
Citation: Yunbo Tu, Shujing Gao, Yujiang Liu, Di Chen, Yan Xu. Transmission dynamics and optimal control of stage-structured HLB model[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5180-5205. doi: 10.3934/mbe.2019259
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Difference equations are the essentials required to understand even the simplest epidemiological model: the SIR-susceptible, infected, recovered-model. This model is a compartmental model, which results in the basic difference equation used to measure the actual reproduction number. It is this basic model that helps us determine whether a pathogen is going to die out or whether we end up having an epidemic. This is also the basis for more complex models, including the SVIR, which requires a vaccinated state, which helps us to estimate the probability of herd immunity.
There has been some recent interest in studying the qualitative analysis of difference equations and system of difference equations. Since the beginning of nineties there has be considerable interest in studying systems of difference equations composed by two or three rational difference equations (see, e.g., [4,5,6,2,8,9,11,10,14,15,17,19,20] and the references therein). However, given the multiplicity of factors involved in any epidemic, it will be important to study systems of difference equations composed by many rational difference equations, which is what we will do in this paper.
In [2], Devault et al. studied the boundedness, global stability and periodic character of solutions of the difference equation
xn+1=p+xn−mxn | (1) |
where
In [20], Zhang et al. investigated the behavior of the following symmetrical system of difference equations
xn+1=A+yn−myn,yn+1=A+xn−mxn | (2) |
where the parameter
Complement of the work above, in [8], Gümüş studied the global asymptotic stability of positive equilibrium, the rate of convergence of positive solutions and he presented some results about the general behavior of solutions of system (2). Our aim in this paper is to generalize the results concerning equation (1) and system (2) to the system of
x(1)n+1=A+x(2)n−mx(2)n,x(2)n+1=A+x(3)n−mx(3)n,…,x(p)n+1=A+x(1)n−mx(1)n,n,m,p∈N0 | (3) |
where
The remainder of the paper is organized as follows. In Section (2), we introduce some definitions and notations that will be needed in the sequel. Moreover, we present, in Theorem (2.4), a result concerning the linearized stability that will be useful in the main part of the paper. Section (3) discuses the behavior of positive solutions of system (3) via semi-cycle analysis method. Furthermore, Section (4) is devoted to study the local stability of the equilibrium points and the asymptotic behavior of the solutions when
In this section we recall some definitions and results that will be useful in our investigation, for more details see [3,7,14,13].
Definition 2.1. (see, [14]) A 'string' of sequential terms
A 'string' of sequential terms
A 'string' of sequential terms
A 'string' of sequential terms
Definition 2.2. (see, [14]) A function
(x(i)μ−¯x(i))(x(i)μ−¯x(i))≤0,i=1,2,…,p. |
We say that a solution
Let
f(i):Ik+11×Ik+12×…×Ik+1p→Ik+1i,i=1,2,…,p, |
where
{x(1)n+1=f(1)(x(1)n,x(1)n−1,…,x(1)n−k,x(2)n,x(2)n−1,…,x(2)n−k,…,x(p)n,x(p)n−1,…,x(p)n−k)x(2)n+1=f(2)(x(1)n,x(1)n−1,…,x(1)n−k,x(2)n,x(2)n−1,…,x(2)n−k,…,x(p)n,x(p)n−1,…,x(p)n−k)⋮x(p)n+1=f(p)(x(1)n,x(1)n−1,…,x(1)n−k,x(2)n,x(2)n−1,…,x(2)n−k,…,x(p)n,x(p)n−1,…,x(p)n−k) | (4) |
where
Define the map
F:I(k+1)1×I(k+1)2×…×I(k+1)p⟶I(k+1)1×I(k+1)2×…×I(k+1)p |
by
F(W)=(f(1)0(W),f(1)1(W),…,f(1)k(W),f(2)0(W),f(2)1(W),…, |
…,f(2)k(W),…,f(p)0(W),f(p)1(W),…,f(p)k(W)), |
where
W=(u(1)0,u(1)1,…,u(1)k,u(2)0,u(2)1,…,u(2)k,…,u(p)0,u(p)1,…,u(p)k)T, |
f(i)0(W)=f(i)(W),f(i)1(W)=u(i)0,…,f(i)k(W)=u(i)k−1,i=1,2,…,p. |
Let
Wn=(x(1)n,x(1)n−1,…,x(1)n−k,x(2)n,x(2)n−1,…,x(2)n−k,…,x(p)n,x(p)n−1,…,x(p)n−k)T. |
Then, we can easily see that system (4) is equivalent to the following system written in vector form
Wn+1=F(Wn),n∈N0. | (5) |
Definition 2.3. (see, [13]) Let
Xn+1=F(Xn)=BXn |
where
Theorem 2.4. (see, [13])
1. If all the eigenvalues of the Jacobian matrix
2. If at least one eigenvalue of the Jacobian matrix
In this section, we discuss the behavior of positive solutions of system (3) via semi-cycle analysis method. It is easy to see that system (3) has a unique positive equilibrium point
Lemma 3.1. Let
Proof. Suppose that
x(j)n0<A+1≤x(j)n0+1 or x(j)n0+1<A+1≤x(j)n0,j=1,2,…,p. |
We suppose the first case, that is,
x(j)n0<A+1≤x(j)n0+m,j=1,2,…,p. |
So, we get from system (3)
x(j)n0+m+1=A+x(j+1)mod(p)n0x(j+1)mod(p)n0+m<A+1,j=1,2,…,p. |
The Lemma is proved.
