Comparison Principle for Weakly Coupled Cooperative Parabolic Systems with Delays

Abstract

In this article, the validity of the comparison principle (CP) for weakly coupled quasi-linear cooperative systems with delays is proven. This is a powerful tool for studying the qualitative properties of the solutions. The CP is crucial in the proofs of the existence and uniqueness of weak solutions to cooperative reaction–diffusion systems presented here. Other direct consequences of the CP are the stability of the solution, the attenuation of long time periods, etc. An example model is given by spatial SEIR models with delays. They are suitable for modeling disease spread in space and time and can be described using a weakly coupled cooperative reaction–diffusion system. In this paper, spatial SEIR models with delays are considered in a continuous space. The emphasis is on the qualitative properties of the solutions, which are important for providing a mathematical basis for the model.

1. Introduction

The aim of this paper is the study of the comparison principle (CP) for weakly coupled quasi-linear cooperative systems with delays. The novelty of this research lies in the application of time delays in the principal part of the system. The validity of the CP is proven in Theorem 1, Section 3. As a consequence of the validity of the CP, results are derived in Theorem 3 on the existence and in Theorem 2 on the uniqueness of weak solutions for the studied systems. The results remain valid with reasonable modifications if the delays are in the other terms of the system.

Let $\Omega$ be a bounded domain in $R^n$ and T be a finite number. Let us consider in $Q=\Omega \times (0,T)$ a strictly parabolic system

$$u_t^l-div{a}^l(x,t+t_l,u^l,D{u}^l)+F^l(x,t,u^1,\dots u^N,D{u}^l)=f^l(x,t)$$

(1)

in Q, where $t_l$ are constants, with the boundary conditions

$$u^l(x,t)=g^l(x,t)$$

(2)

on $\Gamma =\partial \Omega \times (0,T)$.

Let us recall that since system (1) is a strictly parabolic one, for every $u^l$ and ${\xi }^l=({\xi }_1^l,\dots {\xi }_n^l)\in R^n$, $l=1,2,...N$ holds

$$\lambda \left(\left|u\right|\right){\left|{\xi }^l\right|}^2\le \sum _{i,j=1}^n{\displaystyle \frac{\partial a^{li}}{\partial p_j^l}}(x,t,u^1,\dots u^N,p^l){\xi }_i^l{\xi }_j^l\le \Lambda \left(\left|u\right|\right){\left|{\xi }^l\right|}^2.$$

(3)

Here, $\lambda \left(\left|u\right|\right)>0$ and $\Lambda \left(\left|u\right|\right)>0$ are monotonous and continuous functions, $\lambda$ is decreasing, and $\Lambda$ is increasing.

The coefficients $a^l$, $F^l$, $f^l$, and $g^l$ are measurable functions with respect to x and t. In addition, $a^l(x,t,u,p)$ and $F^l(x,t,u,p)$ are locally Lipchitz continuous with respect to $u^l$, u, and p, i.e.,

$$\begin{array}{c}\left|F^l(x,t,u,p)-F^l(x,t,v,q)\right|\le C\left(K\right)\left(\left|u-v\right|+\left|p-q\right|\right),\hfill \\ \\ \left|a^l(x,t,u^l,p)-a^l(x,t,v^l,q)\right|\le C\left(K\right)\left(\left|u^l-v^l\right|+\left|p-q\right|\right)\hfill \end{array}$$

(4)

for every $(x,t)\in \overline{Q}$, $\left|u\right|+\left|v\right|+\left|p\right|+\left|q\right|\le K$ and $l=1,...N$. Furthermore, in $\overline{Q}$, ${\displaystyle \frac{\partial a^{li}}{\partial p_j}},{\displaystyle \frac{\partial a^{li}}{\partial u^l}},{\displaystyle \frac{\partial F^l}{\partial u^i}},{\displaystyle \frac{\partial F^l}{\partial p_i}}\in L_1\left(\overline{Q}\right)$ for $i,j=1,...n$ and $l=1,...N$. Without loss of generality, one may presume that $a^l(x,t,u^l,0)=0$, $F^l(x,t,0,0)=0$ for every $i=1,...n$, $l=1,...N$, $(x,t)\in \overline{Q}$ and $u^l\in R.$

Furthermore, suppose that system (1) is cooperative. Cooperativeness means that in $\overline{Q}\times R^N\times R^n$ the functions

$$F^l(x,t,u^1,\dots u^N,p)$$

(5)

are non-increasing ones with respect to $u^k$ for $l\ne k$.

The interest in the qualitative properties of weakly coupled quasi-linear cooperative systems with delays lies in their useful applications, for instance, in the spatial SEIR models with delays described in Section 2. In these models, the time shift in the spatial propagation of the different compartments is incorporated into the model through the implementation of delays in the principal part of the system.

2. The Model Example

The interest in weakly coupled cooperative parabolic systems with delays is conditioned by some of their applications as spatial SEIR models in epidemiology.

SEIR models have been described in classical monographs on the subject such as [1,2]. For the sake of completeness, we recall the essentials of SEIR models.

An epidemic SEIR model is a compartmental one. In this kind of model, the population is assigned into a number of sets, or compartments—in this case, susceptible (S), exposed (E), infectious (I), and recovered (R).

Migration between compartments is strictly defined. When an infectious person (from compartment I) transfers the pathogen to a susceptible person (from compartment S), the latter contracts the disease and moves to the exposed compartment E. During the latency, or incubation, period of the disease $t_0$, individuals from E are infected but not yet contagious.

