Dynamics of a Diffusive Multigroup SVIR Model with Nonlinear Incidence

Xu, Jinhu; Geng, Yan

doi:https://doi.org/10.1155/2020/8847023

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Research Article | Open Access

Volume 2020 | Article ID 8847023 | https://doi.org/10.1155/2020/8847023

Dynamics of a Diffusive Multigroup SVIR Model with Nonlinear Incidence

Jinhu Xu¹and Yan Geng²

Academic Editor: Dimitri Volchenkov

Received22 Aug 2020

Revised25 Oct 2020

Accepted04 Nov 2020

Published09 Dec 2020

Abstract

In this paper, a multigroup SVIR epidemic model with reaction-diffusion and nonlinear incidence is investigated. We first establish the well-posedness of the model. Then, the basic reproduction number is established and shown as a threshold: the disease-free steady state is globally asymptotically stable if , while the disease will be persistent when . Moreover, applying the classical method of Lyapunov and a recently developed graph-theoretic approach, we established the global stability of the endemic equilibria for a special case.

1. Introduction

As is well-known, mathematical models have played a key role in describing the dynamical evolution of infectious diseases [1–10]. For example, during the special battle for preventing the novel coronavirus (COVID-19) spreading, not only medical and biological research but also theoretical studies based on mathematical modeling may play a crucial role in analyzing and forecasting the spreading of the disease (see, for example, [4–10] and references therein; also, some useful advices for controlling the disease spreading have been given by the researchers).

The spatial heterogeneity has been regarded as an important role that affects the spatial spreading of disease. Therefore, complex models are needed to investigate the spread of diseases. In recent years, reaction-diffusion models involving environmental heterogeneity have been proposed and studied to find reliable measures to control disease spreading [11–22]. In particular, the basic reproduction number is an important threshold value for investigation of the dynamics of epidemic models. Then, Wang and Zhao [14] defined the basic reproduction number and obtained its computation formula for general reaction-diffusion epidemic models. Xu et al. [21] studied an SVIR epidemic model with reaction-diffusion and the existences of travelling waves were investigated, while Zhang and Liu [22] also studied the existence of travelling waves for an SVIR epidemic model but with nonlocal dispersal and time delay.

Notice that the essential assumption in classical compartmental epidemic models is that the individuals are homogeneously mixed, which means each individual has the same probability to get infected. However, infected probability may be different for each individual in terms of the impact of different factors, such as education levels, age, gender, and communities. To overcome this problem, multigroup models have been proposed by many researchers by dividing the individuals into different groups and most of these work focus on the global dynamics of the models (see, for example, [23–30]). Particularly, one of the earliest works of multigroup models was proposed by Lajmanovich and York [23] when studied the gonorrhea disease transmission in a nonhomogeneous population. In order to investigate the global dynamics of multigroup models, a subtle grouping method in estimating the derivatives of Lyapunov functionals guided by graph theory for multigroup models was developed in [28–30]. For example, Kuniya [25] considered a multigroup SVIR epidemic model with vaccination and the global stability of endemic equilibria were established by the method of Lyapunov function. However, spatially heterogeneous was not considered in the model in [25]. To the best of our knowledge, there are few results of multigroup SVIR epidemic model with reaction-diffusion. The existing results are focusing on SIS and SIR models (see [31–34]). On the contrary, incidence rate also plays an important role in modeling the epidemic models, which has been frequently used to describe the nature of certain phenomena and obtained much exact results. In addition, applying general incidence rates can obtain the unification theory by the omission of unessential detail, see, for example, [21, 35–37] and references therein. Hence, inspired by [21, 25] and the above considerations, in this paper, we consider the following multigroup SVIR epidemic model with reaction-diffusion and general incidence rate:with the homogeneous Neumann boundary conditionsand the initial conditionsfor , where is a domain in with smooth boundary and is the outward normal vector to the boundary . Initial functions are nonnegative and continuously defined on . , , , and stand for the densities of the susceptible, vaccinated, infective, and recovered individuals in the -th group at time and spatial location , respectively. is the input rate of in spatial location ; , , , and denote the natural death rates of , , , and in spatial location , respectively; is the death rate induced by the disease in spatial location ; is the vaccination rate of in spatial location ; is the rate of recovery from infection in spatial location ; is the infection rate of infected by in spatial location ; is the infection rate of infected by in spatial location ; and , , , and are the diffusion rate of , , , and in spatial location , respectively. All location-dependent parameters are continuous and strictly positive defined on , and denotes the force of infection. We assume the function satisfies the following properties:

It is natural to assume that due to the fact that disease cannot spread if there is no infection. The disease spreads heavily with the increasing number of infected individuals; thus, it reasonable to suppose . Since the infectivity of infected individuals cannot be unbounded, it should reach a certain level when the infected individuals are heavy. Therefore, the assumption implies that there exists a peak level for the infectivity of the infected individuals at some certain time. Similar to [25, 34], we also assume the -square matrix is nonnegative and irreducible.

