffusion is introduced into the Reaction model (2) we will have a Turing instability for u * .

Conclusion
The model we built and studied can provide capital information on the dynamic of the pandemic: The expression given in (23) enable us to calculate the minimum speed rate for the appearance of a wave spreading the disease, the stability analysis of the equilibrium points have shown different situations that can occur.
If the DFE is stable then the dynamic model remains stable under the condition R 0 ≺ 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: • The contact rate given in β must be reduced. Quarantine and barrier gestures remain the best means so far; • In the absence of an effective vaccine, it is important to reinforce immunity within populations by maintaining symptomatic individuals alive long enough for them to acquire immunity; • From both the transfer rate and the contact rate η we understand that more people should be tested especially the asymptomatic infectious kipping in mind that they seem to be the most dangerous in the spreading of the pandemic since they show no symptoms of the disease.
When R 0 1 we have the existence of an endemic equilibrium point the E.E and when diffusion is introduced there is appearance of a Turing instability.
We have fluctuations in the number of infectious individuals even if R 0 1 i.e the EE will somehow lose its asymptotic behavior.

Simulations
We suppose an asymptomatic infected is likely to be in contact with more people than a symptomatic one because he shows no signs for people to be suspicious. The two infection rates are obtained by β = s 1 .p and η = s 2 .p where p is the percentage of contamination for an infected individual in one day, s 1 is the number of contacts that a symptomatic infected person can meet in one day and s 2 is the number of contacts that an asymptomatic infected person can meet in one day. The initial condition is given by X 0 = ( N −100 N , 50 N , 30 N , 20 N ) and from the literature (see [20]- [26]) we can give the following table containing the estimated values of the parameters.       We can see that even if the number of contacts for symptomatic infected has been strongly reduced, we still have lots of infections due to the asymptomatic infected contact number which is still high. We can also see it slows down the infection waves.