key: cord-0838250-2zwp5lsg
authors: Cherniha, Roman; Davydovych, Vasyl'
title: A reaction-diffusion system with cross-diffusion: Lie symmetry, exact solutions and their applications in the pandemic modeling
date: 2020-12-15
journal: European Journal of Applied Mathematics
DOI: 10.1017/s095679252100022x
sha: 5bc9dd1a1fb6041c2335c0d7f24a12b5d77ba2fc
doc_id: 838250
cord_uid: 2zwp5lsg

A nonlinear reaction-diffusion system with cross-diffusion describing the COVID-19 outbreak is studied using the Lie symmetry method. A complete Lie symmetry classification is derived and it is shown that the system with correctly-specified parameters admits highly nontrivial Lie symmetry operators, which do not occur for all known reaction-diffusion systems. The symmetries obtained are also applied for finding exact solutions of the system in the most interesting case from applicability point of view. It is shown that the exact solutions derived possess all necessary properties for describing the pandemic spread under 1D approximation in space and lead to the distributions, which qualitatively correspond to the measured data of the COVID-19 spread in Ukraine.

The outbreak of the coronavirus called COVID-19 in China has attracted extensive attention of many mathematicians working in mathematical modeling. The first papers were already published in February-April 2020 (see, e.g., [1] [2] [3] [4] [5] [6] [7] . At the present time, the COVID-19 outbreak is already spread over the world as a pandemic. There were 65.5 mln. coronavirus cases and almost 1.5 mln. deaths caused by this coronavirus up to date December 2 [8] .

Nowadays, there are many mathematical models used to describe epidemic processes and they can be found in any book devoted to mathematical models in biology and medicine (see, e.g., [9] [10] [11] [12] and papers cited therein). The paper [13] is one of the first papers in this direction. The authors created a model based on three ordinary differential equations (ODEs), which nowadays is called the SIR model. There are several generalizations of the SIR model and the SEIR model (see the pioneering works [14, 15] ), which involves four ODEs, is the most common among them. These two models are mostly used for numerical simulations in mathematical modeling the COVID-19 outbreak (see, e.g., [2, 4, 5] ).

On the other hand, one may note that the spread of many epidemic processes, including the COVID-19 pandemic, is often highly non-homogenous in space. This fact can be taken in different ways but the most common approach consists in dividing the large domain (say a country) into many small sub-domains (regions of the country) and to apply the standard models based on ODEs to each sub-domain. However, there is another way -to use the reaction diffusion equations in order to model the spread of the infected population as a diffusion process [16, 17] (see also earlier papers cited therein). A possible model was also suggested in our previous work [18] . The model has the form

where the lower subscript t means differentiation with respect to (w.r.t.) this variable, ∆ is the Laplace operator, u = u(t, x, y) and v = v(t, x, y) are two unknown functions, k(t) is the given smooth positive function, d 1 and d 2 are diffusivities. The function u(t, x, y) describes the density (rate) of the infected persons (the number of the COVID-19 cases) in a vicinity of the point (x, y), while v(t, x, y) means the density of the deaths from COVID-19. The diffusivity coefficients d 1 and d 2 describe the random movement of the infected persons, which lead to increasing the pandemic spread. Formally speaking, one may take d 1 = d 2 . However we believe that d 1 > d 2 because the movement of the infected persons leads firstly to higher rate of new COVID-19 cases but only some of them cause new deaths. Each coefficient in the reactions terms, a, b, γ and k(t), has the clear meaning described and verified in [18] (see Pages 2 and 3 therein).

Of course, this model is an essential simplification because many factors causing the spread of COVID-19 are not taking into account. In paper [16] , for example, the authors construct the diffusion model, which is essentially based on the SEIR model. As a result, their model consists of five PDEs, which can be analyzed only using numerical methods. Our idea was to construct a simpler model, which can be solved using analytical approaches, in particular, the Lie symmetry method [19] [20] [21] and to show its applicability for the spread of the coronavirus pandemic. It is interesting to note that equations (3) and (5) from the model developed in [16] under the natural assumptions produce equations with a similar structure to those in (1). In fact, the density of infected population is proportional to that of the total living population and is proportional to the exposed population density (generally speaking, the relevant coefficients are some functions but we keep constants). Having such assumptions, one arrives at the system (in our notations)

The difference between (1) with γ = 1 and (2) consists only in the diffusion terms. In our model, the diffusivity is taking to be a constant, while one is a linear function in paper [16] . Notably, the diffusivity is a time-dependent function in paper [17] .