Members of E transfer into the infectious compartment I after the latency period $t_0$. I contains people who are contagious to the susceptible individuals from S.

Members of I transfer into the removed compartment R in the recovery period $t_1$. It is presumed that members of R develop immunity. R contains recovered people and deceased individuals.

The following illustrates the process:

$$\mathrm{S}⟶\mathrm{E}⟶\mathrm{I}⟶\mathrm{R}$$

Let us denote with $S\left(t\right)$, $E\left(t\right)$, $I\left(t\right)$, and $R\left(t\right)$ measures of the corresponding set S, E, I, and R. $S\left(t\right)$ is the number of susceptible people, $E\left(t\right)$ is the number of exposed people, $I\left(t\right)$ is the number of infected individuals, $R\left(t\right)$ is the number of the recovered or deceased, and $N=S\left(t\right)+E\left(t\right)+I\left(t\right)+R\left(t\right)$ is the total number of the population.

In the classical epidemic SEIR model, the dynamics of disease spread is described by the following system of ordinary differential equations; see [2,3]:

$$\begin{array}{cc}\hfill {\displaystyle \frac{dS\left(t\right)}{dt}}& =\mu N-\mu S\left(t\right)-{\displaystyle \frac{\beta I\left(t\right)S\left(t\right)}{N}}\hfill \\ \hfill {\displaystyle \frac{dE\left(t\right)}{dt}}& ={\displaystyle \frac{\beta I\left(t\right)S\left(t\right)}{N}}-(\mu +\sigma )E\left(t\right)\hfill \\ \hfill {\displaystyle \frac{dI\left(t\right)}{dt}}& =\sigma E\left(t\right)-(\mu +\gamma )I\left(t\right)\hfill \\ \hfill {\displaystyle \frac{dR\left(t\right)}{dt}}& =\gamma I\left(t\right)-\mu R\left(t\right)\hfill \end{array}$$

(6)

where $\beta$, $\gamma$, $\sigma$, and $\mu$ are the parameters. Here, $\beta =\beta \left(t\right)$ is the transmission coefficient; $\gamma =\gamma \left(t\right)$ is the removal coefficient; $\sigma ={\displaystyle \frac{1}{t_0}}$ is a constant and the latency period $t_0$ is medically determined; and $\mu$ is the mortality rate $\mu$, which is considered to be equal to the birth rate.

The main advantage of SEIR models is their ability to predict how a disease spreads over time, for instance, in terms of the total number of infected people at time t and the duration period of the epidemic. SEIR models can illustrate the effects of public health interventions on epidemic dynamics, such as vaccination, social distancing restrictions, etc. Furthermore, they provide a simple direct approximation of the Basic Reproduction Number ${\mathcal{R}}_0={\displaystyle \frac{\beta }{\gamma }}$ and estimations of other epidemiological parameters.

The main disadvantage of the classic SEIR model is that it is described by a system of ordinary differential equations. The solution gives values for $S\left(t\right),E\left(t\right),I\left(t\right)$, and $R\left(t\right)$ at time t in the entire area of study but not the spatial distribution of members of the corresponding compartment.

In order to take into account the spatial distribution of the members of the different compartments, one uses spatial SEIR epidemic models; see [4,5,6,7,8,9,10,11,12,13,14] and others.

The intuitive approach in spatial models is that the process of disease spreading is similar to a chemical reaction, and one may use reaction–diffusion systems (1) and (2).

In contrast to chemical reactions, in spatial epidemic models, one has to take into account the incubation and recovery periods. This is why in a spatial SEIR model, we use a reaction–diffusion system with delays that mark the “shift” in time due to the recovery and incubation periods.

When the epidemic is in full swing, the spatial SEIR model is described by

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial S\left(t,x\right)}{\partial t}}-{\displaystyle \frac{{\partial }^{2}S\left(t,x\right)}{\partial x^2}}& =\mu N-\mu S\left(t,x\right)-{\displaystyle \frac{\beta I\left(t,x\right)S\left(t,x\right)}{N}}\hfill \\ \hfill {\displaystyle \frac{\partial E\left(t,x\right)}{\partial t}}-{\displaystyle \frac{{\partial }^{2}E\left(t,x\right)}{\partial x^2}}& ={\displaystyle \frac{\beta I\left(t,x\right)S\left(t,x\right)}{N}}-(\mu +\sigma )E\left(t,x\right)\hfill \\ \hfill {\displaystyle \frac{\partial I\left(t+t_0,x\right)}{\partial t}}-{\displaystyle \frac{{\partial }^{2}I\left(t+t_0,x\right)}{\partial x^2}}& =\sigma E\left(t,x\right)-(\mu +\gamma )I\left(t,x\right)\hfill \\ \hfill {\displaystyle \frac{\partial R\left(t+t_0+t_1,x\right)}{\partial t}}-{\displaystyle \frac{{\partial }^{2}R\left(t+t_0+t_1,x\right)}{\partial x^2}}& =\gamma I\left(t,x\right)-\mu R\left(t,x\right)\hfill \end{array}$$

(7)

where $t_0>0$ is the incubation period, and $t_1>0$ is the recovery period. The above system is similar to that studied in [4], but the latter does not use delays to include the incubation and recovery periods.

