Citation: Xiaojun Liu, Ikbel Souli, Mohamad-Amr Chamaa, Thomas Lendormi, Claire Sabourin, Yves Lemée, Virginie Boy, Nizar Chaira, Ali Ferchichi, Pascal Morançais, Jean-Louis Lanoisellé. Effect of thermal pretreatment at 70 °C for one hour (EU hygienization conditions) of various organic wastes on methane production under mesophilic anaerobic digestion[J]. AIMS Environmental Science, 2018, 5(2): 117-129. doi: 10.3934/environsci.2018.2.117
[1] | Chentong Li, Jinyan Wang, Jinhu Xu, Yao Rong . The Global dynamics of a SIR model considering competitions among multiple strains in patchy environments. Mathematical Biosciences and Engineering, 2022, 19(5): 4690-4702. doi: 10.3934/mbe.2022218 |
[2] | Maoxing Liu, Yuhang Li . Dynamics analysis of an SVEIR epidemic model in a patchy environment. Mathematical Biosciences and Engineering, 2023, 20(9): 16962-16977. doi: 10.3934/mbe.2023756 |
[3] | Siyu Liu, Yong Li, Yingjie Bi, Qingdao Huang . Mixed vaccination strategy for the control of tuberculosis: A case study in China. Mathematical Biosciences and Engineering, 2017, 14(3): 695-708. doi: 10.3934/mbe.2017039 |
[4] | Lili Liu, Xi Wang, Yazhi Li . Mathematical analysis and optimal control of an epidemic model with vaccination and different infectivity. Mathematical Biosciences and Engineering, 2023, 20(12): 20914-20938. doi: 10.3934/mbe.2023925 |
[5] | Mostafa Adimy, Abdennasser Chekroun, Claudia Pio Ferreira . Global dynamics of a differential-difference system: a case of Kermack-McKendrick SIR model with age-structured protection phase. Mathematical Biosciences and Engineering, 2020, 17(2): 1329-1354. doi: 10.3934/mbe.2020067 |
[6] | Holly Gaff, Elsa Schaefer . Optimal control applied to vaccination and treatment strategies for various epidemiological models. Mathematical Biosciences and Engineering, 2009, 6(3): 469-492. doi: 10.3934/mbe.2009.6.469 |
[7] | Kento Okuwa, Hisashi Inaba, Toshikazu Kuniya . Mathematical analysis for an age-structured SIRS epidemic model. Mathematical Biosciences and Engineering, 2019, 16(5): 6071-6102. doi: 10.3934/mbe.2019304 |
[8] | Eunha Shim . A note on epidemic models with infective immigrants and vaccination. Mathematical Biosciences and Engineering, 2006, 3(3): 557-566. doi: 10.3934/mbe.2006.3.557 |
[9] | Muntaser Safan . Mathematical analysis of an SIR respiratory infection model with sex and gender disparity: special reference to influenza A. Mathematical Biosciences and Engineering, 2019, 16(4): 2613-2649. doi: 10.3934/mbe.2019131 |
[10] | Steady Mushayabasa, Drew Posny, Jin Wang . Modeling the intrinsic dynamics of foot-and-mouth disease. Mathematical Biosciences and Engineering, 2016, 13(2): 425-442. doi: 10.3934/mbe.2015010 |
With the development of transportation and urbanisation, population migration across regions becomes more frequent, and more and more rural population crowded into cities. The increasing mobility among regions might lead to the spread of the infectious diseases regionally and globally much faster than ever before [19]. For example, SARS was first reported in Guangdong Province of China in November of 2002, and in late June of 2003, the emerging infectious disease had spread to 32 countries and regions due to the human mobility [21,25]. In February 2014 Ebola virus appeared in Guinea and then due to the human mobility the disease spread very quickly to other countries including the United States, Spain and the United Kingdom et al [12], and has caused about 6070 reported deaths and 17145 reported cases of Ebola virus disease up to December 3,2014 according to the report from the World Health Organization (WHO) [6]. All the above facts show that the population dispersal can affect transmission dynamics of the infectious diseases.
In the recent years, the impact of population dispersal has received increasing attention, and many mathematical patch models are formulated to investigate this hot issue (see [24,3,14] and the references cited therein). Here, the patches can be cities, towns, states, countries or other appropriate community divisions. Wang and Zhao [24] proposed an epidemic model with population dispersal to describe the dynamics of disease spread between two patches and n patches. Arino and van den Driessche [3] developed a multi-city epidemic model to analyze the spatial spread of infectious diseases. In 2011, Gao and Ruan [15] formulated an SIS patch model with non-constant transmission coefficients to investigate the effect of media coverage and human movement on the spread of infectious diseases among patches, and soon after, Gao and Ruan [13] proposed a multi-patch model to study the impacts of population dispersal on the spatial spread of malaria between patches. All the above mathematical models have provided useful information about the effect of host mobility on transmission dynamics of infectious diseases, but almost these models do not include the control measures, such as vaccination in it.
There is no doubt that the top priority of global public security is to prevent and contain the spread of infectious diseases. Thus it is important to study how to control the spread of infectious diseases in patchy environment and how the increasing mobility of hosts affects the current public health security. In this paper, we will use a mathematical model to explore this important issue. As we all know, vaccination is one of the most effective biological means of containing the outbreak of infectious diseases, which inoculates antigenic material into the individuals to stimulate immune system to develop adaptive immunity to a pathogen. Since Edward Jenner, the founder of vaccinology, inoculated a 13 year-old-boy with vaccinia virus (cowpox) and demonstrated immunity to smallpox [18] in 1796, vaccination has played an important role in controlling and preventing the outbreak of infectious diseases. The widespread immunity due to vaccination is largely responsible for the worldwide eradication of smallpox and the restriction of infectious diseases, such as polio, measles, and tetanus from much of the world [17]. Over the past two decades, many modeling studies have been conducted the effect of vaccination on transmission dynamics of infectious diseases (see [1,2] and reference therein). However, most of the epidemic models with vaccination are formulated in an isolated patch, ignoring spatial heterogeneity both for populations and disease transmissions.