The remainder of this paper is organized as follows. In Section 2, a complete Lie symmetry classification (LSC) (group classification) of system (1) is derived. In particular, we have proved that there are systems with correctly-specified parameters d 1 , d 2 , a and γ when the system in question admits highly nontrivial Lie symmetry, which have no analogs for other known reaction-diffusion systems. The results were obtained using the Lie-Ovsiannikov method [19] , which is a combination of the classical Lie method and the technique for finding equivalence transformations (ETs). The modern description of this method, its extension and applications can be found in [20] (Chapter 2).

In Section 3, exact solutions of the specified system of the form (1) are constructed using its Lie symmetry operators. In particular, the traveling wave type solution is derived and its applicability is extensively discussed. It is shown that this exact solution describes adequately the spread of the coronavirus pandemic provided the 1D approximation of space is assumed. Finally, we discuss the main results of the paper in the last section.

In this section, it is identified that the basic system (1) for the pandemic modeling possesses a very reach Lie symmetry depending on the parameters γ, d 1 and d 2 and the function k(t). First of all, we note that for the LSC we need only the restrictions d 2 1 + d 2 2 = 0 (otherwise the system in question degenerates into the ODE system, which was solved in [18] ), k = 0 (otherwise the system in question seems to be useless for applications) and b = 0 and γ = 0, −1 (otherwise the system in question is linear, hence is also integrable).

First of all, we present a statement about the group of ETs of system (1) . For this purpose we apply the technique, which was developed in [22, 23] (see also Section 2.3 in [20] ).

The group of the continues ETs transforming system (1) to that with the same structure, i.e.,

is the infinite-parameter Lie group generated by the transformations

Here α i (i = 0, 1, 2) and β j (j = 0, . . . , 4) are the real group parameters with the restrictions β 1 β 2 β 4 = 0, β 3 > 0, and f (x, y) is an arbitrary smooth function.

In order to obtain system (3) with the nonnegative parameters (this is the biologically motivated requirement explained above), the additional restrictions β 1 > 0 and β 4 > 0 should take place.

Sketch of Proof of Theorem 1 is based on the known technique for constructing the group of ETs. It is nothing else but a modification of the classical Lie method. In the case of system (1), one should start from the infinitesimal operator

being ξ 0 , ξ 1 , η 1 , η 2 and ζ to-be-determined functions, while µ i (i = 1, . . . , 5) to-be-determined constants. The operator E involves the additional terms with the coefficients µ i and ζ, because d 1 , d 2 , a, b, γ and k(t) should be treated as a new variables. In order to find the operator E, we should apply Lie's invariance criteria to the system of equations consisting of (1) and a set of differential consequences of k(t) w.r.t. the variables t, x, u, v. Of course, each consequence is equal to zero, excepting ∂k ∂t = k ′ (t) (the latter is not useful because it is identity). As a result, we obtain a multicomponent system consisting equations from (1) and primitive equations like ∂k ∂x = 0. Applying to this system Lie's invariance criteria, i.e., the second prolongation of the infinitesimal operator E, for deriving the system of determining equations, the coefficients ξ 0 , ξ 1 , η 1 , η 2 , ζ and µ i (i = 1, . . . , 5) were found. They have the form

where C i , i = 1, . . . , 8 are arbitrary constants and h(x, y) is an arbitrary smooth function. The operator (5) with the coefficients (6) generates the Lie group (4) .

The sketch of the proof is now completed.

In order to provide a complete LSC of system (1), one should to identify the principal algebra of invariance (see definition, for example, in [20] , page 23) from the very beginning. In fact, system (1) involves an arbitrary function k and several parameters (some of them can vanish). Thus, it should be considered as a class of systems of partial differential equations (PDEs), if one is going to provide a rigorous LSC.