The dynamics of disease spreading at the beginning of an epidemic is different. Since the volume of I is small, the use of the Michaelis–Menten equation on the right-hand side is reasonable. The analogue to chemical reactions is Michaelis–Menten-type modeling of catalyst or enzyme reactions, where the quantity of enzymes is small. This is why a spatial SEIR epidemic model with a low rate of infection is

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial S\left(t,x\right)}{\partial t}}-{\displaystyle \frac{{\partial }^{2}S\left(t,x\right)}{\partial x^2}}& =\mu N-\mu S\left(t,x\right)-{\displaystyle \frac{\beta S\left(t,x\right)}{N(I\left(t,x\right)-S\left(t,x\right))}}\hfill \\ \hfill {\displaystyle \frac{\partial E\left(t,x\right)}{\partial t}}-{\displaystyle \frac{{\partial }^{2}E\left(t,x\right)}{\partial x^2}}& ={\displaystyle \frac{\beta S\left(t,x\right)}{N(I\left(t,x\right)-S\left(t,x\right))}}-(\mu +\sigma )E\left(t,x\right)\hfill \\ \hfill {\displaystyle \frac{\partial I\left(t+t_0,x\right)}{\partial t}}-{\displaystyle \frac{{\partial }^{2}I\left(t+t_0,x\right)}{\partial x^2}}& =\sigma E\left(t,x\right)-(\mu +\gamma )I\left(t,x\right)\hfill \\ \hfill {\displaystyle \frac{\partial R\left(t+t_0+t_1,x\right)}{\partial t}}-{\displaystyle \frac{{\partial }^{2}R\left(t+t_0+t_1,x\right)}{\partial x^2}}& =\gamma I\left(t,x\right)-\mu R\left(t,x\right)\hfill \end{array}$$

(8)

where $t_0>0$ is the incubation period, and $t_1>0$ is the recovery period. System (8) is similar to that for enzyme reaction–diffusion—the low rate of infection corresponds to a low quantity of an enzyme.

In order to describe an epidemic from the beginning to the end, it is reasonable to employ full-range spatial SEIR epidemic models of the type

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial S\left(t,x\right)}{\partial t}}-{\displaystyle \frac{{\partial }^{2}S\left(t,x\right)}{\partial x^2}}& =\mu N-\mu S\left(t,x\right)-F(\beta ,N,S\left(t,x\right),I\left(t,x\right))\hfill \\ \hfill {\displaystyle \frac{\partial E\left(t,x\right)}{\partial t}}-{\displaystyle \frac{{\partial }^{2}E\left(t,x\right)}{\partial x^2}}& =F(\beta ,N,S\left(t,x\right),I\left(t,x\right))-(\mu +\sigma )E\left(t,x\right)\hfill \\ \hfill {\displaystyle \frac{\partial I\left(t+t_0,x\right)}{\partial t}}-{\displaystyle \frac{{\partial }^{2}I\left(t+t_0,x\right)}{\partial x^2}}& =\sigma E\left(t,x\right)-(\mu +\gamma )I\left(t,x\right)\hfill \\ \hfill {\displaystyle \frac{\partial R\left(t+t_0+t_1,x\right)}{\partial t}}-{\displaystyle \frac{{\partial }^{2}R\left(t+t_0+t_1,x\right)}{\partial x^2}}& =\gamma I\left(t,x\right)-\mu R\left(t,x\right)\hfill \end{array}$$

(9)

where $t_0>0$ is the incubation period, and $t_1>0$ is the recovery period. Function F is smooth, and

$$F(\beta ,N,S\left(t,x\right),I\left(t,x\right))={\displaystyle \frac{\beta S\left(t,x\right)}{N(I\left(t,x\right)-S\left(t,x\right))}}$$

(10)

when $I\left(t,x\right)\le C_1$ and

$$F(\beta ,N,S\left(t,x\right),I\left(t,x\right))={\displaystyle \frac{\beta I\left(t,x\right)S\left(t,x\right)}{N}}$$

(11)

when $I\left(t,x\right)\ge C_2$. The constants $C_1$ and $C_2$ depend on the pathogen and transmission coefficient, and determination of these constants is a major obstacle in problem (9). Another obstacle is the collection of high-quality data.

Unlike classic SEIR models, the spatial ones require initial and boundary conditions. If there is no disease at the beginning of our study, the initial conditions are $S(0,x)=E(0,x)=I(0,x)=R(0,x)=0$. If the pathogen is imported, then we have Neumann boundary conditions

$${\displaystyle \frac{dE(t,x_0)}{dn}},{\displaystyle \frac{dI(t,x_0)}{dn}}$$

(12)

where $x_0$ is entrance point. If the disease starts within the region, we have the right-hand side f of system (1).

3. The Comparison Principle for Cooperative Parabolic Systems with Delays

The following theorem states that the comparison principle holds for cooperative weakly coupled parabolic systems with delays. For more detailed proofs, see [15].

Theorem 1. 

Let $u,v\in W_{\infty }^{1,1}\left(Q\right)\cap C\left(\overline{Q}\right)$ be weak sub- and super-solutions to system (1) and (2). If (3)–(5) hold and $u\le v$ on Γ, then $u\le v$ in Q.

Proof. 