Because the last equation of model (1) is decoupled from other equations, we indeed need to study the following subsystem of (1):

The rest of this paper is organized as follows. In Section 2, some preliminaries are introduced for the well-posedness of the model. In Section 3, we define the basic reproduction number . In Section 4, the threshold dynamics are established in terms of . An special case is performed as a supplementary to the theoretical results in Section 5. A brief conclusion ends the paper.

2. Well-Posedness

Throughout this paper, we denote and . Let with the norm and . It is easy to see that is a strongly ordered Banach space. Let with norm , where , . Denote be the positive cone of . For convenience, set for . , , and . Denotebe the semigroup associated with , , and , respectively, subject to the Neumann boundary condition. Let be the Green function associated with , , and , respectively, subject to the Neumann boundary condition. Then, for any and , we have

Applying Corollary 7.2.3 in [38], we know that, for each is compact and strongly positive. Then, there exist constants (), satisfying for each , where denotes the principle eigenvalue of , , and subject to the Neumann boundary condition. Definefor , , and . Set be the solution of model (5) with initial function , and then, model (5) can be rewritten asfor and . By virtue of Corollary 4 in [39], we have the following.

Lemma 1. For , model (5) has a unique mild solution onand. Moreover, this solution is a classical solution.
Then, we give the existence of solutions of model (5).

Theorem 1. The model (5) has a unique solution on with . Furthermore, the solution semiflow of model (5) defined byadmits a global compact attractor.

Proof. Suppose to the contrary that , then as by Theorem 2 in [39]. It follows from the first equation of model (5) thatBy the comparison principle and Lemma 2 in [40], there exists a constant such that for , . Furthermore, similar procedure can be applied to the second equation of model (5), and then, there exists a constant such that for , . Hence, from the third equation of model (5), we haveNow, we consider the following comparison system:It follows from the standard Krein–Rutman theorem (see [41]) that the eigenvalue problem of system (13) admits a principle eigenvalue with a strongly positive eigenfunction . Thus, system (13) has a solution for , where is a positive constant, satisfying for . By using the comparison principle, we haveThus, there exists a constant such thatwhich leads to a contradiction. Hence, the global existence of is derived.
Now, we are in the position to show the solution is also ultimately bounded. Indeed, it follows from the comparison principle, (11), and Lemma 2 in [40] that there exist and such that , , . Moreover, from the second equation of model (5) and applying similar procedures, we can find and , satisfying for , .
DenoteThen, we haveThus, there exist and such that for any . Next, we denote be the eigenvalue of subject to the Neumann boundary condition with eigenfunction , which satisfies . From Chapter 5 in [42], one obtainsSince is uniformly bounded, there exists constant such that for . Moreover, assume are eigenvalues of subject to Neumann boundary condition, which satisfies . By Theorem 2.4.7 in Wang [43], one gets for all . Since decreases like , there exists such thatLet . For all , by (9) and (4), we havewhich yields thatThus, the above discussion implies that the system (5) is point dissipative. Furthermore, by Theorem 2.2.6 in [44], the solution semiflow is compact for any . Therefore, it follows from Theorem 3.4.8 in [45] that has a global compact attractor in .

3. Basic Reproduction Number

For each , consider the following subsystem of model (5):

It follows from the Lemma 2.2 in [40] that the following systemadmits a unique positive steady state which satisfies the equationwith for , which is globally asymptotically stable in . Hence, the second equation of system (22) is asymptotic to

By Lemma 2.2 in [40] and Corollary 4.3 [46], there exists a globally asymptotically stable steady state . Therefore, concluding from the above discussion, we know that model (5) admits a unique disease-free steady state with , , and . Furthermore, if all the parameters of model (5) are positive constants, then we have and .

Linearizing model (5) at , we obtain the linearized system

Substituting into (26), we obtain

From Theorem 7.6.1 in [38], we have the following result.

Lemma 2. The eigenvalue problem (27) admits a principle eigenvalue with a strictly positive eigenfunction.

Denote . Assume that the distribution of initial infection is . Then, is the distribution of new infective part as time evolves. Therefore, we useto describe the total distribution of the new infective numbers produced during the infection period.