The main purpose of the paper is to formulate an SIR epidemic model to study the impact of vaccination on transmission dynamics of infectious disease in patchy environment and the impact of the increasing mobility of hosts on the current immunization strategy. The paper is organized as follows. In Section 2, based on the SIR model with birth targeted vaccination we propose an SIR epidemic model with vaccination in patchy environment. In Section 3, we mainly present some preliminary results and derive the reproduction number. A classification of the equilibria of system on two patches and its the local dynamical behavior is provided in Section 4. We conclude with some numerical simulations in Section 5 and give a brief conclusion in the final section.
In this section, we employ an
First, let us formulate a model for the spread of the disease in the
We assume that the hosts are recruited at a rate
Based on the transfer diagram 1, the spread of an infectious disease in the
{dSidt=(1−pi)μiNi−βiIiNiSi−μiSi,dIidt=βiIiNiSi−(μi+γi)Ii,dRidt=piμiNi+γiIi−μiRi. | (1) |
When
mij=−ln(1−lij)1d, i,j=1,2,⋯,n, i≠j. | (2) |
Then the dynamics of the hosts with migration is governed by the following model:
{dSidt=(1−pi)μiNi−βiIiNiSi−μiSi+n∑j≠i(mjiSj−mijSi),dIidt=βiIiNiSi−(μi+γi)Ii,dRidt=piμiNi+γIi−μiRi+n∑j≠i(mjiRj−mijRi),Ni=Si+Ii+Ri,i=1,2,⋯,n. | (3) |
In this paper, we will use the system (3) to investigate the effect of vaccination on transmission dynamics of infectious disease in patchy environment and the impact of the increasing mobility of hosts on the current immunization strategy.
We first introduce some notations which will be used throughout this paper. Let
Let
Γ={(S1,I1,R1,⋯,Sn,In,Rn)∈R3n+:n∑i=1(Si+Ii+Ri)≤N(0),i=1,2,⋯,n}, |
is positively invariant with respect to system (3).
Define movement matrix
M=(∑nj≠1m1j−m21⋯−mn1−m12∑nj≠2m2j⋯−mn2⋮⋮⋱⋮−m1n−m2n⋯∑nj≠nmnj). | (4) |
In this paper, we always assume that the movement matrix is irreducible, that is, the graph of the patches are strongly connected through the movement of hosts with respect to disease. If the movement matrix is reducible, the system may be decoupled into several samll systems (see [11] and reference therein).
To find the disease-free equilibrium with all
{(1−pi)μiNi−μiSi+n∑j≠i(mjiSj−mijSi)=0,piμiNi−μiRi+n∑j≠i(mjiRj−mijRi)=0,Ni=Si+Ri,i=1,2,⋯,n, | (5) |
or in the form of matrix systems
{Diag((1−p)∗μ)N−(M+Diag(μ))S=0,Diag(p∗μ)N−(M+Diag(μ))R=0,MN=0, | (6) |
where
We first solve the third equation of (5) or (6) which independent of the first two equations. Applying the results presented in Lemma 2.1 [16], the general solution to the third equation of (6) can be given as
N0≜(N01,N02,⋯,N0n)T=N(0)∑ni=1cii(c11,c22,⋯,cnn)T. |
Substituting
S0≜(S01,S02,⋯,S0n)=(M+Diag(μ))−1Diag((1−p)∗μ)N0,R0≜(R01,R02,⋯,R0n)=(M+Diag(μ))−1Diag(p∗μ)N0. | (7) |
Since all off-diagonal entries of matrix
In absence of infectious disease, adding the three equations of system (3) together leads to
dNidt=n∑j≠i(mjiNj−mijNi), i=1,2,⋯,n, | (8) |
or in the form of matrix system
dN(t)dt=−MN(t). | (9) |
It follows from Theorem 2.1 in [4] that the positive equilibrium
{(S−S0)′=−(M+Diag(μ))(S−S0),(R−R0)′=−(M+Diag(μ))(R−R0). | (10) |
Since the Gerschgorin circular disc theorem implies that matrix
Theorem 3.1. System (3) always has a disease-free equilibrium
S0=(M+Diag(μ))−1Diag((1−p)∗μ)N0,R0=(M+Diag(μ))−1Diag(p∗μ)N0,N0=N(0)∑ni=1cii(c11,c22,⋯,cnn)T, |
and
Γ0={(S1,⋯,Sn,I1,⋯,In,R1,⋯,Rn):n∑i=1(Si+Ri)=N(0),Ii=0,i=1,2,⋯,n}. |
Note that the system (3) has
F=Diag(β1S01N01,β2S02N02,⋯,βnS0nN0n) and V=Diag(μ+γ). |
From literature [10], the reproduction number
Rv=ρ(FV−1)=max | (11) |
where
\mathfrak{R}_{vi}=\frac{\beta_i}{\mu_i+\gamma_i}\frac{S_i^0}{N_i^0}, | (12) |
which represents the reproduction number in the
Theorem 3.2. The disease-free equilibrium
In the special case
\mathfrak{R}_v=(1-p_1)\frac{\beta_1}{\mu_1+\gamma_1}. | (13) |
which represents the numbers of secondary cases directly produced by infectious diease during the period of infection in a susceptible population.