Let $w_{+}=max(w,0)$, $w=u-v$. Suppose $w_{+}$ is non-equivalent to 0 in Q. Then, $w\in W_{\infty }^{1,1}\left(Q\right)\cap C\left(\overline{Q}\right)$, and it is a weak sub-solution of

$$D_t{w}^l-\sum _{i=1}^n{D}_i\left(\sum _{j=1}^n{B}_j^{li}{D}_j{w}^l+B_0^{li}{w}^l\right)+\sum _{k=1}^N{E}_k^l{w}^k+{\sum }_{i=1}^n{H}_i^l{D}_i{w}^l=0$$

(13)

in Q with non-positive boundary data on $\partial Q$. Here,

$$B_j^{li}={\int }_0^1{\displaystyle \frac{\partial a^{li}}{\partial p_j}}(x,t+t_l,P^l)ds,B_0^{li}={\int }_0^1{\displaystyle \frac{\partial a^{li}}{\partial u^l}}(x,t+t_l,P^l)ds,$$

(14)

$$E_k^l={\int }_0^1{\displaystyle \frac{\partial F^l}{\partial u^k}}(x,t,Q^l)ds,H_i^l={\int }_0^1{\displaystyle \frac{\partial F^l}{\partial p_i}}(x,t,Q^l)ds,$$

(15)

$P^l=\left(v^l+s(u^l-v^l),D{v}^l+sD(u^l-v^l)\right)$ and $Q^l=\left(v+s(u-v),D{v}^l+sD(u^l-v^l)\right)$. Then, $w\left(x\right)$ satisfies the following integral inequality:

$$0\ge {\int }_{Q_{\tau }}\left[-w^l{\eta }_t^l+B_j^{li}{w}_{x_j}^l{\eta }_{x_i}^l+B_0^{li}{w}^l{\eta }_{x_i}^l+E_k^l{w}^k{\eta }^l+H_i^l{w}_{x_i}^l{\eta }^l\right]dxdt+{\int }_{\Omega }{w}^l{\eta }^{l}dx$$

(16)

for every non-negative test function $\eta \in W_c^{1,1}(\Omega )\cap C\left(\overline{Q}\right)$ and every $\tau \in [0,T]$.

Consider the test function $\eta (x,t)=w_{+}(x,t)e^{-Lt}$ for some constant $L\ge {\displaystyle \frac{2C^2\left(K_1\right)}{\lambda \left(K_1\right)}}+2(n^2+1)C\left(K_1\right)+2$, where $K_1=sup\left(\left|u\right|+\left|v\right|+\left|Du\right|+\left|Dv\right|\right)$ and $C\left(K_1\right)$ is the constant from (4). Then,

$$-{\int }_{Q_{\tau }}{w}^l(x,t)D_t\left(w_{+}^l(x,t)e^{-Lt}\right)dxdt={\int }_{Q_{\tau }}\left[L{w}^l{w}_{+}^l{e}^{-Lt}-w^l{D}_t\left(w_{+}^l\right)e^{-Lt}\right]dxdt=$$

$$={\int }_{Q_{\tau }}\left[L{w}_{+}^l{w}_{+}^l{e}^{-Lt}\right]dxdt-{\int }_{Q_{\tau }}\left[{\displaystyle \frac{1}{2}}{\displaystyle \frac{\partial {\left(w_{+}^l\right)}^2}{\partial t}}{e}^{-Lt}\right]dxdt=$$

$$={\int }_{Q_{\tau }}\left[L{\left(w_{+}^l\right)}^2{e}^{-Lt}\right]dxdt-{\left.{\displaystyle \frac{1}{2}}\left({\int }_{\Omega }\left[{\left(w_{+}^l\right)}^2.e^{-Lt}\right]dx\right)\right|}_0^{\tau }-{\int }_{Q_{\tau }}\left[{\displaystyle \frac{L}{2}}{\left(w_{+}^l\right)}^2.e^{-Lt}\right]dxdt$$

$$={\int }_{Q_{\tau }}\left[{\displaystyle \frac{L}{2}}{\left(w_{+}^l\right)}^2.e^{-Lt}\right]dxdt-{\displaystyle \frac{1}{2}}\left({\int }_{\Omega }{\left[w_{+}^l(x,\tau )\right]}^2.e^{-L\tau }dx\right)$$

because $w^l{w}_{+}^l=w_{+}^l{w}_{+}^l$, and according to the initial conditions, we have $w_{+}^l(x,0)=0$.

Therefore, (16) can be written as

$$0\ge {\int }_{Q_{\tau }}\left[{\displaystyle \frac{L}{2}}{\left(w_{+}^l\right)}^2.e^{-Lt}\right]dxdt-{\displaystyle \frac{1}{2}}\left({\int }_{\Omega }{\left[w_{+}^l(x,\tau )\right]}^2.e^{-L\tau }dx\right)+$$

$$+{\int }_{Q_{\tau }}\left[B_j^{li}{D}_j{w}^l{D}_i\left(w_{+}^l\right)e^{-Lt}+B_0^{li}{w}^l{D}_i\left(w_{+}^l\right)e^{-Lt}+E_k^l{w}^k.w_{+}^l{e}^{-Lt}+H_i^l{D}_i\left(w^l\right)w_{+}^l{e}^{-Lt}\right]dxdt+$$

$$+{\int }_{\Omega }{\left[w_{+}^l(x,\tau )\right]}^2{e}^{-L\tau }dx$$

Taking into account that $E_k^l\le 0$ for $l\ne k$ and $E_k^l{w}^k{w}_{+}^l\ge E_k^l{w}_{+}^k{w}_{+}^l$, according to (16), we obtain

$$0\ge {\displaystyle \frac{1}{2}}{\int }_{\Omega }\sum _{l=1}^N{\left(w_{+}^l(x,\tau )\right)}^2{e}^{-L\tau }dx+$$