Lemma 3.2. Let
Proof. Assume that there exists
x(j)n0,x(j)n0+2,…,x(j)n0+m−1<A+1≤x(j)n0+1,x(j)n0+3,…,x(j)n0+m,j=1,2,…,p, | (6) |
or
x(j)n0+1,x(j)n0+3,…,x(j)n0+m<A+1≤x(j)n0,x(j)n0+2,…,x(j)n0+m−1,j=1,2,…,p. | (7) |
We will prove the case (6). The case (7) Is identical and will not be included. According to system (3) we obtain
x(j)n0+m+1=A+x(j+1)mod(p)n0x(j+1)mod(p)n0+m<A+1,j=1,2,…,p, |
and
x(j)n0+m+2=A+x(j+1)mod(p)n0+1x(j+1)mod(p)n0+m+1>A+1,j=1,2,…,p, |
The result proceeds by induction. Thus, the proof is completed.
Lemma 3.3. System (3) has no nontrivial periodic solutions of (not necessarily prime) period
Proof. Suppose that
(α(1)1,α(2)1,…,α(p)1),(α(1)2,α(2)2,…,α(p)2),…,(α(1)m,α(2)m,…,α(p)m),(α(1)1,α(2)1,…,α(p)1),… |
is a
(x(1)n−m,x(2)n−m,…,x(p)n−m)=(x(1)n,x(2)n,…,x(p)n),n≥0. |
So, the equilibrium solution
Lemma 3.4. All non-oscillatory solutions of system (3) converge to the equilibrium
Proof. We assume there exists non-oscillatory solutions of system (3). We will prove this lemma for the case of a single positive semi-cycle, the situation is identical for a single negative semi-cycle, so it will be omitted. Assume that
x(j)n+1=A+x(j+1)mod(p)n−mx(j+1)mod(p)n≥A+1,j=1,2,…,p, |
So, we get
A+1≤x(j)n≤x(j)n−m,n≥0,j=1,2,…,p | (8) |
From (8), there exists
limn→+∞x(j)nm+i=δ(j)i. |
Hence,
(δ(1)0,δ(2)0,…,δ(p)0),(δ(1)1,δ(2)1,…,δ(p)1),…,(δ(1)m−1,δ(2)m−1,…,δ(p)m−1) |
is a periodic solution of (not necessarily prime period) period
Theorem 4.1. Suppose
ⅰ): If
limn→+∞x(j)2n=+∞,limn→+∞x(j)2n+1=A. |
ⅱ): If
limn→+∞x(j)2n=A,limn→+∞x(j)2n+1=+∞. |
Proof. (ⅰ): From (3), for
x(i)1=A+x(i+1)mod(p)−mx(i+1)mod(p)0<A+1x(i+1)mod(p)0<A+(1−A)=1,x(i)2=A+x(i+1)mod(p)1−mx(i+1)mod(p)1>A+x(i+1)mod(p)1−m>x(i+1)mod(p)1−m>11−A. |
By induction, for n
x(i)2n−1<1,x(i)2n>11−A. | (9) |
So, from (3) and (9), we have
x(i)2n=A+x(i+1)mod(p)2n−1−mx(i+1)mod(p)2n−1>A+x(i+1)mod(p)2n−1−m>2A+x(i+1)mod(p)2n−3−m>3A+x(i+1)mod(p)2n−5−m>⋯ |
So
x(i)2n>nA+x(i+1)mod(p)0. | (10) |
By limiting the inequality (10), we get
limn→∞x(i)2n=∞. | (11) |
On the other hand, from(3), (9) and (11), we get
limn→∞x(i)2n+1=limn→∞(A+x(i+1)mod(p)2n−mx(i+1)mod(p)2n)=A. |
(ⅱ): The proof is similar to the proof of (ⅰ).
