On the effect of transition from a model with concentrated parameters to a model with distributed parameters

We note from a general point of view that adding diffusion terms to ordinary differential equations, for example, to logistic ones, can in some cases improve sufficient conditions for the stability of a stationary solution. We give examples of models in which the addition of diffusion terms to ordinary differential equations changes the stability conditions of a stationary solution.


Introduction
Systems of differential equations simulate the growth of phenomena of various types, and for all such systems, studies of the stability of stationary solutions play an important role. These studies have a long history. In many cases, such models are based on ordinary differential equations. Although the theory of systems of ordinary differential equations has long been classical, interest in it does not fade away. In the last few decades, this is also due to the fact that such systems have found applications in modelling biological and social systems. From relatively recent works on mathematical biology, it is possible to indicate in this regard [1][2][3][4][5][6].
In the work [7], a model of the origin and development of currents in painting, based on equations of the same type, is considered.
In this paper, we consider a certain class of mathematical models with partial differential equations (models with distributed parameters), which are obtained from models with ordinary differential equations (models with concentrated parameters) by adding the so-called diffusion terms. The tendency of such sophistications of mathematical models can be traced in some works related to modelling the growth and distribution of populations, the growth and spread of infections, and the growth of tumors. In this regard, see first of all the monograph [8]. In the work [9], a diffusion model of a malignant tumor is presented.
The mathematical model of glioma growth is based on the classical definition of cancer as uncontrolled proliferation of cells with the potential for invasion and metastasis, simplified for gliomas, which practically do not metastasize. This model is governed by the equation (see [10]) where u(x, t) defines the concentration of malignant cells at location x and time t, M is the random motility coefficient defining the net rate of migration of the tumor cells, τ represents is the Laplace operator. The term M ∆u is usually called the diffusion term. We are interested in the stability of stationary solutions of diffusion models. This issue is discussed in the book [8]. This book states that adding diffusion terms can change the stability of a stationary solution both for the worse and for the better. For models of a certain type, we try to concretize sufficient conditions for the stability of stationary solutions.

Materials and methods
We consider the initial-boundary value problem for the system of partial differential equations: where Ω is a bounded domain with a piecewise smooth boundary Γ = ∂Ω, − → ν is a unit external normal vector to the boundary ∂Ω of the domain Ω, Let us consider a special kind of functions F s (u) = F s (u 1 , . . . , u m ) : Let w = (w 1 (x), . . . , w m (x)) be a stationary solution of system (2), that is, the solution of the system satisfying the boundary conditions In this paper, we study the stability of a stationary solution of system (2).
be a vector of deviations from a stationary solution. We substitute u = w + z in system (2). Then After identical transformations, we obtain the equality Taking into account (6), the last equality is converted to the form We multiply this equality by z s and integrate over the domain Ω. We obtain: Assuming that the deviations z s are small enough, we discard the monomials of degree higher than 2 of these deviations, then we get: We apply the first Green formula for the Laplace operator to the first term on the right side of this equation. For two functions f ∈ C 1 (Ω) and g ∈ C 2 (Ω) ∩ C 1 (Ω), the Green formula takes the form (see [11]) where − → ν is a unit external normal vector to Γ. Here dΓ is an arc element of the boundary Γ = ∂Ω. Substituting f = z s , g = z s into (8), we obtain: where the second term on the right side of the equation is a surface (for n ≥ 3) or contour (for n = 2) integral of the first kind over the boundary of the domain Ω or the sum of non-negative values at the ends of the interval Ω in the case of n = 1 ; g(z s ) = 0 for µ s = 0 or for η s = 0; if η s = 0, then g(z s ) = µ s z 2 s /η s . In all cases g(z s ) ≥ 0. Summing m equalities (9), we obtain where We put Θ sk = (β sk + β ks )/2.
Then equality (10) can be rewritten as Transformation (12) is introduced for the transition from an unsymmetric quadratic form to a symmetric one. Obviously, the negative definiteness of a quadratic form m s=1 m k=1 Θ sk z k z s dx (14) will ensure the negativity of the left side of equality (13), and, therefore, the stability of the stationary solution.
In the absence of diffusion terms, that is, when the variables x 1 , . . . , x n are included in equations (2) as parameters whose derivatives are not contained in these equations. This is the case of a model with concentrated parameters. Let that is, we proceed to the consideration of the diffusion model with distributed parameters. In this case, it is possible to weaken the sufficient condition for the stability of a stationary solution.
For this purpose, we use the Steklov-Poincare-Friedrichs inequality (see [12] p. 150, [13] p. 62) where d = diam Ω is a diameter of the domain Ω. Therefore, Now we can assert that a sufficient condition for the stability of a stationary solution is the negative definiteness of the quadratic form In order to demonstrate how the properties of the model change when introducing distributed parameters by adding diffusion terms, we consider the case b sk = 0, f (x) = 0, x ∈ Ω, s, k = 1, . . . , m.
In this case, the vector w = 0 is a stationary solution of the system both for the case of concentrated parameters (15) and for the case of distributed parameters (16). However, the situations are fundamentally different. In the diffusionless case, the zero vector is not a stable solution. If all the equations of the system contain diffusion terms, that is, quadratic form (18) will take the form and, obviously, will be negatively defined, consequently, the zero solution will be stable. Another interesting example is provided by the Hotelling equation where u is an unknown function, u = u(x 1 , x 2 , t) ∈ C 2 (Ω) C 1 (Ω) for any t > 0, is the Laplace operator, A, B, ξ are given positive constants. This equation describes population growth and distribution. In this case, the values included in the equation have the following meaning: x 1 , x 2 are the geographical coordinates, A is the population growth rate, B is the migration rate, ξ is the coefficient of the saturated population density, u is the population density, t is the time parameter. This model takes into account migration processes. Population growth is modelled as a logistic process. Migration processes are described using Fourier's Law of Heat Conduction. Let w(x 1 , x 2 ) be a stationary solution of the Hotelling equation, that is, the solution of the equation The above method leads to the conclusion that the condition is sufficient for the stability of the stationary solution w(x 1 , x 2 ) [14] (see also [15], where this result was generalized). It is interesting to note that in the diffusion case (when B = 0) the zero stationary solution can be both stable and instable, what is determined by the size of the domain Ω.

Results and discussion
Let us consider the basic SIR model for the control of endemic infections (see [1]). This model assumes vaccination at birth at constant coverage p, which is reminiscent of a situation where a mandatory immunization program exists. The resulting model is as follows: where S, I, R denote the fractions of individuals who are, respectively, susceptible to acquiring infection, infective, i.e. able to retransmit infection to others, and removed because of e.g. immunity acquired after recovery. The infective fraction I is also called the infection prevalence.
The function β(t) denotes the transmission rate which is typically time-dependent. The other demo-epidemiological parameters are: γ = + ζ, > 0 which denotes both the birth and death rates, assumed identical to ensure that the population is stationary over time, and ζ > 0 which is the rate of recovery from infection. The equality allows to omit the third equation. In the most well-known case of constant transmission rate β(t) = β, the SIR model admits a disease-free equilibrium point DFE = (1 − p, 0, p), which is stable if β(1 − p) < γ and unstable otherwise. Taking into account (23), we add the diffusion terms and consider the system where S = S(x 1 , x 2 , t) = S(x, t), I = I(x 1 , x 2 , t) = I(x, t), is the Laplace operator.