$$+{\int }_{Q_{\tau }}\left[\sum _{l=1}^N\left(\sum _{i,j=1}^n{B}_j^{li}{D}_j{w}^l{D}_i{w}_{+}^l+\sum _{i=1}^n{B}_0^{li}{w}^l{D}_i{w}_{+}^l\right)\right]e^{-Lt}dxdt+$$

$$+{\int }_{Q_{\tau }}\left[\sum _{l=1}^N\left(\sum _{k=1}^N{E}_k^l{w}^k{w}_{+}^l+\sum _{i=1}^n{H}_i^l{w}_{+}^l{D}_i{w}^l+{\displaystyle \frac{L}{2}}{\left(w_{+}^l\right)}^2\right)\right]e^{-Lt}dxdt\ge$$

$$\ge {\int }_{Q_{\tau }}\left[\sum _{i,j=1}^n\sum _{l=1}^N{B}_j^{li}{D}_j{w}_{+}^l{D}_i{w}_{+}^l+\sum _{i=1}^n\sum _{l=1}^N{B}_0^{li}{w}_{+}^l{D}_i{w}_{+}^l\right]e^{-Lt}dxdt+$$

$$+{\int }_{Q_{\tau }}\left[\sum _{k,l=1}^N{E}_k^l{w}_{+}^k{w}_{+}^l+\sum _{i=1}^n\sum _{l=1}^N{H}_i^l{w}_{+}^l{D}_i{w}_{+}^l+\sum _{l=1}^N{\displaystyle \frac{L}{2}}{\left(w_{+}^l\right)}^2\right]e^{-Lt}dxdt=$$

$$={\int }_{Q_{\tau }}<B\xi ,\xi >dxdt,$$

where $B=A+{\displaystyle \frac{L}{2}}{I}_0$, $\xi =({\xi }^1,\dots {\xi }^N)$, ${\xi }^l=(D{w}_{+}^l,w_{+}^l)$. $I_0$ is an $(n+1)N\times (n+1)N$ diagonal matrix, and all of its elements are 0, but $I_0^{mm}=1$ for $m=(n+1)s$, $s=1,\dots n+1$ (every $(n+1)$-th element of the diagonal equals 1).

$A=\mathbf{A}+{\mathbf{A}}^t$ is a symmetrized $(n+1)N\times (n+1)N$ matrix, where

$$\mathbf{A}=\left(\begin{array}{ccc}{\displaystyle \frac{\partial a^{li}}{\partial p_j^s}}& & {\displaystyle \frac{\partial F^l}{\partial p_j^s}}\\ & & \\ {\displaystyle \frac{\partial a^{li}}{\partial u^s}}& & {\displaystyle \frac{\partial F^l}{\partial u^s}}\end{array}\right)$$

for $i,j=1,\dots n,$ and $l,s=1,\dots N$.

For a sufficiently large constant L, the matrix B defines the positive-defined quadratic form. Therefore, from ${\int }_Q<B\xi ,\xi >dxdt\le 0$, it follows that $<B\xi ,\xi >=0$ in $\Omega$, i.e.,

$$\sum _{i,j=1}^n\sum _{l=1}^N{B}_j^{li}{D}_j{w}_{+}^l{D}_i{w}_{+}^l+\sum _{i=1}^n\sum _{l=1}^N{B}_0^{li}{w}_{+}^l{D}_i{w}_{+}^l+\sum _{k,l=1}^N{E}_k^l{w}_{+}^k{w}_{+}^l+$$

(17)

$$+\sum _{i=1}^n\sum _{l=1}^N{H}_i^l{w}_{+}^l{D}_i{w}_{+}^l+\sum _{l=1}^N{\displaystyle \frac{L}{2}}{\left(w_{+}^l\right)}^2=0.$$

Then, according to (3), we have

$$\sum _{i,j=1}^n\sum _{l=1}^N{B}_j^{li}{D}_j{w}_{+}^l{D}_i{w}_{+}^l=\sum _{i,j=1}^n\sum _{l=1}^N{\int }_0^1{\displaystyle \frac{\partial a^{li}}{\partial p_j}}(x,t,\left(v^l+s(w^l),D{v}^l+sD(w^l)\right))D_j{w}_{+}^l{D}_i{w}_{+}^{l}ds=$$

$$=\sum _{l=1}^N{\int }_0^1\left(\sum _{i,j=1}^n{\displaystyle \frac{\partial a^{li}}{\partial p_j}}{D}_j{w}_{+}^l{D}_i{w}_{+}^l\right)ds\ge \lambda \sum _{l=1}^N{\left|D{w}_{+}^l\right|}^2$$

where c is a constant such that ${\left(B_j^{li}\right)}^2<c$ for every $i,j=1,\dots n$, and $\lambda$ is defined in (3).