The principal algebra of invariance of system (1) is infinite-dimensional Lie algebra generated by the operators:

where F (x, y) is an arbitrary smooth function.

The proof of this statement can be derived in different two ways. The direct approach consists in application of the classical Lie method to system (1), assuming that all parameters are arbitrary. The second way is useful if the group of ETs is known. So, having Theorem 1, we simply calculate when transformations (4) transform (1) in itself, i.e., system (3) coincides with (1) . The result immediately leads to formulae (7) . Now we present two main theorems, which completely solve the LSC problem for (1). It turns out that there are two essentially different cases, d 1 = 0 and d 1 = 0, leading to absolutely different results.

Theorem 3 System (1) with d 1 = 0 admits the extension of the principal algebra (7) only in five cases. These cases and the corresponding Lie symmetry operators are as follows

Here p is arbitrary constant.

Any other system (1) with d 1 = 0 admitting an extension of the principal algebra (7) is reduced by an ET from (4) to one of the listed in cases 1)-5).

Remark 2 Using the simple ET from (4), one can set k 0 k(t) (k 0 is an arbitrary constant) instead of k(t) in each case of Theorem 3 without any changes in Lie symmetry operators. It is useful from the applicability point of view.

Theorem 4 System (1) with d 1 = 0 (then automatically d 2 = 0) admits the extension of the principal algebra (7) only in four cases. In each case the additional operators have the structure

where the functions ξ 1 , ξ 2 form an arbitrary solution of the famous Cauchy-Riemann system

and G is an arbitrary solution of the linear first-order PDE

In the operator X, the functions ξ 0 and η 1 depending on the parameters γ and a have the forms : 1) if γ = 1 and a = 0 then

2) if γ = 1 and a = 0 then

3) if γ = 1 and a = 0 then

4) if γ = 1 and a = 0 then

Here α 1 and α 2 are arbitrary parameters while the functions f 1 , f 2 and g are such that the identity ξ 0 t = −η 1 u should take place.

Proofs of Theorems 3 and 4 are based on the technique, which is a combination of the classical Lie method and the group of ETs. This technique is often called the Lie-Ovsiannikov method because L.V. Ovsyannikov was the first who applied such technique for solving the LSC problem (group classification problem) for a class of the nonlinear heat equations [19] . Here we present the proof of Theorems 4 because that is more complicated comparing with the proof of Theorem 3.

Proof of Theorem 4. System (1) with d 1 = 0, d 2 = 0 can be rewritten as

using an appropriate ET from (4). As usually, we start from the most general form of a Lie symmetry operator

In order to find all Lie symmetry operators of the form (12) of system (11) one should apply the following invariance criterion :

where operators X 1 and X 2 are the first and second prolongations of the operator X and the manifold M is defined by the system of equations

It should be stressed, that the manifold M involves not only the equations of the system in question but also the first-order consequences of the first equation of (11). These consequences guarantee a complete solving the LSC problem. Notably, such peculiarity does not occur for scalar PDEs but one was noted for some systems of PDEs involving equations of different order (see, e.g., the relevant discussion in [21] Section 1.2.5).

Having the correctly-defined manifold M, the invariance criterion (13) after rather standard calculations leads to the system of determining equations as follows

Equations (14)- (15) can be easily integrated, hence the general form of the infinitesimal operator (12) can be specified as (8). Now substituting the function

into equation (19) and splitting the equation obtained w.r.t. the variable v, we arrive at the linear equation (10) for G(t, x, y, u) and the equation

Obviously, equations (16) coincide with (9) . Thus, we need only to solve the overdetermined system of equations (17), (18) and (20) w.r.t. the functions ξ 0 and η 1 . This is a nontrivial task because the function ξ 0 depends on the dependent variable u in contrast to the standard situation for the systems of reaction-diffusion equations (see [24, 25] and the papers cited therein). Since ξ 0 u = 0, we used the differential consequence of equations (17) and (18) w.r.t. the variables t and u. Differentiating equation (17) w.r.t. t, taking the second-order consequence of equation (18) w.r.t. u and making the relevant calculations, we were able to derive the simple relation

Substituting the derivatives ξ 0 tu and ξ 0 tt derived from (21) into equation (20) one arrives at the classification equation

Thus, the following two cases must be examined separetely : γ = 1 and γ = 1.

