 Research
 Open Access
Threshold dynamical analysis on a class of agestructured tuberculosis model with immigration of population
 Lili Liu^{1}Email author,
 Xinzhi Ren^{2} and
 Zhen Jin^{1}
https://doi.org/10.1186/s136620171295y
© The Author(s) 2017
 Received: 16 May 2017
 Accepted: 27 July 2017
 Published: 29 August 2017
Abstract
Some studies show that latency and relapse, especially the agedependent latency and relapse, may affect the transmission dynamics of tuberculosis model. Meanwhile, the immigration of infected individuals induces the loss of diseasefree steady state and hence no basic reproduction number. In our work, a class of agestructured tuberculosis model with immigration is proposed, where the new individuals can immigrate into the susceptible, latent, infectious and removed compartments. We show that the endemic steady state is unique and globally asymptotically stable by using the Lyapunov functional. Numerical simulations are given to support our theoretical results.
Keywords
 tuberculosis model
 agestructured
 immigration
 global stability
 Lyapunov functional
1 Introduction
Tuberculosis (TB), mainly caused by Mycobacterium tuberculosis, is a widespread infectious disease and has become a global public health issue. Despite various treatment strategies and beneficial policies on TB patients, the current global TB remains a leading cause of death from an infectious disease. According to reports, there were one death in five in England in the 17th century [1]. About \(8.7 \times10^{6}\) cases of TB globally is estimated in 2011, in which India has the largest total infected population, with an estimated 2.2 million new cases; China has the second largest TB epidemic, with more than \(1.3 \times10^{6}\) new cases every year [2].
Mathematical models have been a useful tool to understand and analyze the transmission dynamics of TB and other infectious diseases. In [3], a SEI type of TB model with a general contact rate is considered, and the global stability of equilibria is derived. In [4], a TB model with early and late latent stages is introduced to discuss effectiveness of treating TB patients at different stages. The reader can refer to more related mathematical models for TB; see [5–7]. It is well known that TB experiences a latent phase as well as a relapse phase which are the removed individuals who have been previously infected, but revert back to the infectious compartments due to the reactivation (see [8, 9]). Hence, in this paper, we focus on an SEIRtype model.
In the modern world, the worldwide transportation leads to tremendous movement of individuals and it is inevitable that infectious diseases may be introduced into a population from outside the population. Although most of the developed countries have screening policies for new immigrants, latent tuberculosis may take long time to become infectious and latent tuberculosis individuals may travel, thus, they are usually not detected by the TB screening. Furthermore, the removed tuberculosis patients who often have higher relapse rate may travel into another region; or the infectious individuals may also travel. In [19], McCluskey et al. proposed a TB model with immigration of both latent and infective compartments. Later, In [20], Guo et al. introduced a TB model with treatment and immigration into the latent compartment. It is well known that the model will no longer have the diseasefree equilibrium and the basic reproduction number, and there will always have a unique endemic equilibrium which is globally asymptotically stable by using a global Lyapunov function. More related work can be found in [21–25] and references therein.
In this paper, we propose and investigate a TB model with immigration into the four compartments. We also incorporate into the continuous agedependent in latent and removed compartments. It is a generalization of model proposed in [10]. According to mathematical analysis, we show that TB always exists in a region and the endemic equilibrium is unique and globally asymptotically stable. This paper is organized as follows. In the next section, we will formulate the model. In Section 3, we show the mathematical wellposedness of our model. In Section 4, we investigate the asymptotic smoothness of the semiflow generated by our model and the existence of compact attractor. In Section 5, we show our dynamics results, including the existence and global stability of the unique endemic steady state. Some simulations and conclusion are provided in Section 6 and Section 7, respectively.
2 Model formulation
This section we devote to formulating our model.
Assumption 2.1
 (A_{1}):

All constant parameters \(\Lambda_{s},\Lambda_{i},\mu_{s},\mu _{e},\mu_{i},\mu_{r},k>0\).
 (A_{2}):

The functions \(\sigma(a),\gamma(b)\in L^{\infty}(\mathbb {R}^{+},\mathbb{R}^{+})\), and denote \(\sigma^{\mathrm{inf}}\), \(\gamma ^{\mathrm{inf}}\) and \(\sigma^{\mathrm{sup}}\), \(\gamma^{\mathrm{sup}}\) as the essential infimums and the essential supremums of σ and γ, respectively.
 (A_{3}):