Furthermore, according to the Cauchy inequality, we have

$$\sum _{i=1}^n\sum _{l=1}^N\left(B_0^{li}+H_i^l\right)w_{+}^l{D}_i{w}_{+}^l\le {\displaystyle \frac{1}{2}}\sum _{i=1}^n\sum _{l=1}^N\left[{\displaystyle \frac{{\left(B_0^{li}+H_i^l\right)}^2}{\epsilon }}{\left(w_{+}^l\right)}^2+\epsilon {\left(D_i{w}_{+}^l\right)}^2\right]\le$$

(18)

$$\le {\displaystyle \frac{c_{5}n}{2\epsilon }}\sum _{l=1}^N{\left(w_{+}^l\right)}^2+{\displaystyle \frac{c_5\epsilon }{2}}\sum _{l=1}^N{\left|D{w}_{+}^l\right|}^2,$$

where ${\left(B_0^{li}+H_i^l\right)}^2<c_5$. Therefore, (17) becomes

$$\lambda \sum _{l=1}^N{\left|D{w}_{+}^l\right|}^2\le \sum _{i,j=1}^n\sum _{l=1}^N{B}_j^{li}{D}_j{w}_{+}^l{D}_i{w}_{+}^l=$$

(19)

$$=-\sum _{i=1}^n\sum _{l=1}^N\left(B_0^{li}+H_i^l\right)w_{+}^l{D}_i{w}_{+}^l-\sum _{k,l=1}^N{E}_k^l{w}_{+}^k{w}_{+}^l-\sum _{l=1}^N{\displaystyle \frac{L}{2}}{\left(w_{+}^l\right)}^2\le$$

$$\le \left|\sum _{i=1}^n\sum _{l=1}^N\left(B_0^{li}+H_i^l\right)w_{+}^l{D}_i{w}_{+}^l\right|-\sum _{k,l=1}^N{E}_k^l{w}_{+}^k{w}_{+}^l-\sum _{l=1}^N{\displaystyle \frac{L}{2}}{\left(w_{+}^l\right)}^2\le$$

$$\le \sum _{i=1}^n\sum _{l=1}^N\left|\left(B_0^{li}+H_i^l\right)w_{+}^l{D}_i{w}_{+}^l\right|+{\displaystyle \frac{c_6}{2}}\sum _{k,l=1}^N\left[\epsilon {\left(w_{+}^k\right)}^2+{\displaystyle \frac{1}{\epsilon }}{\left(w_{+}^l\right)}^2\right]-\sum _{l=1}^N{\displaystyle \frac{L}{2}}{\left(w_{+}^l\right)}^2\le$$

$$\le {\displaystyle \frac{c_{5}n}{2\epsilon }}\sum _{l=1}^N{\left(w_{+}^l\right)}^2+{\displaystyle \frac{c_5\epsilon }{2}}\sum _{l=1}^N{\left|D{w}_{+}^l\right|}^2+{\displaystyle \frac{c_{6}N}{2}}\left[\epsilon +{\displaystyle \frac{1}{\epsilon }}\right]\sum _{l=1}^N{\left(w_{+}^l\right)}^2-\sum _{l=1}^N{\displaystyle \frac{L}{2}}{\left(w_{+}^l\right)}^2,$$

where $c_6>{sup}_{\Omega }{E}_k^l$ for every $l,k=1,..N$ and $c_5$ is the constant form (18).

Hence,

$$\lambda \sum _{l=1}^N|D{w}_{+}^l{|}^2\le {\displaystyle \frac{c_{5}n}{2\epsilon }}\sum _{l=1}^N{\left(w_{+}^l\right)}^2+{\displaystyle \frac{c_5\epsilon }{2}}\sum _{l=1}^N{\left|D{w}_{+}^l\right|}^2+{\displaystyle \frac{c_{6}N}{2}}\left[\epsilon +{\displaystyle \frac{1}{\epsilon }}\right]\sum _{l=1}^N{\left(w_{+}^l\right)}^2-\sum _{l=1}^N{\displaystyle \frac{L}{2}}{\left(w_{+}^l\right)}^2$$

or

$$\lambda \sum _{l=1}^N|D{w}_{+}^l{|}^2-{\displaystyle \frac{c_5\epsilon }{2}}\sum _{l=1}^N{\left|D{w}_{+}^l\right|}^2\le {\displaystyle \frac{c_{5}n}{2\epsilon }}\sum _{l=1}^N{\left(w_{+}^l\right)}^2+{\displaystyle \frac{c_{6}N}{2}}\left[\epsilon +{\displaystyle \frac{1}{\epsilon }}\right]\sum _{l=1}^N{\left(w_{+}^l\right)}^2-\sum _{l=1}^N{\displaystyle \frac{L}{2}}{\left(w_{+}^l\right)}^2$$

and for $0<\epsilon <min\left\{{\displaystyle \frac{2\lambda }{c_5}},{\displaystyle \frac{2(c_{5}n+2c_{6}N)}{L}},1\right\}$, we obtain

$$\sum _{l=1}^N{\left|D{w}_{+}^l\right|}^2\le C_4\sum _{l=1}^N{\left(w_{+}^l\right)}^2.$$

The constant

$$C_4=\underset{\epsilon >0}{inf}{\displaystyle \frac{c_{5}n+c_{6}N({\epsilon }^2+1)-L\epsilon }{(2\lambda -c_5\epsilon )\epsilon }}$$

depends only on n, N, $\Omega$, $\lambda \left(K_1\right)$, and $C\left(K_1\right)$.

Thus, the inequality

$${\left|Dlog\left(1+{\displaystyle \frac{1}{\epsilon }}{\sum }_{l=1}^N{w}_{+}^l\right)\right|}^2\le N.C_4{\displaystyle \frac{{\sum }_{l=1}^N{\left(w_{+}^l\right)}^2}{{\left(\epsilon +{\sum }_{l=1}^N{w}_{+}^l\right)}^2}}\le N^3{C}_4.$$

holds for every $\epsilon >0$.