Open Problem. Investigate the asymptotic behavior of the system (3) when
Lemma 4.2. Suppose
Proof. Let
x(j)i∈[L,LL−1],i=1,2,…,m+1,j=1,2,…,p, |
where
L=min{α,ββ−1}>1,α=min1≤j≤m+1{x(1)j,x(2)j,…,x(p)j}, |
β=max1≤j≤m+1{x(1)j,x(2)j,…,x(p)ji}. |
So, we get
L=1+LL/(L−1)≤x(j)m+2=1+x(j+1)mod(p)1x(j+1)mod(p)m+1≤LL−1, |
thus, the following is obtained
L≤x(j)m≤LL−1. |
By induction, we get
x(j)i∈[L,LL−1],j=1,2,…,p,i=1,2,… | (12) |
Theorem 4.3. Suppose
lim infn→+∞x(i)n=lim infn→+∞x(j)n,i,j=1,2,…,p,lim supn→+∞x(i)n=lim supn→+∞x(j)n,i,j=1,2,…,p. |
Proof. From (12), we can set
Li=limn→∞supx(i)n,mi=limn→∞infx(i)n,i=1,2,…,p. | (13) |
We first prove the theorem for
L1≤1+L2m2,L2≤1+L1m1,m1≥1+m2L2,m2≥1+m2L2, |
which implies
L1m2≤m2+L2≤m1L2≤m1+L1≤m2L1 |
thus, the following equalities are obtained
m2+L2=m1+L1,L1m2=m1L2. |
So, we get that
Li=Lj,mi=mj,i,j=1,2,…,p−1, |
From system (3), we have
Lp−1≤1+Lpmp,Lp≤1+Lp−1mp−1,mp−1≥1+mpLp,mp≥1+mpLp, |
hence, we get
Lp−1mp≤mp+Lp≤mp−1Lp≤mp−1+Lp−1≤mpLp−1, |
consequently, the following equalities are obtained
mp+Lp=mp−1+Lp−1,Lp−1mp=mp−1Lp. |
So, we get that
Theorem 4.4. Assume that
Proof. The linearized equation of system (3) about the equilibrium point
Xn+1=BXn |
where
B=(JAOO…OOOJAO…OOOOJA…OO⋮⋮⋮⋮⋮⋮OOOO…JAAOOO…OJ) |
where
J=(00…0010…00⋮⋮⋱⋮⋮00…10),O=(00…0000…00⋮⋮⋱⋮⋮00…00), | (14) |
A=(−1A+10…01A+100…00⋮⋱…⋮⋮00…00). | (15) |
Let
D=diag(d1,d2,…,dpm+p) |
be a diagonal matrix where
0<ε<A−1(m+1)(A+1). | (16) |
It is obvious that
DBD−1=(J(1)A(1)OO…OOOJ(2)A(2)O…OOOOJ(3)A(3)…OO⋮⋮⋮⋮⋮⋮OOOO…J(p−1)A(p−1)A(p)OOO…OJ(p)), |
where
J(j)=(00…00d(j−1)m+j+1d(j−1)m+j0…00⋮⋮⋱⋮⋮00…d(j−1)m+m+jd(j−1)m+m+j−10),j=0,1,…,p, |
A(j)=(−1A+1djdjm+j+10…01A+1djdjm+j+100…00⋮⋱…⋮⋮00…00),j=0,1,…,p−1, |
and
A(p)=(−1A+1d(p−1)m+pd10…01A+1d(p−1)m+pdm+100…00⋮⋱…⋮⋮00…00). |
From
A(p)=(−1A+1d(p−1)m+pd10…01A+1d(p−1)m+pdm+100…00⋮⋱…⋮⋮00…00). |
Moreover, from
1A+1+1(1−(m+1)ε)(A+1)<1(1−(m+1)ε)(A+1)+1(1−(m+1)ε)(A+1)<2(1−(m+1)ε)(A+1)<1. |
It is common knowledge that
1A+1+1(1−(m+1)ε)(A+1)<1(1−(m+1)ε)(A+1)+1(1−(m+1)ε)(A+1)<2(1−(m+1)ε)(A+1)<1. |
We have that all eigenvalues of
To prove the global stability of the positive equilibrium, we need the following lemma.