In the special case of no movement between patches (i.e.,
\mathfrak{R}_v= \max\{\mathfrak{R}_{v1},\mathfrak{R}_{v2},\cdots,\mathfrak{R}_{vn}\}, | (14) |
with
Theorem 3.3. If
Proof. For any equilibrium
\left\{ \begin{array}{ll} \displaystyle {\rm Diag}((1-p)*\mu) {\bf{N}}-{\rm Diag}(\mu+\gamma) {\bf{I}}-(M+{\rm Diag}(\mu)){\bf{S}}=0,\\[2ex] \displaystyle\displaystyle {\rm Diag}({\bf{I}})({\bf{S}}-B{\bf{N}})=0,\\[2ex] \displaystyle {\rm Diag}(p*\mu) {\bf{N}}+{\rm Diag}(\gamma) {\bf{I}}-(M+{\rm Diag}(\mu) {\bf{R}}=0,\\[2ex] \displaystyle {\bf{N}} = {\bf{S}}+{\bf{I}}+{\bf{R}}. \end{array}\right. | (15) |
where
Adding the first three equations of (15) together yields
{\bf{N}}-{\bf{I}}=k(c_{11},c_{22},\cdots,c_{nn})^T, |
where
\begin{array}{rl} {\bf{N}}={\bf{N}}^0+(\mathbb{E}-C){\bf{I}}, \end{array} | (16) |
where
\begin{array}{rl} {\bf{S}}=&{\bf{S}}^0+(M+{\rm Diag}(\mu))^{-1}\Big({\rm Diag}(({\bf{1}}-{\bf{p}})*\mu)(\mathbb{E}-C)-{\rm Diag}(\mu+\gamma)\Big){\bf{I}}, \end{array} | (17) |
where
Substituting (16), (17) into the second equation of (15), the system of equation (15) can be reduced to the following equation with one single equation of
\begin{array}{l} \displaystyle {\rm Diag}({\bf{I}})\Big({\bf{S}}^0-B{\bf{N}}^0-((M+{\rm Diag}(\mu))^{-1}{\rm Diag}(({\bf{1}}-{\bf{p}})\mu)C-B C){\bf{I}}\\ -(B+(M+{\rm Diag}(\mu))^{-1}({\rm Diag}(\mu+\gamma)-{\rm Diag}(({\bf{1}}-{\bf{p}})\mu)){\bf{I}}\Big)=0. \end{array} | (18) |
In the following, we only need to solve (18) for
Since
\begin{array}{rl} &\displaystyle(M+{\rm Diag}(\mu))^{-1}{\rm Diag}(({\bf{1}}-{\bf{p}})*\mu)C-BC \\ &=\displaystyle (M+{\rm Diag}(\mu))^{-1}\left({\rm Diag}(({\bf{1}}-{\bf{p}})*\mu){\bf{N}}^0-(M+{\rm Diag}(\mu))B{\bf{N}}^0\right)\frac{1}{N(0)}{\bf{1}} \\ &=\displaystyle \left({\bf{S}}^0-B{\bf{N}}^0\right)\frac{1}{N(0)}{\bf{1}}, \end{array} | (19) |
and the expression for
{\bf{S}}^0-B{\bf{N}}^0=\displaystyle \left( \begin{array}{c} \displaystyle\frac{\mu_1+\gamma_1}{\beta_1}N_1^0(\mathfrak{R}_{v1}-1)\\ \displaystyle\frac{\mu_2+\gamma_2}{\beta_2}N_2^0(\mathfrak{R}_{v2}-1)\\ \vdots\\ \displaystyle\frac{\mu_n+\gamma_n}{\beta_n}N_n^0(\mathfrak{R}_{vn}-1) \end{array} \right). | (20) |
Therefore, the equation (18) can be expressed as
{\rm Diag}({\bf{I}})(M+{\rm Diag}(\mu))^{-1}({\bf{b}}-A{\bf{I}})=0, | (21) |
where
\label{2} {\bf{b}}\triangleq(b_1,b_2,\cdots,b_n)^T=(M+{\rm Diag}(\mu))({\bf{S}}^0-B{\bf{N}}^0), |
and
\label{1} \begin{array}{rl} A\triangleq (a_{ij})_{n\times n}=&{\rm Diag}(\gamma+{\bf{p}}\mu)+(M+{\rm Diag}(\mu))B\\[2ex] &\displaystyle +(M+{\rm Diag}(\mu))\left({\bf{S}}^0-B{\bf{N}}^0\right)\frac{1}{N(0)}{\bf{1}}.\\[2ex] \end{array} |
Note that
It is easily to see that
In this section, we mainly consider the dynamic behaviors for system (3) with
\label{dfe2} \begin{array}{l} \displaystyle S_1^0=\frac{((1-p_1)(\mu_1\mu_2+\mu_1m_{21})+(1-p_2)\mu_2m_{12})m_{21}N(0)} {(\mu_1\mu_2+\mu_1m_{21}+\mu_2m_{12})(m_{12}+m_{21})},\\[2ex] \displaystyle S_2^0=\frac{((1-p_1)\mu_1m_{21}+(1-p_2)(\mu_1\mu_2+\mu_2m_{12})m_{12}N(0)} {(\mu_1\mu_2+\mu_1m_{21}+\mu_2m_{12})(m_{12}+m_{21})},\\[2ex] \displaystyle R_1^0=\frac{(p_1(\mu_1\mu_2+\mu_1m_{21})+p_2\mu_2m_{12})m_{21}N(0)} {(\mu_1\mu_2+\mu_1m_{21}+\mu_2m_{12})(m_{12}+m_{21})},\\[2ex] \displaystyle R_2^0=\frac{(p_1\mu_1m_{21}+p_2(\mu_1\mu_2+\mu_2m_{12})m_{12}N(0)} {(\mu_1\mu_2+\mu_1m_{21}+\mu_2m_{12})(m_{12}+m_{21})}.\\[2ex] \end{array} |
From (11) and (12), the control reproduction number for this case can be given by
\mathfrak{R}_v=\max\{\mathfrak{R}_{v1},\mathfrak{R}_{v2}\}, | (22) |
where
\begin{array}{l} \displaystyle\mathfrak{R}_{v1}=\frac{\beta_1}{\mu_1+\gamma_1}\frac{(1-p_1)(\mu_1\mu_2+\mu_1m_{21})+(1-p_2)\mu_2m_{12}}{\mu_1\mu_2+\mu_1m_{21}+\mu_2m_{12}},\\ \displaystyle \mathfrak{R}_{v2}=\frac{\beta_2}{\mu_2+\gamma_2}\frac{(1-p_1)\mu_1m_{21}+(1-p_2)(\mu_1\mu_2+\mu_2m_{12})}{\mu_1\mu_2+\mu_1m_{21}+\mu_2m_{12}}, \end{array} | (23) |
represent the control reproduction number correspond to the sub-patch 1 and 2, respectively.
Like in the single patch model (1) or many other epidemic models, we have the global stability of the disease-free equilibrium for system (3) with
Theorem 4.1. If
The proof of Theorem (4.1) is analogous to those of Theorem 2.4 in Gao and Ruan [15] and Theorem 3.2 in Sun et al. [23]. We omit the details here.