Integrating the above inequality into $\Omega$ and taking into account that $\sum w_{+}^l=0$ on $\partial \Omega$, we obtain

$${\left|log\left(1+{\displaystyle \frac{1}{\epsilon }}{\sum }_{l=1}^N{w}_{+}^l\right)\right|}^2\le N^3.C_4.diam\Omega ,$$

which is impossible for $\epsilon \to 0$. □

A direct consequence of the validity of the CP is the uniqueness of the solution of (1) and (2).

Theorem 2. 

If (3)–(5) hold for system (1) and (2), then the solution is unique.

The proof for Theorem 2 is trivial. If u and v are solutions of (1) and (2), then one may consider u a sub-solution and v a super-solution for (1) and (2). Since $u=v=g(h,t)$ on $\Gamma =\partial \Omega \times (0,T)$, according to the CP, $u\le v$ in $\Omega$. On the other hand, v is a sub-solution and u is a super-solution for (1) and (2) as well, and according to the CP, $u\ge v$ in $\Omega$.

For sake of simplicity, we present the existence theorem for a simpler case that covers the model example; in particular, suppose

$$div{a}^l(x,t+t_l,u^l,D{u}^l)=\sum _{i,j=1}^N{D}_i\left(a_{ij}^l(x,t)D_j{u}^l\right),$$

$$F^l(x,t,u^1,\dots u^N,p^l)=\sum _{i=1}^N{a}_i^l(x,t)p_i^l+F_1^l(x,t,u^1,\dots u^N)$$

for every $l=1,...N$. Then, system (1) becomes

$$u_t^l-\sum _{i,j=1}^N{D}_i\left(a_{ij}^l(x,t)D_j{u}^l\right)+\sum _{i=1}^N{a}_i^l(x,t)D_i{u}^l+F_1^l(x,t,u^1,\dots u^N)=f^l(x,t)$$

(20)

for $l=1,...N$, with the initial data $g(x,0)$ and null boundary data.

Nevertheless, some additional conditions are required. First of all, f is supposed to be a bounded function, i.e., a positive constant C exists such that

$$|f^l(x,t)|\le C$$

in $\overline{Q}$ for every $l=1,...N$.

The coefficients $a_{ij}^l(x,t)$ are $C^{1+\alpha }\left(\overline{Q}\right)$ smooth with respect to t. $a_i^l$, $F_1^l$, and $f^l$ are Holder continuous in $\overline{Q}$ with the Holder constant ${\alpha }^{\prime }<1$ and

$$\underset{t\in [0,T]}{max}\sum _{i=1}^n{\parallel a_i^l\parallel }_{L_{q/2}(\Omega )}<\infty$$

for $q>n$. In addition, $g^l(x,0)\in C^{\alpha +2}\left(\overline{\Omega }\right)$.

Then, the following theorem holds:

Theorem 3. 

Suppose (3)–(5) hold for system (20). Let $v(x,t)$ be a smooth super-solution, and $w(x,t)$ be a smooth sub-solution of (20), $g(x,t)=0$ on Γ. Then, there is a solution $u(x,t)\in {\left(W_c^{1,1}\left(Q\right)\right)}^N$ for (20) with null boundary data.

Proof. 

1. The function $f^l(x,t)-F_1^l(x,t,u^1,...u^N)+\sigma u^l$ is non-decreasing with respect to $u^k$ for a sufficiently large constant $\sigma >0$. Indeed, according to condition (5), the functions $-F_1^l$ are non-decreasing with respect to $u^k$ for $k\ne l$, $k=1,\dots n$. Since $F_1^l(x,t,u^1,...u^N)$ are independent on $D{u}^l$, for $k=l$, then according to (4), $f^l(x,t)-F_1^l(x,t,u^1,...,u^l,...u^N)+\sigma u^l-f^l(x,t)+F_1^l(x,t,u^1,...,v^l,...u^N)-\sigma v^l\ge (\sigma -c_3)(u^l-v^l)\ge 0$ in Q for $\sigma >c_3$, where $c_3$ is the constant from (4). $c_3$ depends only on the constant M such that $M>ma{x}_{\overline{Q}}\left|u(x,t)\right|$.

Hence, the functions $f^l(x,t)-F_1^l(x,t,u^1,...u^N)+\sigma u^l$ are non-decreasing with respect to $u^l$ for $\sigma \ge c_3$.

2. We construct through induction a sequence of vector-functions $u_0,u_1,\dots u_j,\dots$. The first term is $u_0=w(x,t)$. The term $u_k\in W_2^{1,1}\left(Q\right)$ defines through induction $u_{k+1}$ as the solution of the problem

$$D_t{u}_{k+1}^l-\sum _{i,j=1}^N{D}_i\left(a_{ij}^l(x,t)D_j{u}_{k+1}^l\right)+\sum _{i=1}^N{a}_i^l(x,t)D{u}_{k+1}^l+\sigma u_{k+1}^l=$$

(21)

$$=f^l(x,t)-F_1^l(x,t,u_k^1,...u_k^N)+\sigma u_k^l$$

in Q

$$u_{k+1}^l(x,t)=0$$

(22)

in $\partial Q$ for every $l=1,\dots N$.

Let the left-hand side of (21) be $A^l(x,t,u,\sigma )$ and the right-hand side be $B^l(x,t,u,\sigma )$ for $l=1,\dots N$.

System (21) is decomposable into N independent equations that are solvable in $W_2^{1,1}\left(Q\right)$ using Theorem 5 from [16], p. 127.

