1. Introduction
In recent years, reaction-diffusion systems on complex networks have been used to study epidemic processes [1] [2] . A network is mathematically a graph
, which contains a set
of vertices and a set E of edges. If vertices x and y are connected by an edge (also called adjacent), we write
. G is called a finite-dimensional graph if it has a finite number of edges and vertices. A graph G is weighted if each adjacent x and y is assigned a weight function
. Here
is a symmetric, nonnegative and bounded function, and
if and only if
.
A graph
is called connected, if for every pair of vertices
, there exists a sequence (called a path) of vertices
such that
for
. For a finite subset
, let
denote the boundary of
and
denote the interior of
, which are defined by
(1)
respectively. Throughout this paper,
is assumed to be a connected weighted finite-dimensional graph without self-loops. We also assume that
is a nonempty connected subset.
In this paper, we use discrete p-Laplacian operators defined on a network to describe the movements of mosquitoes in each vertex which depend on the topological structure of the network. In order to describe our problem more conveniently, we first introduce the following discrete p-Laplacian operators defined on a network.
Definition 1.1 For a function
and
, the graph p-Laplacian
on
is defined by
(2)
For
, the degree
on
is defined by
.
When
, it is called the discrete Laplacian
on
, which is defined by
(3)
Recently, classical Laplacian Δ is substituted by the discrete Laplacian
in graph Laplacian problems, and various methods and techniques to study the existence and qualitative properties of solutions have been developed [2] - [7] . Here we should emphasize that the discrete p-Laplacian operator
is actually nonlinear, which is different from the classical Laplacian Δ or the discrete Laplacian
or the discrete Laplacian
.
We consider the following nonlinear SIR model with p-Laplacian defined on networks
(4)
Here S, I represent the population sizes of susceptible and infectious compartments, respectively. The recovery R is omitted, due to the fact that
is assumed to be a constant. Parameter d is the diffusion rate of individuals,
indicates the recruitment rate of S and parameter
the contact rate between susceptible and infectious populations. Population S and I die at a rate of
and
, respectively, here
is the additional death rate caused by infectious disease.
In this paper, when
, we overcome the difficulties caused by the nonlinear operators p-Laplacian
and study the global stability for the solution of system (4). First, we prove the Green formula of nonlinear operators
. Then we construct the maximum principle and stronge maximum principle of the graph Laplacian equations. With the help of the priori estimate, we present the global existence result. At last, we investigate the asymptotical behavior of the system by the method of Lyapunov function.
2. Preliminaries
Lemma 2.1 (Green Formula). For any functions
, then
holds. In particular, in case of
, the following holds
(5)
Proof 1 Using (2), we have
From the above equality, we deduce
, which completes the proof.
It’s worth noting that the existence of nonlinear operators
causes difficulties when we construct the Maximun principle of system (4).
Lemma 2.2 (Maximun Principle). Suppose that
and K are constants. For any
, assume that
is continuous with respect to t in
, is differentiable with respect to t in
, and further satisfies
(6)
then
in
.
Proof 2 By setting
, where
is a positive constant satisfying
, we deduce
. Thus we have
(7)
Notice that
are continuous on
for each
and
is finite, we can find
such that
.
For the case that
, in view of the boundary condition of u in (6), we have
. Thus we have
in
, which implies
in
.
For the case that
, the above equation implies
for any
. In view of the definition of
, we have
(8)
Meanwhile it follows from the differentiability of
in
that
(9)
By substituting (8) and (9) into (7), we have
. Noting that
, we deduce
, which means
. Therefore, we have
in
. That is
in
.
Lemma 2.3 (Strong Maximun Principle). Suppose that
and K are constants. For any
, assume that
is continuous with respect to t in
, is differentiable with respect to t in
, and satisfies (6). If
for some
, then
in
.
Proof 3 Using the above maximum principle, we have
in
. By setting
, where
is defined as in the proof of Lemma 2.2, which satisfying
, we have
(10)
Notice that
are continuous on
for each
and
is finite, we deduce that
. Plugging (2) into (10), we have
(11)
Since
, (11) implies that
(12)
We prove the result by contradiction. If
in
cannot hold, there would exists a point
such that
, which implies
. By (7), we have
(13)
Since v is differential with respect to t in
, it follows that
. Thus (13) implies that
(14)
By (2), we also have
. In view of
and
, we have
. The above inequality implies that
(15)
On the other hand, since
is connected, for
, there exists a path
. By (15), we obtain that
. Employing the above argument repeatedly, we shall induce
in order, which contradicts with (12). The proof is completed.
Using the strong maximum principle, we easily have the following Lemma.
Lemma 2.4 Suppose that for each
,
is differentiable in
. Assume that
are constants. If w satisfies
(16)
or
(17)
then
uniformly for
. (18)
Lemma 2.5 Let
be a solution to the system (4) defined for
for any
. There exist a constant M such that
,
for
.
Proof 4 Applying Lemma 2.2 to the system (4), we obtain
,
for
. Consequently,
satisfies
for
with
for
. By choosing
, and applying Lemma 2.2 again, we have
for
.
Owing to the priori estimate of Lemma 2.5, we present the following global existence theorem.
Theorem 2.6 System (4) possesses a unique solution for all
.
3. Global Stability of the Disease-Free and Endemic Equilibria
Theorem 3.1 Define the basic reproduction number
, and the endemic equilibrium
with
,
. The disease- free equilibrium
is globally asymptotically stable if
, but the endemic equilibrium
is globally asymptotically stable if
.
Proof 5 We first prove the case of the disease-free equilibrium
. By Lemma 2.5, we have
for
. Then we find that S satisfies
(19)
Applying Lemma 2.4, for any small
, there exists
such that
(20)
Moreover, we have
(21)
Plugging (20) into system (4), we see that I satisfies
Since
, we choose
such that
.
Following Lemma 2.4, we have
uniformly in
. In view of the positivity of I, we have
uniformly in
. Consequently, for any small
, there exists
such that
, for
. Plugging this into system (4), we see that S satisfies
It follows from Lemma 2.4 that, for any small
, there exists
such that
for
. Due to the arbitrariness of
and
, it follows immediately that
(22)
Combining (21) with (22), we obtain
uniformly for
. Thus we prove that
is globally asymptotically stable.
We show the endemic equilibrium
by using Lyapunov functions. Define a Lyapunov function
Then
for all
, and
if and only if
. We can compute
(23)
In view of Lemma 2.1, it is easy to see that
, thus we have
(24)
Similarly, we have
. Plugging this and (24) into (23), we have
(25)
for all
. By applying the Lyapunov-LaSalle invariance principle [8] , we have
uniformly for
, which complete the proof.
4. Conclusion
Define the basic reproduction number
, and the endemic equilibrium
with
,
. We prove that the disease-free equilibrium
is globally asymptotically stable if
, while
the endemic equilibrium
is globally asymptotically stable if
. Our results extend the stability results of SIR models with graph Laplacian (
) studied in [2] to general graph p-Laplacian with
.
Acknowledgements
The work is partially supported by National Natural Science Foundation of China (11771380) and Natural Science Foundation of Jiangsu Province (BK20191436).