Following Theorem 4.1 and the proof of Theorem 4.1, when
For convenience of presentation, set
\begin{array}{l} \xi_1=\mu_1\mu_2+\mu_1m_{21}+\mu_2m_{12}, \ \xi_2=(\gamma_2+p_2\mu_2)\beta_2, \ \xi_3=(\gamma_1+p_1\mu_1)\beta_1, \\ \xi_4=(\mu_1+\gamma_1)(\mu_2+\gamma_2)(\mu_2+m_{21})+(\mu_1+\gamma_1)\xi_2+(\mu_2+\gamma_2)^2m_{12},\\ \xi_5=(\mu_1+\gamma_1)(\mu_2+\gamma_2)(\mu_1+m_{12})+(\mu_2+\gamma_2)\xi_3+(\mu_1+\gamma_1)^2m_{21}, \end{array} |
and define
\begin{array}{rl} \displaystyle \mathfrak{\bar{R}}_{v1}=\frac{((\mu_2+\gamma_2)(\mu_2+m_{21})+\xi_2)(1-p_1)\mu_1\beta_1+(\mu_2+\gamma_2)^2m_{12}\beta_1} {(\mu_1+\gamma_1)((\mu_2+\gamma_2)\xi_1+(\mu_1+m_{12})\xi_2)},\\ \displaystyle \mathfrak{\bar{R}}_{v2}=\frac{((\mu_1+\gamma_1)(\mu_1+m_{12})+\xi_3)(1-p_2)\mu_2\beta_2+(\mu_1+\gamma_1)^2m_{21}\beta_2} {(\mu_2+\gamma_2)((\mu_1+\gamma_1)\xi_1+(\mu_2+m_{21})\xi_3)}, \end{array} | (24) |
which can be considered as a second threshold for epidemic invasion of sub-populations 1 and 2, respectively.
Theorem 4.2. The system (3) can have other three equilibria, and we have the following results:
1. Boundary equilibria
\label{en:boundaryI2}\begin{array}{l} \displaystyle \hat{S}_1\mspace{-3mu}=\mspace{-3mu}\frac{((\mu_1+\gamma_1)(\mu_2+m_{21})\mspace{-3mu}+\mspace{-3mu}m_{12}(1-p_2)\mu_2)(\mu_1+\gamma_1)m_{21}N(0)} {(\mu_1\mspace{-3mu}+\mspace{-3mu}\gamma_1)(m_{12}\xi_1\mspace{-3mu}+\mspace{-3mu} (\mu_2\mspace{-3mu}+\mspace{-3mu}m_{21})m_{21}\beta_1)\mspace{-3mu}+\mspace{-3mu}m_{12} (\mu_2\mspace{-3mu}+\mspace{-3mu}m_{21})\xi_3\mspace{-3mu}+\mspace{-3mu}m_{12}m_{21} (1-p_2)\mu_2\beta_1},\\ \displaystyle\hat{S}_2\mspace{-3mu}=\mspace{-3mu}\frac{((\mu_1\mspace{-3mu}+\mspace{-3mu}\gamma_1)^2m_{21}\mspace{-3mu} +\mspace{-3mu}((\mu_1\mspace{-3mu}+\mspace{-3mu}\gamma_1)(\mu_1\mspace{-3mu} +\mspace{-3mu}m_{12})\mspace{-3mu}+\mspace{-3mu}(\gamma_1\mspace{-3mu} +\mspace{-3mu}p_1\mu_1)\beta_1)(1\mspace{-3mu}-\mspace{-3mu}p_2)\mu_2)m_{12}N(0)} {(\mu_1\mspace{-3mu}+\mspace{-3mu}\gamma_1)(m_{12}\xi_1\mspace{-3mu}+\mspace{-3mu}(\mu_2\mspace{-3mu}+\mspace{-3mu}m_{21})m_{21}\beta_1)\mspace{-3mu}+\mspace{-3mu}m_{12}(\mu_2\mspace{-3mu}+\mspace{-3mu}m_{21})\xi_3\mspace{-3mu}+\mspace{-3mu}m_{12}m_{21} (1-p_2)\mu_2\beta_1},\\ \displaystyle\hat{I}_1\mspace{-3mu}=\mspace{-3mu}\frac{(\mu_1\mspace{-3mu}+\mspace{-3mu}\gamma_1)\xi_1m_{21}(\mathfrak{R}_{v1}-1)N(0)} {(\mu_1\mspace{-3mu}+\mspace{-3mu}\gamma_1)(m_{12}\xi_1\mspace{-3mu}+\mspace{-3mu} (\mu_2\mspace{-3mu}+\mspace{-3mu}m_{21})m_{21}\beta_1)\mspace{-3mu}+\mspace{-3mu}m_{12} (\mu_2\mspace{-3mu}+\mspace{-3mu}m_{21})\xi_3\mspace{-3mu}+m_{12}m_{21} (1-p_2)\mu_2\beta_1},\\ \displaystyle\hat{R}_1\mspace{-3mu}=\mspace{-3mu}\frac{((\mu_2\mspace{-3mu}+\mspace{-3mu}m_{21})(\xi_3-(\mu_1\mspace{-3mu}+\mspace{-3mu}\gamma_1)\gamma_1) \mspace{-3mu}+\mspace{-3mu}m_{12}(\mu_1\mspace{-3mu}+\mspace{-3mu}\gamma_1)p_2\mu_2)m_{21}N(0)} {(\mu_1\mspace{-3mu}+\mspace{-3mu}\gamma_1)(m_{12}\xi_1\mspace{-3mu}+\mspace{-3mu}(\mu_2\mspace{-3mu}+\mspace{-3mu}m_{21})m_{21}\beta_1)\mspace{-3mu}+\mspace{-3mu}m_{12}(\mu_2\mspace{-3mu}+\mspace{-3mu}m_{21})\xi_3\mspace{-3mu}+\mspace{-3mu}m_{12}m_{21} (1-p_2)\mu_2\beta_1},\\ \displaystyle\hat{R}_2\mspace{-3mu}=\mspace{-3mu}\frac{(m_{21}(\xi_3-(\mu_1\mspace{-3mu}+\mspace{-3mu}\gamma_1)\gamma_1)\mspace{-3mu}+\mspace{-3mu}((\mu_1\mspace{-3mu}+\mspace{-3mu}\gamma_1)(\mu_1\mspace{-3mu}+\mspace{-3mu}m_{12}) \mspace{-3mu}+\mspace{-3mu}(\gamma_1\mspace{-3mu}+\mspace{-3mu}p_1\mu_1)\beta_1)p_2\mu_2)m_{12}N(0)} {(\mu_1\mspace{-3mu}+\mspace{-3mu}\gamma_1)(m_{12}\xi_1\mspace{-3mu}+\mspace{-3mu}(\mu_2\mspace{-3mu}+\mspace{-3mu}m_{21})m_{21}\beta_1)\mspace{-3mu}+\mspace{-3mu}m_{12}(\mu_2\mspace{-3mu}+\mspace{-3mu}m_{21})\xi_3\mspace{-3mu}+\mspace{-3mu}m_{12}m_{21} (1-p_2)\mu_2\beta_1}, \end{array} |
and
\label{en:boundaryI1} \begin{array}{l} \displaystyle \displaystyle \bar{S}_1\mspace{-3mu}=\mspace{-3mu}\frac{((\mu_2\mspace{-3mu}+\mspace{-3mu}\gamma_2)^2m_{12}\mspace{-3mu}+\mspace{-3mu}((\mu_2\mspace{-3mu}+\mspace{-3mu}\gamma_2)(\mu_2\mspace{-3mu}+\mspace{-3mu}m_{21})\mspace{-3mu}+\mspace{-3mu}(\gamma_2\mspace{-3mu}+\mspace{-3mu}p_2\mu_2)\beta_2)(1-p_1)\mu_1)m_{21}N(0)} {(\mu_2\mspace{-3mu}+\mspace{-3mu}\gamma_2)(m_{21}\xi_1\mspace{-3mu}+\mspace{-3mu}(\mu_1\mspace{-3mu}+\mspace{-3mu}m_{12})m_{12}\beta_2)\mspace{-3mu}+\mspace{-3mu}m_{21}(\mu_1\mspace{-3mu}+\mspace{-3mu}m_{12})\xi_2\mspace{-3mu}+\mspace{-3mu}m_{12}m_{21} (1-p_1)\mu_1\beta_2}, \\ \displaystyle \bar{S}_2\mspace{-3mu}=\mspace{-3mu}\frac{((\mu_2\mspace{-3mu}+\mspace{-3mu}\gamma_2)(\mu_1\mspace{-3mu}+\mspace{-3mu}m_{12})\mspace{-3mu}+\mspace{-3mu}m_{21}(1-p_1)\mu_1)(\mu_2\mspace{-3mu}+\mspace{-3mu}\gamma_2)m_{12}N(0)} {(\mu_2\mspace{-3mu}+\mspace{-3mu}\gamma_2)(m_{21}\xi_1\mspace{-3mu}+\mspace{-3mu}(\mu_1\mspace{-3mu}+\mspace{-3mu}m_{12})m_{12}\beta_2)\mspace{-3mu}+\mspace{-3mu}m_{21}(\mu_1\mspace{-3mu}+\mspace{-3mu}m_{12})\xi_2\mspace{-3mu}+\mspace{-3mu}m_{12}m_{21} (1-p_1)\mu_1\beta_2},\\ \displaystyle\bar{I}_2\mspace{-3mu}=\mspace{-3mu}\frac{(\mu_2\mspace{-3mu}+\mspace{-3mu}\gamma_2)\xi_1m_{12}(\mathfrak{R}_{v2}-1)N(0)} {(\mu_2\mspace{-3mu}+\mspace{-3mu}\gamma_2)(m_{21}\xi_1\mspace{-3mu}+\mspace{-3mu}(\mu_1\mspace{-3mu}+\mspace{-3mu}m_{12})m_{12}\beta_2)\mspace{-3mu}+\mspace{-3mu}m_{21}(\mu_1\mspace{-3mu}+\mspace{-3mu}m_{12})\xi_2\mspace{-3mu}+\mspace{-3mu}m_{12}m_{21} (1-p_1)\mu_1\beta_2},\\%[11pt] \end{array} |
\begin{array}{l} \displaystyle \bar{R}_1\mspace{-3mu}=\mspace{-3mu}\frac{(m_{12}(\xi_2-(\mu_2\mspace{-3mu}+\mspace{-3mu}\gamma_2)\gamma2)\mspace{-3mu}+\mspace{-3mu}((\mu_2\mspace{-3mu}+\mspace{-3mu}\gamma_2)(\mu_2\mspace{-3mu}+\mspace{-3mu}m_{21}) \mspace{-3mu}+\mspace{-3mu}(\gamma_2\mspace{-3mu}+\mspace{-3mu}p_2\mu_2)\beta_2)p_1\mu_1)m_{21}N(0)} {(\mu_2\mspace{-3mu}+\mspace{-3mu}\gamma_2)(m_{21}\xi_1\mspace{-3mu}+\mspace{-3mu}(\mu_1\mspace{-3mu}+\mspace{-3mu}m_{12})m_{12}\beta_2)\mspace{-3mu}+\mspace{-3mu}m_{21}(\mu_1\mspace{-3mu}+\mspace{-3mu}m_{12})\xi_2\mspace{-3mu}+\mspace{-3mu}m_{12}m_{21} (1-p_1)\mu_1\beta_2},\\ \displaystyle\bar{R}_2\mspace{-3mu}=\mspace{-3mu}\frac{((\mu_1\mspace{-3mu}+\mspace{-3mu}m_{12})(\xi_2-(\mu_2\mspace{-3mu}+\mspace{-3mu}\gamma_2)\gamma_2) \mspace{-3mu}+\mspace{-3mu}m_{21}(\mu_2\mspace{-3mu}+\mspace{-3mu}\gamma_2)p_1\mu_1)m_{12}N(0)} {(\mu_2\mspace{-3mu}+\mspace{-3mu}\gamma_2)(m_{21}\xi_1\mspace{-3mu}+\mspace{-3mu}(\mu_1\mspace{-3mu}+\mspace{-3mu}m_{12})m_{12}\beta_2)\mspace{-3mu}+\mspace{-3mu}m_{21}(\mu_1\mspace{-3mu}+\mspace{-3mu}m_{12})\xi_2\mspace{-3mu}+\mspace{-3mu}m_{12}m_{21} (1-p_1)\mu_1\beta_2}.\\%[11pt] \end{array} |
2. System (3) has a unique endemic equilibrium
\begin{split} \displaystyle S_1^*=&\frac{((\mu_1+\gamma_1)((\mu_2+\gamma_2)(\mu_2+m_{21})+\xi_2)+(\mu_2+\gamma_2)^2m_{12})(\mu_1+\gamma_1)m_{21}N(0)} {m_{21}\beta_1\xi_4+m_{12}\beta_2\xi_5},\\ \displaystyle S_2^*=&\frac{((\mu_2+\gamma_2)((\mu_1+\gamma_1)(\mu_1+m_{12})+\xi_3)+(\mu_1+\gamma_1)^2m_{21})(\mu_2+\gamma_2)m_{12}N(0)} {m_{21}\beta_1\xi_4+m_{12}\beta_2\xi_5},\\ \displaystyle I_1^*=&\frac{(\mu_1+\gamma_1)((\mu_1+m_{12})\xi_2+(\mu_2+\gamma_2)\xi_1)(\mathfrak{\bar{R}}_{v1}-1)m_{21}N(0)} {m_{21}\beta_1\xi_4+m_{12}\beta_2\xi_5},\\ \displaystyle I_2^*=&\frac{(\mu_2+\gamma_2)((\mu_2+m_{21})\xi_3+(\mu_1+\gamma_1)\xi_1)(\mathfrak{\bar{R}}_{v2}-1)m_{12}N(0)} {m_{21}\beta_1\xi_4+m_{12}\beta_2\xi_5},\\ \displaystyle R_1^*=&\frac{(\xi_2 +(\mu_2+ \gamma_2) (\mu_2 + m_{21}) ) (\xi_3- \gamma_1(\mu_1 + \gamma_1) ) m_{21} N (0)} {m_{21}\beta_1\xi_4 + m_{12}\beta_2\xi_5}\\ &+\frac{m_{12}(\mu_1 + \gamma_1) (\xi_2 - \gamma_2(\mu_2 + \gamma_2) ) m_{21} N (0)}{m_{21}\beta_1\xi_4 + m_{12}\beta_2\xi_5},\\ \displaystyle R_2^*=&\frac{ (\xi_3 + (\mu_1 + \gamma_1) (\mu_1 + m_{12}) ) (\xi_2 - \gamma_2 (\mu_2 + \gamma_2) )m_{12} N (0)} {m_{21}\beta_1\xi_4 + m_{12}\beta_2\xi_5}\\ &+\frac{ m_{21}(\mu_2 + \gamma_2) (\xi_3 - \gamma_1(\mu_1 + \gamma_1) ) m_{12} N (0)} {m_{21}\beta_1\xi_4 + m_{12}\beta_2\xi_5}.\\%[9pt] \end{split} |
It follows from the expressions of
\label{d} \begin{array}{l} \displaystyle \mathfrak{\bar{R}}_{v1}-\mathfrak{R}_{v1}=\frac{(\mu_2+\gamma_2)(\gamma_2+p_2\mu_2)m_{12}\beta_1} {(\mu_1+\gamma_1)((\mu_2+\gamma_2) \xi_1+(\mu_1+m_{12})\xi_2)}(1-\mathfrak{R}_{v2}),\\ \displaystyle \mathfrak{\bar{R}}_{v2}-\mathfrak{R}_{v2}=\frac{(\mu_1+\gamma_1)(\gamma_1+p_1\mu_1)m_{21}\beta_2} {(\mu_2+\gamma_2)((\mu_1+\gamma_1) \xi_1+(\mu_2+m_{21})\xi_3)}(1-\mathfrak{R}_{v1}). \end{array} |
Thus, if
Using
\left\{ \begin{array}{ll} \displaystyle \frac{dS_1}{dt}=(1-p_1)\mu_1 N_1-\beta_1\frac{I_1}{N_1} S_1-\mu_1 S_1+m_{21}S_2-m_{12} S_1,\\ \displaystyle \frac{dS_2}{dt}=(1-p_2)\mu_2 (N(0)-N_1)-\beta_2\frac{I_2}{N(0)-N_1} S_2-\mu_2 S_1+m_{12}S_1-m_{21} S_2,\\ \displaystyle \frac{dI_1}{dt}=\beta_1\frac{I_1}{N_1} S_1-(\mu_1+\gamma_1) I_1, \displaystyle \frac{dI_2}{dt}=\beta_2\frac{I_2}{N(0)-N_1} S_2-(\mu_2+\gamma_2) I_2,\\ \displaystyle \frac{dN_1}{dt}=m_{21}(N(0)-N_1-I_2)-m_{12}(N_1-I_1), \end{array}\label{model:n2}\right. | (25) |
which can be used to study the local behavior of system (3) near the boundary equilibria. By considering the linear system for (25), we have the following theorems.