In the case γ = 1, one immediately obtains

Differentiating equation (22) w.r.t. the variable u and substituting the expression obtained for η 1 uu into equation (17) , one arrives at the equation

Equations (22) and (23) can be easily integrated. The general solutions are

if a = 0 and

if a = 0. Here f (t) and g(t) are arbitrary smooth functions at the moment. In order to find the functions f (t) and g(t), one needs to substitute (24) and (25) into equation (18) . As a result, cases 1) and 2) of Theorem 4 were identified.

In the case γ = 1, equations (17), (18) and (21) take the forms

while equation (20) is satisfied identically. Integrating the last two equations of system (26), we arrive exactly at Cases 3) (for a = 0) and 4) (a = 0) of Theorem 4. Notably, we used the transformations u * = a−bu u e at (in the case a = 0) and u * = 1 u − bt (in the case a = 0) for solving the second equation of system (26) .

As one can note, the functions ξ 0 and η 1 involve arbitrary functions f 1 , f 2 and g in cases 3) and 4) of Theorem 4. However, they should be specified from the equation ξ 0 t = −η 1 u . At the final stage, both functions, ξ 0 and η 1 , should be inserted into (10) . The equation obtained is an integrable first-order PDE and its general solution is easily constructed in an explicit form provided the function k(t) is given. Thus, all the coefficients of operator (8) are identified.

The proof is now complete.

Let us present examples of highly nontrivial Lie symmetries in cases 3) and 4) (see Theorem 4). Setting f 1 = 1 into expressions arising in cases 3) and 4) and using the equation ξ 0 t = −η 1 u , one can specify ξ 0 and η 1 as follows

in cases 3) and 4), respectively (here α 0 , α 1 , and α 2 are arbitrary constants). Now one should use the functions ξ 0 and η 1 for finding the function G from equation (10) . In order to avoid cumbersome formulae we additionally set α 0 = α 1 = α 2 = 0 (just for simplicity), take the particular solution ξ 1 = y, ξ 2 = −x of the Cauchy-Riemann system (9) and fix the function k(t) : k(t) = e −2at (case 3) ) and k(t) = t −2 (case 4)). As a result, one arrives at the Lie symmetry operators

and

in cases 3) and 4), respectively. Here H is an arbitrary smooth function.

Theorems 3 and 4 allows us to reduce the basic system (1) to that of lower dimensionality. In fact, using the Lie symmetry operators (or their linear combinations) listed in Theorems 3 and 4 one can reduce (1) to the corresponding (1 + 1)-dimensional system and the latter to an ODE system. Here we examine only two cases in order to show that those lead to useful exact solutions.

First of all, one may simplify the nonlinear system (1) using the ETs (4) with the correctlyspecified parameters

to the form

Here and in what follows we preserve the old notations for all the variables. Example 1. Let us apply the operator ∂ y from the principal algebra (7) for reduction of the basic system (29) . Obviously, this operator produces the trivial ansatz u = u(t, x), v = v(t, x), so that we arrive at the system

This system is nothing else but the initial system under assumption that the distribution of the infected persons is one-dimensional in space (i.e., the diffusion w.r.t. the axis y is very small). In this case, the distribution of the total number of deaths will be also one-dimensional.