\(\sigma(a)\) and \(\gamma(b)\) are Lipschitz continuous on \(\mathbb{R}^{+}\) with Lipschitz coefficients \(M_{\sigma}\) and \(M_{\gamma}\), respectively.
 (A_{4}):

\(\int_{0}^{\infty}\sigma(a)\,da=0\) and \(\int_{0}^{\infty}\gamma(b)\,db=0\).
 (A_{5}):

\(\Lambda_{e},\Lambda_{r}\in L^{1}(\mathbb{R}^{+},\mathbb{R}^{+})\), and denote \(\bar{\Lambda}_{e}=\int_{0}^{\infty}\Lambda_{e}(a)\,da\), \(\bar{\Lambda}_{r}=\int_{0}^{\infty}\Lambda_{r}(b)\,db\).
 (A_{6}):

\(\bar{\Lambda}_{e},\bar{\Lambda}_{r}\in\mathbb{R}^{+}\), and \(\bar{\Lambda}_{e}+\bar{\Lambda}_{r}>0\).
3 The wellposedness of system (2.1)
This section is devoted to the positivity and boundedness of solutions.
Proposition 3.1
 (i)
Ω is positively invariant for Φ, that is, \(\Phi (t,x_{0})\in\Omega\), for \(\forall t\geq0\) and \(x_{0}\in\Omega\);
 (ii)
Φ is point dissipative and Ω attracts all points in X.
Proof
Moreover, it follows from (3.12) that \(\limsup_{t\rightarrow \infty}\\Phi(t,x_{0})\_{X}\leq{\Lambda^{*}}/{\mu^{*}}\) for any \(x_{0}\in X\). Therefore, Φ is point dissipative and Ω attracts all points in X. This completes the proof. □
Combining Assumption 2.1 and Proposition 3.1, we have the following two propositions.
Proposition 3.2
 (i)
\(0\leq S(t),\int_{0}^{\infty}e(t,a)\,da,I(t),\int_{0}^{\infty}r(t,b)\,db\leq M\);
 (ii)
\(e(t,0)\leq\beta M^{2}, r(t,0)\leq kM\).
Proposition 3.3
 (i)
\(\Phi(\mathbb{R}^{+},B)\) is bounded;
 (ii)
Φ is eventually bounded on B;
 (iii)
if \(M\geq{\Lambda^{*}}/{\mu^{*}}\) is a bound for B, then M is also a bound for \(\Phi(\mathbb{R}^{+},B)\);
 (iv)
given any \(L\geq{\Lambda^{*}}/{\mu^{*}}\), there exists \(T=T(B,M)\) such that L is a bound for \(\Phi(t,B)\) whenever \(t\geq T\).
Now, we give the following proposition on the asymptotic lower bounds for system (2.1).
Proposition 3.4
Proof
4 Asymptotic smoothness
In order to obtain global properties of the semiflow \(\{\Phi(t)\} _{t\geq0}\), it is important to prove that the semiflow is asymptotically smooth.
We introduce two definitions and a useful lemma.
Definition 4.1
A function \(Y:\mathbb{R}\rightarrow X\) is called a total trajectory of Φ if Y satisfies \(\Phi_{s}(Y(t))=Y(t+s)\) for all \(t\in\mathbb {R}\) and \(s\geq0\).
Definition 4.2
Remark 4.1
For any point \(y_{0}\in A\), it follows from Definitions 4.1 and 4.2, and [18], Theorem 1.40, that there exists a total trajectory \(Y(\cdot)\) with \(y(0)=y_{0}\) and \(y(t)\in A\) for all \(t\in\mathbb{R}\).
Lemma 4.1
[13] Let \(D\subseteq\mathbb{R}\). For \(j=1,2\), suppose \(f_{j}:D\rightarrow \mathbb{R}\) is a bounded Lipshitz continuous function with bound \(K_{j}\) and Lipschitz coefficient \(M_{j}\). Then the product function \(f_{1}f_{2}\) is Lipschitz with coefficient \(K_{1}M_{2}+K_{2}M_{1}\).
In order to prove the asymptotic smoothness of the semiflow, we will apply the following result, which is a special case of [18], Theorem 2.46.
Lemma 4.2
 (i)
\(\lim_{t\rightarrow\infty}\operatorname{diam}\Phi_{1}(t,B)=0\);
 (ii)
for \(t\geq0\), \(\Phi_{2}(t,B)\) has compact closure.
The following result is used to verify (ii) of Lemma 4.2, which is based on Theorem B.2 in [18].
Lemma 4.3
 (i)
\(\sup_{f\in K}\int_{0}^{\infty}f(a)\,da<\infty\);
 (ii)
\(\lim_{h\rightarrow\infty}\int_{h}^{\infty}f(a)\,da\rightarrow0\) uniformly in \(f\in K\);
 (iii)
\(\lim_{h\rightarrow{0^{+}}}\int_{0}^{\infty}f(a+h)f(a)\,da\rightarrow0\) uniformly in \(f\in K\);
 (iv)