3. Since B is an increasing function with respect to u, according to the CP, the sequence $u_0,u_1,\dots u_j,\dots$ satisfies

$$u_0^l\le u_1^l\le \dots \le u_{k+1}^l\le \dots$$

(23)

4. The equation $u_{k+1}(x,t)\le v(x,t)$ holds for every k.

5. The sequence $\left\{u_k\right\}$ is bounded in Q and monotonously increasing. Thus, a function $u(x,t)$ exists such that $u_k(x,t)\to u(x,t)$ pointwise in Q. Moreover, $\left\{u_k\right\}$ is uniformly bounded and equicontinuous in $\overline{Q}$ since $u_k^l(x,t)$ is Holder continuous and $|u_k^l(x,t)-u_k^l(x_0,t)|\le c_4\left(\right|x-x_0{|}^{\beta }+|t-t_0{|}^{\beta /2})$ for every $l=1,\dots N$. Then, according to the Arzelà—Ascoli theorem, there is a sub-sequence $\left\{u_{k_j}\right\}$ that converges uniformly with $u\in C\left(\overline{Q}\right)$. Without loss of generality, we denote $\left\{u_{k_j}\right\}$ as $\left\{u_k\right\}$.

Since $u\in C\left(\overline{Q}\right)$ and every $\left\{u_{k_j}\right\}$ satisfies boundary conditions (22), then u satisfies (22) as well.

The functions $u_{k+1}(x,t)$ are monotonous, and $u(x,t)$ is continuous; therefore, $\left\{u_k^2\right\}\to u^2$ in Q. Then, according to Theorem 5 in [17], p. 648, it follows that $u_k\to u(x,t)$ in ${\left(L^2\left(Q\right)\right)}^N$.

6. Analogously to 5, $\left\{D_{x_i}{u}_k\right\}$ is uniformly bounded and equicontinuous in $\overline{Q}$, and according to the Arzelà—Ascoli theorem, one obtains $u\in C^1\left(\overline{Q}\right)$.

7. The function $u(x,t)$ is a solution of (20) and (22).

For every $\eta (x,t)=({\eta }^1(x,t),...{\eta }^N(x,t))\in {\left(W_c^{1,1}\left(Q\right)\right)}^N$,

$${\int }_{\Omega }{u}_{k+1}^l(x,t){\eta }^l(x,t)dx-{\int }_{Q_t}{u}_{k+1}^l(x,t)D_t{\eta }^l(x,t)dxdt+$$

$$+{\int }_{Q_t}\left(\sum _{i,j=1}^N{a}_{ij}^l(x,t)D_j{u}_{k+1}^l{D}_i{\eta }^l(x,t)+\sum _{i=1}^N{a}_i^l(x,t)D{u}_{k+1}^l{\eta }^l(x,t)+\sigma u_{k+1}^l{\eta }^l(x,t)\right)dxdt=$$

$$={\int }_{Q_t}(f^l(x,t)-F_1^l(x,t,u_k^1,...u_k^N)+\sigma u_k^l){\eta }^l(x,t)dxdt$$

holds, and for $k\to \infty$,

$${\int }_{\Omega }{u}^l(x,t){\eta }^l(x,t)dx-{\int }_{Q_t}{u}^l(x,t)D_t{\eta }^l(x,t)dxdt+$$

$$+{\int }_{Q_t}\left(\sum _{i,j=1}^N{a}_{ij}^l(x,t)D_j{u}^l{D}_i{\eta }^l(x,t)+\sum _{i=1}^N{a}_i^l(x,t)D{u}^l{\eta }^l(x,t)\right)dxdt=$$

$$={\int }_{Q_t}(f^l(x,t)-F_1^l(x,t,u^1,...u^N)){\eta }^l(x,t)dxdt.$$

Therefore, $u(x,t)\in {\left(W_c^{1,1}\left(Q\right)\right)}^N$ is a weak solution of (20) and (22). □

It is easy to check that systems (8)–(10) are cooperative ones. Therefore, the CP holds for them, and they are unique weak solutions. What is more, it is not hard to prove that the solutions are $C^1$ smooth.

4. Discussion

The use of spatial SEIR models provides much more precise information on disease spread than the classical ones. It provides the opportunity to create situational maps of the disease spread in space and time, to carry out digital simulations, to measure the effect of different counter-epidemic measures, etc. Solving the inverse problem allows for the detection of the initial infiltration of the pathogen into society. Therefore, spatial SEIR models have the potential to improve epidemic control.

The main practical disadvantage of the presented SEIR model with delays is that it is a continuous one. Closer to the real world would be a patchy model or applying systems (8), (9), or (10) to graphs. Work on the validity of the CP in graphs is still in progress.

5. Conclusions

Spatial SEIR models provide powerful tools for modeling epidemic spreads in space and time. Studying the qualitative properties of the solutions is important to ensure the mathematical background of the model. The validity of the CP for system (1) and (2) asserts the existence and uniqueness of a smooth solution to the system, attenuation of the solution for long time periods, the behavior after small perturbations, etc.

It is easy to check that systems (7)–(9) are cooperative ones and that the conditions of Theorems 1–3 hold for spatial SEIR models. Therefore, the problem is well posed, and there is a unique smooth solution. Furthermore, since the CP holds, small perturbations in the initial and boundary data yield small perturbations in the solution. Therefore, the problem is stable, and the computation errors are manageable.