Making the further plausible assumption d 1 ≫ d 2 , i.e., the space diffusion of the infected persons leads mostly to increasing the total number of the COVID-19 cases and not so much to new deaths, we may put D = 0. Following our previous paper [18] , we specify the function k(t) = k 0 e −αt (hereafter k 0 > 0, α > 0). Thus, system (30) takes the form

Using Theorem 3 (see case 4) therein), one notes that system (31) admits the Lie symmetry ∂ t − α a v∂ v . So, taking the linear combination of this operator and the operator ∂ x

we obtain the ansatz

Substituting (32) into (31) , one arrives at the ODE system

The first equation in (33) is the known second-order ODE, which arises in many applications (e.g., for study the Fisher equation and its natural generalizations [26] ). The general solution of this ODE can be presented only in parametric form (see, e.g., [27] ), which is not useful for further analysis. However, an exact solution in the explicit form can be constructed for the correctly-specified parameter c = γ+4 √ 2(γ+2)

. To the best of our knowledge, this parameter and the relevant solution was established in [28] for the first time (see more references in [29] ). As a result, we obtain the traveling front solution of the first equation in system (31):

Notably (34) with A < 0 is still a solution, however, one possesses a singularity. It should be also noted that the basic system (1) and its particular cases derived above are invariant under the discrete transformation x → −x, therefore we may put ω = x − γ+4 √ 2(γ+2) t in what follows.

Having the function u in the explicit form (34) , one easily derives the function v from the second equation of system (31) :

where g(x) is an arbitrary smooth function. The integral in right hand side of (35) cannot be expressed in the terms of elementary functions for arbitrary parameters a, α and γ, therefore we study below a particular case. In order to avoid cumbersome formulae, let us set γ = 1 in system (31) . In this case, the above exact solution takes the form

Remark 3 The expression for u(t, x) in (36) presents the well-known traveling front of the famous Fisher equation u t = u xx + u(1 − u), which was firstly identified in [30] .

The integral in right hand side of (36) can be expressed in the terms of elementary functions for several values of the parameter α a . Taking α a = 5 6 , for example, we obtain

Now we turn to a possible interpretation of solution (37) . First of all, the functions u and v should be nonnegative for any t > 0 and x ∈ I (here I ⊂ R) because they represent the densities. Obviously, the functions u is always positive. It is easily seen that each function g(x) satisfying the inequality

guarantees also nonnegativity of v. In particular, one may take the function

which guarantees that the zero density of the deaths in the initial time t = 0, i.e., v(0, x) = 0. Examining the space interval I = [x 1 , x 2 ], x 1 < x 2 , we can calculate the total number of the COVID-19 cases and deaths on this interval as follows

So, substituting solution (37) into (39), we arrive at the formulae

Obviously the functions U(t) and V (t) are increasing and bounded, because

g(x)dx as t → +∞. Moreover, taking the appropriate function g(x), we can guarantee that

Thus, one may claim that the exact solution (37) possesses all necessary properties for the description of the distribution of the COVID-19 cases and the deaths from this virus in timespace (under 1D approximation). Examples of the exact solution (37) with the specified parameters and the corresponding functions (40) are presented in Fig. 1 and Fig. 2 . Notably, the parameters a and k 0 were taken approximately the same as in [18] . It follows from Fig. 1 that the spread of the COVID-19 cases in space has the form of a traveling wave and this coincides (at least qualitatively) with the real situation in many countries. In Ukraine, for example, the COVID-19 pandemic started in the western part and then spread to the central and eastern parts of Ukraine (the major exception was only the capital Kyiv, in which the total number of the COVID-19 cases was high from the very beginning). The distribution of deaths in space has more complicated behavior (see the right plot in Fig. 1 ). On the other hand, it is easily seen from Fig. 1 and Fig. 2 that v(t, x) ≪ u(t, x) what is in agreement with the measured data in many countries [8] . Notably, the behavior of the function v(t, x) can be essentially changed by the appropriate choice of the function g(x).

Example 2. Let us apply the operator y∂ x − x ∂ y from the principal algebra (7) for reduction of the basic system (29) . Obviously, this operator produces the well-known ansatz u = u(t, r), v = v(t, r), r 2 = x 2 + y 2 , i.e., we examine the radially-symmetric case. In this case, we arrive at the system

Making the same assumptions k(t) = k 0 e −αt and d 1 ≫ d 2 , i.e., D = 0, as in Example 1, we obtain the system

We have proved that system (41) again admits the Lie symmetry ∂ t − α a v∂ v (however, the operator of the space translation ∂ r is absent in this case). So, using this symmetry, one obtains the ansatz

Substituting (42) into (41), one arrives at the system

In contrast to the first equation in (33) , the first ODE in the above system is much more complicated. To the best of our knowledge, exact solutions of this equation are unknown.