Lemma 4.5. Suppose
Proof. Let
x(j)i∈[L,LL−A],i=1,2,…,m+1,j=1,2,…,p, |
where
L=min{α,ββ−1}>1,α=min1≤j≤m+1{x(1)j,x(2)j,…,x(p)j}, |
β=max1≤j≤m+1{x(1)j,x(2)j,…,x(p)ji}. |
So, we get
L=A+LL/(L−A)≤x(j)m+2=A+x(j+1)mod(p)1x(j+1)mod(p)m+1≤LL−1, |
thus, the following is obtained
L≤x(j)m≤LL−1. |
By induction, we get
x(j)i∈[L,LL−1],j=1,2,…,p,i=1,2,… | (17) |
Theorem 4.6. Assume that
Proof. Let
limn→∞(x(1)n,x(2)n,…,x(p)n)=(A+1,A+1,…,A+1). |
To do this, we prove that for
limn→∞x(i)n=A+1. |
From Lemma (4.5), we can set
Li=limn→∞supx(i)n,mi=limn→∞infx(i)n,i=1,2,…,p. | (18) |
So, from (3) and (13), we have
Li≤A+L(i+1)mod(p)m(i+1)mod(p),mi≥A+m(i+1)mod(p)L(i+1)mod(p). | (19) |
We first prove the theorem for
AL1+m1≤L1m2≤Am2+L2,AL2+m2≤L2m1≤Am1+L1. |
So,
AL1+m1−(Am1+L1)≤Am2+L2−(AL2+m2), |
hence
(A−1)(L1−m1+L2−m2)≤0, |
since
L1−m1+L2−m2=0, |
we know that
ALp+mp≤Lpm1≤Am1+L1,AL1+m1≤L1mp≤Amp+Lp. |
So,
ALp+mp−(Amp+Lp)≤Am1+L1−(AL1+m1), |
Thus, the following inequality is obtained
(A−1)(Lp−mp+L1−m1)≤0, |
since
Li=mi,=1,2,…,p. |
Therefore every positive solution
In this section, we estimate the rate of convergence of a solution that converges to the equilibrium point
Xn+1=(A+Bn)Xn | (20) |
where
‖Bn‖→0, when n→∞ | (21) |
where
Theorem 5.1. (Perron's first Theorem, see [16]) Suppose that condition (21) holds. If
ρ=limn→+∞‖Xn+1‖‖Xn‖ |
exists and is equal to the modulus of one of the eigenvalues of matrix
Theorem 5.2. (Perron's second Theorem, see [16]) Suppose that condition (21) holds. If
ρ=limn→+∞(‖Xn‖)1n |
exists and is equal to the modulus of one of the eigenvalues of matrix
Theorem 5.3. Assume that a solution
en=(e(1)ne(1)n−1⋮e(1)n−m⋮e(p)ne(p)n−1⋮e(p)n−m)=(x(1)n−¯x(1)x(1)n−1−¯x(1)⋮x(1)n−m−¯x(1)⋮x(p)n−¯x(p)x(p)n−1−¯x(p)⋮x(p)n−m−¯x(p)) |
of every solution of system (3) satisfies both of the following asymptotic relations:
limn→+∞‖en+1‖‖en‖=|λiJF((¯x(1),¯x(2),…,¯x(p)))|,i=1,2,…,m |
limn→+∞(‖en‖)1n=|λiJF((¯x(1),¯x(2),…,¯x(p)))|,i=1,2,…,m |
where
Proof. First, we will find a system that satisfies the error terms. The error terms are given as
x(j)n+1−¯x(j)=m∑i=0(j)A(1)i(x(1)n−i−¯x(1))+m∑i=0(j)A(2)i(x(2)n−i−¯x(2))+⋯+m∑i=0(j)A(1)i(x(p)n−i−¯x(p)), | (22) |
for
e(j)n=x(j)n−¯x(j),j=1,2,…,p |
Then, system (22) can be written as
e(j)n+1=m∑i=0(j)A(1)ie(1)n−i+m∑i=0(j)A(2)ie(2)n−i+⋯+m∑i=0(j)A(1)ie(p)n−i |
where
e(j)n+1=m∑i=0(j)A(1)ie(1)n−i+m∑i=0(j)A(2)ie(2)n−i+⋯+m∑i=0(j)A(1)ie(p)n−i |