\(\lim_{h\rightarrow{0^{+}}}\int_{0}^{h} f(a)\,da\rightarrow0\) uniformly in \(f\in K\).
Based on Lemmas 4.2 and 4.3, we have the following theorem.
Theorem 4.1
The semiflow \(\{\Phi(t)\}_{t\geq0}\) is asymptotically smooth.
Proof
Now, we show that \(\Phi_{2}(t,B)\) has compact closure. According to Proposition 3.3, \(S(t)\) and \(I(t)\) remain in the compact set \([0,\Lambda^{*}/\mu^{*}]\subset[0,M]\), where \(M\geq\Lambda ^{*}/\mu^{*}\) is a bound for B. Thus, it is only to show that \(\tilde {y}_{2}(t,a)\) and \(\tilde{y}_{4}(t,b)\) satisfy conditions (i)(iv) in Lemma 4.3.
Propositions 3.1 and 3.3 and Theorem 4.1 show that Φ is point dissipative, eventually bounded on bounded sets, and asymptotically smooth. Thus, following [18], Theorem 2.33, we have the following proposition on the existence of a global attractor.
Theorem 4.2
The semigroup \(\{\Phi(t)\}_{t\geq0}\) has a global attractor \(\mathcal {A}\) contained in X, which attracts the bounded sets of X.
5 Dynamical results
This section is devoted to the existence and global stability of the steady state.
5.1 Existence of steady state
From the above discussions, we have the following theorem.
Theorem 5.1
System (2.1) always has a unique endemic steady state \(T^{*}=(S^{*}, e^{*}(\cdot), I^{*}, r^{*}(\cdot))\), and \(T^{*}\in\mathcal{A}\).
5.2 Global stability
This subsection is devoted to showing the global stability of \(T^{*}\), which implies that the attractor \(\mathcal{A}\) only contains the unique endemic steady state.
The following lemma will be useful in the proof of our main result.
Lemma 5.1
Proof
Based on the above preparations, we show our main result.
Theorem 5.2
The unique endemic steady state \(T^{*}\) is globally asymptotically stable, and \(\mathcal{A}=\{T^{*}\}\).
Proof
6 Numerical simulations
6.1 The long time behaviors of system (2.1)
6.2 The age distribution of the latent and removed populations
6.3 The stationary distribution
7 Conclusion and discussion
In this paper, we proposed and investigated a class of agestructured SEIR epidemic model with immigration. We show that, for all parameter values, the endemic steady state is unique and globally asymptotically stable by using the Lyapunov functional.
All simulation results show that the immigration of individuals leads to the insight that the TB cannot be fully eliminated from the population and will eventually reach a steady endemic level. The age of latent and removed leads to stationary patterns. This implies that TB will exist if either the immigration of population is allowed or there are TB infections in the region. Thus, in order to eradicate the TB, there are two choices: one is to prohibit the immigration of infected individuals, which is difficult to achieve; the other is to clear away the TB in any one region.
Declarations
Acknowledgements
This work was supported by the National Nature Science Foundation of China under Grant Nos. 11601293 and 11331009, Shanxi Scientific Data Sharing Platform for Animal Diseases (201605D121014), and the science and technology innovation team of Shanxi province (201605D13104406). We are very grateful to the anonymous referees and editor for their careful reading, valuable comments and helpful suggestions, which have helped to improve the presentation of this work significantly.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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