It can be only noted that φ = 1 is the steady-state solution. As a result, we arrive at the space-homogeneous solution of the nonlinear system (41)

Of course, the restriction α < 0 should take place because the function v means the density. On the other hand, it means the exponential growing of the function v what is rather unrealistic because one obtains the total extinction of the population in question for a finite time.

The main part of this paper is devoted to the LSC of the class of reaction-diffusion system with the cross-diffusion (1). The system in question was suggested in work [18] as the natural generalization of the mathematical model for describing the COVID-19 outbreak. Firstly, we present a statement about the group of ETs of system (1) (see Theorem 1) in order to establish possible relations between systems that admit equivalent invariance algebras. Secondly, we find the principal algebra of system (1), i.e., the maximal invariance algebra of this system with arbitrary coefficients (see Theorem 2) . And lastly, we present two main theorems (Theorems 3 and 4) describing reaction-diffusion systems of the form (1) admitting nontrivial Lie symmetry, i.e., present the LSC of system (1). In Section 3, we demonstrate that the Lie symmetries identified in Section 2 are useful for finding exact solutions, which can describe the spread of the COVID-19 pandemic.

From the mathematical point of view, the most interesting Lie symmetry operators of system (1) occur when d 1 = 0 and are presented in Theorem 4. One sees that the coefficient ξ 0 of the infinitesimal operator X (see(8)) depends on the variable u (excepting case 2). Moreover, this dependence is nonlinear. To the best of our knowledge, it is the first example of such dependence for systems of evolution equations, in particular, reaction-diffusion systems. We assume that such unusual Lie symmetry of system (1) with d 1 = 0 can be a consequence of its integrability. In fact, one may consider the first equation as an ODE with the variables x and y as parameters. Solving this ODE, one obtains u(t, x, y) = a 1/γ e at C(x, y) a + b C γ (x, y) e aγt − 1 −1/γ , where C(x, y) is an arbitrary function. Substituting this expression for u into the second equation of the system, one again obtains the integrable ODE to find the function v.

Finally, it should be pointed out that Lie symmetries operators, which are nonlinear w.r.t. unknown functions were recently identified for a simplification of the Shigesada-Kawasaki-Teramoto system in [25] (see Section 3). Such peculiarity of Lie symmetry also occurs for a special Schrödinger type equation [31] , which can be rewritten in the form of a reactiondiffusion system with the cross-diffusion. However, the coefficient ξ 0 (see operator (12) ) in all known Lie symmetries of a wide range of reaction-diffusion systems [25, [31] [32] [33] [34] [35] [36] (see more references in Chapter 2 of [37] ) do not depend on the unknown function(s) in contrast to those in Theorem 4 and examples (27)- (28) . Moreover, we may conclude that the well-known 'people theorem' stating that the coefficient ξ 0 in each Lie symmetry of an arbitrary scalar evolution PDE of the order two and higher can depend only the time variable (no dependence on space variables and/or dependent variable!), cannot be generalized on the systems of evolution equations without additional restrictions. The problem how to define these restrictions is an open question.

From the applicability point of view, the most interesting system of the form (1) admitting nontrivial Lie symmetry is presented in case 4) of Theorem 3. Here the function k(t) of system (1) has the form that can be useful for describing the COVID-19 outbreak [18] . Moreover, the diffusivity d 2 = 0 as it is stated in [16] . In Section 3, we demonstrate how the Lie symmetries obtained can be applied for constructing of exact solutions. Furthermore we prove that an exact solution (with correctly-specified parameters) possesses all necessary properties for the description of the distribution of the COVID-19 cases and the deaths from this virus in time and space. Although it was done under 1D space approximation, this solution can be useful for the prediction of the COVID-19 pandemic if its spread has a favorite direction (a typical example is Ukraine). Of course, one needs to identify all the parameters in system (1) in order to calculate correct numbers of the COVID-19 cases and make a plausible prediction but this lays beyond the scope of this work.

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