and the others parameters
If we consider the limiting case, It is obvious then that
e(j)n+1=m∑i=0(j)A(1)ie(1)n−i+m∑i=0(j)A(2)ie(2)n−i+⋯+m∑i=0(j)A(1)ie(p)n−i |
That is
e(j)n+1=m∑i=0(j)A(1)ie(1)n−i+m∑i=0(j)A(2)ie(2)n−i+⋯+m∑i=0(j)A(1)ie(p)n−i |
where
en+1=(A+Bn)en |
where
A=JF((¯x(1),¯x(2),…,¯x(p)))=(JA(1)nOO…OOOJA(2)nO…OOOOJA(3)n…OO⋮⋮⋮⋮⋮⋮OOOO…JA(p−1)nA(p)nOOO…OJ) |
Bn=(JAOO…OOOJAO…OOOOJA…OO⋮⋮⋮⋮⋮⋮OOOO…JAAOOO…OJ) |
where
A(j)n=(α(j)n0…0β(j)n00…00⋮⋱…⋮⋮00…00),j=1,2,…,p. |
and
en+1=(JAOO…OOOJAO…OOOOJA…OO⋮⋮⋮⋮⋮⋮OOOO…JAAOOO…OJ)(e(1)ne(1)n−1⋮e(1)n−m⋮e(p)ne(p)n−1⋮e(p)n−m) |
and
In this section we will consider several interesting numerical examples to verify our theoretical results. These examples shows different types of qualitative behavior of solutions of the system (3). All plots in this section are drawn with Matlab.
Exemple 6.1. Let
x(1)n+1=1.2+x(2)n−1x(2)n,x(2)n+1=A+x(3)n−1x(3)n,…,x(10)n+1=1.2+x(1)n−1x(1)n,n∈N0 | (23) |
with
Exemple 6.2. Consider the system (23) with
Exemple 6.3. Consider the system (23) with
Exemple 6.4. Let
x(1)n+1=A+x(2)n−5x(2)n,x(2)n+1=A+x(3)n−5x(3)n,x(3)n+1=A+x(4)n−5x(4)n,x(4)n+1=A+x(1)n−5x(1)n,n∈N0 | (24) |
with
Exemple 6.5. Consider the system (24) with
Exemple 6.6. Consider the system (24) with
In the paper, we studied the global behavior of solutions of system (3) composed by
The findings suggest that this approach could also be useful for extended to a system with arbitrary constant different parameters, or to a system with a nonautonomous parameter, or to a system with different parameters and arbitrary powers. So, we will give the following some important open problems for difference equations theory researchers.
Open Problem 1. study the dynamical behaviors of the system of difference equations
x(1)n+1=A1+x(2)n−mx(2)n,x(2)n+1=A2+x(3)n−mx(3)n,…,x(p)n+1=Ap+x(1)n−mx(1)n,n,m,p∈N0 |
where
Open Problem 2. study the dynamical behaviors of the system of difference equations
x(1)n+1=αn+x(2)n−mx(2)n,x(2)n+1=αn+x(3)n−mx(3)n,…,x(p)n+1=αn+x(1)n−mx(1)n,n,m,p∈N0 |
where
Open Problem 3. study the dynamical behaviors of the system of difference equations
x(1)n+1=A1+(x(2)n−m)p1(x(2)n)q1,x(2)n+1=A2+(x(3)n−m)p2(x(3)n)q2,…,x(p)n+1=Ap+(x(1)n−m)pp(x(1)n)qp, |
where
This work was supported by Directorate General for Scientific Research and Technological Development (DGRSDT), Algeria.
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