Theorem 4.3. If
Proof. Evaluating system (25) at boundary equilibrium
\begin{split} J\mspace{-2mu}(\hat{E})\mspace{-5mu}=\mspace{-5mu}\left(\begin{array}{cccccc} -\mspace{-4mu}2\mu_1\mspace{-5mu} -\mspace{-5mu}\gamma_1\mspace{-5mu}-\mspace{-5mu}m_{12}\mspace{-3mu}&m_{21}& \mspace{-3mu}-\mspace{-3mu}\beta_1\mspace{-3mu}\frac{\hat{S}_1}{\hat{N}_1}&0& (\mspace{-3mu}1\mspace{-5mu}-\mspace{-5mu}p_1)\mu_1\mspace{-5mu}+\mspace{-5mu} (\mu_1\mspace{-5mu}+\mspace{-5mu}\gamma_1)\mspace{-3mu}\frac{\hat{I}_1}{\hat{N}_1}\\ m_{12}& \mspace{-4mu}-\mspace{-4mu}\mu_2\mspace{-5mu}-\mspace{-5mu}m_{21}&0&-\beta_2\frac{\hat{S}_2}{\hat{N}_2}&-(1-p_2)\mu_2\\ \beta_1\frac{\hat{I}_1}{\hat{N}_1}&0&0&0&-(\mu_1+\gamma_1)\frac{\hat{I}_1}{\hat{N}_1}\\ 0&0&0&\beta_2\mspace{-4mu}\frac{\hat{S}_2}{\hat{N}_2}\mspace{-5mu}-\mspace{-5mu} \mu_2\mspace{-5mu}-\mspace{-5mu}\gamma_2&0\\ 0&0&m_{12}&-m_{21}&-(m_{12}+m_{21}) \end{array}\right). \end{split} |
It is clearly that one of the eigenvalue of
\lambda_1=\beta_2\frac{\hat{S}_2}{\hat{N}_2}-(\mu_2+\gamma_2)=(\mu_2+\gamma_2)(\mathfrak{\bar{R}}_{v2}-1), |
where
\lambda^4+\alpha_3\lambda_3+\alpha_2\lambda_2+\alpha_1\lambda_1+\alpha_0=0, | (26) |
where
\begin{array}{rl} \alpha_0=&m_{12}(\mu_1\mu_2+\mu_1m_{21}+\mu_2m_{12}(\mu_1+\gamma_1)\hat{N}_1^3\hat{I}_1+m_{12}m_{21}(1-p_2)\mu_2\hat{N}_1^4\\ &+(\mu_1\gamma_1)^2(\mu_2+m_{21})m_{21}\hat{N}_1^4+(\gamma_1+p_1\mu_1)(\mu_2+m_{21})m_{12}\hat{N}_1^4>0,\\ \alpha_1=&(\mu_1+\gamma_1)(\mu_1+\mu_2+m_{12}+m_{21})m_{12}\hat{N}_1^2\hat{I}_1+((2\mu_1+\gamma_1)m_{21}^2+\mu_2m_{12}^2)\hat{N}_1^2\\ &+(\mu_2(\mu_1+\gamma_1)^2+(m_{12}+2m_{21})\gamma_1^2+2\mu_1(\mu_1+\mu_2)+(4\mu_1+\mu_2)\gamma_1)\hat{N}_1^2\\ &+(m_{21}(2\mu_1+\mu_2+\gamma_1)+\mu_1(\gamma_1+p_1\mu_1+p_1\gamma_1)+\mu_2(\gamma_1+2\mu_1))\hat{N}_1^2>0,\\ \alpha_2=&m_{12}(\mu_1+\gamma_1)\hat{N}_1\hat{I}_1+(m_{12}^2+m_{21}^2+(\mu_1+\gamma_1)^2+\mu_2\gamma_1+2\mu_1\mu_2)\hat{N}_1^2\\ &+(m_{21}(2\gamma_1+4\mu_1+\mu_2) +m_{12}(2m_{21}+\gamma_1+2\mu_1+2\mu_2))\hat{N}_1^2>0,\\ \alpha_3=&(2\mu_1+\mu_2+\gamma_1+2m_{12}+2m_{21})\hat{N}_1>0. \end{array} |
By Routh-Hurwitz theorem, (26) has roots with negative real parts only requires that
\begin{split} \alpha_1\alpha_2\mspace{-4mu}-\mspace{-4mu}\alpha_0\alpha_3 &= m_{12}^2(\mu_1+\gamma_2)^2(\mu_1+\mu_2+m_{12}+m_{21})\hat{I}_1^2\hat{N}_1^3 +(\mu_1+\gamma_1)(m_{12}^3+m_{21}^3\\ &+\mu_1\gamma_1^2 +2\mu_1^2+\gamma_1+\mu_1^3+2\mu_2\gamma_1^2+4\mu_1\mu_2\gamma_1+2\mu_1^2\mu_2 +\mu_2^2\gamma_1+\mu_1\mu_2^2\\ &+\mspace{-3mu}(3\mu_1\mspace{-3mu}+\mspace{-3mu}2\mu_2\mspace{-3mu}+\mspace{-3mu}\gamma_1\mspace{-3mu}+\mspace{-3mu}2m_{21})m_{12}^2\mspace{-3mu}+\mspace{-3mu}(5\mu_1+2\mu_2+3\gamma_1)m_{21}^2 +(5\mu_1^2 \mspace{-3mu}+\mspace{-3mu}7\mu_1\gamma_1\\ &+\mspace{-3mu}2\gamma_1^2+4\mu_2\gamma_1+6\mu_1\mu_2+\mu_2^2)m_{21}^2+(3\mu_1^2+2\gamma_1^2+\mu_2^2+3m_{21}^2+p_1\mu_1^2\\ &+\mspace{-3mu}4\mu_1\mu_2 \mspace{-3mu}+\mspace{-3mu}4(2\mu_1\mspace{-3mu}+\mspace{-3mu}\mu_2\mspace{-3mu}+\mspace{-3mu}\gamma_1)m_{21}+(p_1+4)\mu_1\gamma_1+2\mu_2\gamma_1)m_{12})m_{12}\hat{N}_1^4\hat{I}_1\\ &+\mspace{-3mu}( m_{21}^4(2\mu_1\mspace{-3mu}+\mspace{-3mu}\gamma_1)\mspace{-3mu}+\mspace{-3mu}\mu_2m_{12}^4\mspace{-3mu} +\mspace{-3mu}2m_{21}^3(2\mu_1\mspace{-3mu}+\mspace{-3mu}\gamma_1)(2\mu_1\mspace{-3mu}+\mspace{-3mu} \gamma_1\mspace{-3mu}+\mspace{-3mu}\mu_2) m_{21}^2(2\mu_1\mspace{-3mu}+\mspace{-3mu}\gamma_1) \end{split} |
\begin{split}&\times\mspace{-3mu}(4(\mu_1\mspace{-3mu}+\mspace{-3mu}\gamma_1)^2\mspace{-3mu}+\mspace{-3mu} 3\mu_2\gamma_1\mspace{-3mu}+\mspace{-3mu}6\mu_1\mu_2\mspace{-3mu}+\mspace{-3mu}\mu_2^2) \mspace{-3mu}+\mspace{-3mu}(\mu_1+\gamma_1)^2\mu_2(\gamma_1^2+(2\mu_1+\mu_2)\gamma_1\\ &+\mspace{-3mu}(\mu_1\mspace{-3mu}+\mspace{-3mu}2\mu_2)\mu_1) \mspace{-3mu}+\mspace{-3mu}m_{12}^3(\gamma_1^2\mspace{-3mu}+\mspace{-3mu}p_1\mu_1^2 \mspace{-3mu}+\mspace{-3mu}4\mu_1\mu_2+2\mu_2^2+(\mu_1+p_1\mu_1+2\mu_2)\gamma_1\\ &+\mspace{-3mu}(2\mu_1\mspace{-3mu}+\mspace{-3mu}\gamma_1\mspace{-3mu}+\mspace{-3mu}3\mu_2)m_{21}) \mspace{-3mu}+\mspace{-3mu}m_{21}(2\gamma_1^4\mspace{-3mu}+\mspace{-3mu}4\gamma_1^3(2\mu_1 \mspace{-3mu}+\mspace{-3mu}\mu_2)\mspace{-3mu}+\mspace{-3mu}4\mu_1\gamma_1(2\mu_1^2\mspace{-3mu}+\mspace{-3mu}5\mu_1\mu_2\\ &+\mu_2^2)+\gamma_1^2 (12\mu_1+16\mu_1\mu_2+\mu_2^2)+2\mu_1^2(\mu_1^2+4\mu_1\mu_2+2\mu_2^2)) m_{12}^2(\gamma_1^3\\ &+3m_{21}^2(2\mu_1+\mu_2+\gamma_1)+\gamma_1^2(3\mu_2+(p_1+3)\mu_1)+\gamma_1(8\mu_1\mu_2+2\mu_2^2\\ &+\mspace{-4mu}(2\mspace{-4mu}+\mspace{-4mu}3p_1)\mu_1^2) \mspace{-3mu}+\mspace{-3mu}2\mu_1(p_1\mu^2\mspace{-4mu}+\mspace{-4mu} 3\mu_2(\mu_1\mspace{-4mu}+\mspace{-4mu}\mspace{-3mu}\mu_2))\mspace{-3mu}+\mspace{-3mu}m_{21}(3\gamma_1^2 \mspace{-3mu}+\mspace{-3mu}6\mu_1^2\mspace{-3mu}+\mspace{-3mu}2(p_2\mspace{-3mu}+\mspace{-3mu}7)\mu_1\mu_2\\ &+\mspace{-3mu}3\mu_2^2\mspace{-3mu}+\mspace{-3mu}2\gamma_1(4\mu_1\mspace{-3mu}+\mspace{-3mu} (p_2\mspace{-3mu}+\mspace{-3mu}3)\mu_2))) (\gamma_1^4\mspace{-3mu}+\mspace{-3mu}m_{21}(3\gamma_1\mspace{-3mu}+\mspace{-3mu}6\mu_1 \mspace{-3mu}+\mspace{-3mu}\mu_2)\mspace{-3mu}+\mspace{-3mu}\gamma_1^3((p_1\mspace{-3mu}+\mspace{-3mu}3)\mu_1\\ &+\mspace{-3mu}2\mu_2)\mspace{-3mu}+\mspace{-3mu}\gamma_1^2(3(p_1\mspace{-3mu}+\mspace{-3mu}1)\mu_1^2+8\mu_1\mu_2+2\mu_2^2) +\mu_1\gamma_1((1+3p_1)\mu_1^2+10\mu_1\mu_2\\ &+\mspace{-3mu}(7\mspace{-3mu}-\mspace{-3mu}p_1)\mu_2^2)\mspace{-3mu}+\mspace{-3mu} \mu_1^2(p_1\mu_1^2\mspace{-3mu}+\mspace{-3mu}4\mu_1\mu_2\mspace{-3mu}+\mspace{-3mu}(6-p_2)\mu_2^2) +m_{21}^2(4\gamma_1^2+(14-p_1)\mu_1^2\\ &+\mspace{-3mu}2(p_2\mspace{-3mu}+\mspace{-3mu}7)\mu_1\mu_2\mspace{-3mu}+\mspace{-3mu}\mu_2^2\mspace{-3mu}+\mspace{-3mu} \gamma_1((15\mspace{-3mu}-\mspace{-3mu}p_1)\mu_1\mspace{-3mu}+\mspace{-3mu}2(p_2\mspace{-3mu}+\mspace{-3mu}3)\mu_2)) \mspace{-3mu}+\mspace{-3mu}m_{21}(\gamma_1^2((15\mspace{-3mu}+\mspace{-3mu}p_1)\mu_1\\ &+\mspace{-3mu}4\gamma_1^3\mspace{-3mu}+\mspace{-3mu}(6\mspace{-3mu}+\mspace{-3mu}p_2)\mu_2)\mspace{-3mu}+\mspace{-3mu}\gamma_1((17\mspace{-3mu}+\mspace{-3mu}3p_1)\mu_1^2\mspace{-3mu}+\mspace{-3mu}(21-2p_1\mspace{-3mu}+\mspace{-3mu}3p_2)\mu_1\mu_2 \mspace{-3mu}+\mspace{-3mu}(p_2\mspace{-3mu}+\mspace{-3mu}2)\mu_2^2)\\ &+\mspace{-3mu}\mu_1(2(p_1+3)\mu_1^2+(19-2p_1+2p_2)\mu_1\mu_2+(p_2+7)\mu_2^2))) )\hat{N}_1^5>0. \end{split} |
Then all solution of (26) have negative real parts. Therefore, based on the above discussion, we know that if
Similar results hold for boundary equilibrium
Theorem 4.4. If
To complement the mathematical analysis carried out in the previous sections, we now investigate some of the numerical properties of system (3). We take the default parameter values as:
Time evolution of system (3) in the special case of
The theoretical and numerical results all show that
To explore the effect of vaccination and migration, we also compare the second peak size and second peak time with various vaccination coverage and migration rate. For the case of
We also compared the residual values of the first peak size to investigate the impact of vaccination and migration, which shown in the histogram 6. The results show that migration can reduce the first peak size for each patches and the entire population as long as the migration rate
In this paper, we proposed a multi-patch SIR model with vaccination to study the influence of vaccination coverage and human mobility on disease transmission. Our theoretical results show that the control reproduction number
In our model, we assume that the infective do not move between patches, corresponding to either a very severe disease so that infective are not able to move or move is forbidden in order to control outbreak of disease. In the further, we can generalize the current model with infective move between patches.
This project has been partially supported by grants from National Natural Science Foundation of China (Nos. 11671206,11271190) and Scientific Research Innovation Project of Jiangsu Province (No. KYZZ15_0130). We also thank two anonymous referees for their helpful comments.
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