According to [14], the basic reproduction number is defined by , where is the spectral radius of the operator . Furthermore, following Theorem 3 in [14], we have the following lemma.

Lemma 3. The principle eigenvalue and have the same sign, and the disease-free steady state is locally asymptotically stable.

4. Extinction/Persistence Result

Based on the discussion above, we now investigate the extinction and persistence of the disease.

Theorem 2. If , then the disease-free steady state is globally asymptotically stable.

Proof. By Lemma 2, we have when . Thus, there exists a small enough such that . According to Theorem 1, there exists a such that and () for all . Thus, from the third equation of model (5) and assumption (4) thatlet be the eigenfunction corresponding to the principal eigenvalue . Assume thatwhere is a constant. With the aid of comparison principle, we can obtainThis yields uniformly for . Thus, the model (5) is asymptotic to (22). Then, by Lemma 2.2 in [40] and Corollary 4.3 in [46], we have and .
Before proving the main results on disease persistence, we first need to establish the following lemma.

Lemma 4. Let is the solution of model (5) with .(i)For any , we always have and for all , and there exist constants such that(ii)If there exists such that , then we have , , .

Proof. (i)It follows from Theorem 1 that there exists and such that Then, by the first equation of model (5) and the assumption (4), we have Thus, by Lemma 2.2 in [40], it follows that the comparison system admits a unique positive steady state which is globally asymptotically stable in . By the standard parabolic comparison theorem, there exists a constant such that is uniformly for . Moreover, it follows from the second equation of model (5) that Similarly, there exists a constant such that is uniformly for .(ii)Assume for some . According to Theorem 1, it follows from model (5) that Thus, the strong maximum principle (see, e.g., [47], Theorem 4) and Hopf boundary lemma (see, e.g., [47], Theorem 3) imply that , , and . Now, we are in position to state the main results of this section.

Theorem 3. If , then there exists a constant such that, for any with , solution of model (5) satisfiesuniformly for all . Moreover, model (5) has at least one endemic steady state .

Proof. DefineAccording to Lemma 4, for any , we have , , and . Thus, , which implies that is the invariant set for solution semiflow of model (5). Defineand be the omega limit set of the orbit . We first prove the following claim.

Claim 1. , . Since , we have . Thus, , . Then, model (5) reduces to model (22); by Lemma 2.2 in [40] and Corollary 4.6 in [46], we have and uniformly for . Hence, , ; i.e., the claim holds.
Since , we have by Lemma 3. Using the continuity of , there exists a sufficient small enough positive constant such that .

Claim 2. is uniform weak repeller for in the sense thatSuppose, by contradiction, there exists a such thatThen, there exists an enough large such thatTherefore, from model (5) and assumptions (4), we haveSet be the strongly positive eigenfunction associated with principle eigenvalue . Since for all and , there exists a constant such that for . It is clear that is a solution of the following linear system:Applying the comparison principle, we haveDue to , we conclude that is unbounded. This is a contradiction.
Define a continuous function byClearly, and has the property that if or and , then for all . Hence, for the semiflow , is a generalized distance function [48]. From Claim 1, it follows that any forward orbit of in converges to . Moreover, Claim 2 implies that is isolated in and , where is the stable set of . Furthermore, there is no cycle in from to . It then follows from Theorem 3 in [48] that there exists a constant such that for all . This implies the uniform permanence of . By Lemma 4 and Theorem 4.7 in [49], we know that has a positive steady state of model (5). The proof is complete.

5. Global Dynamics for a Special Case

In this section, we consider a special case with all the parameters of model (5) as constants, except for the diffusion coefficients. Then, model (5) degenerates into the following model:with the homogeneous Neumann boundary conditions (2) and the initial conditions (3). We mainly focus on the global stability of the endemic steady states of model (48). The proofs of the main results applying the Lyapunov functions and a subtle grouping technique guided by graph theory were developed in [28–30].

It follows from the Lemma 2 in [13] that model (48) has a disease-free steady state , where and with and . Define matriceswhere , then we have

Then, it follows from [26] that the basic reproduction number is defined as the spectral radius of , i.e., . As a consequence of Theorems 2 and 3, we can obtain the following results without proofs.

Theorem 4. The disease-free steady state of model (48) is globally asymptotically stable when .

Theorem 5. If , then there exists a constant such that, for any with , solution of model (48) satisfiesuniformly for all . Moreover, model (48) has at least one endemic steady state.

Furthermore, set the endemic steady state with , , and satisfyingFor the global stability of the endemic steady state , we have the following conclusion.

Theorem 6. If , then the endemic steady state is globally asymptotically stable.

Proof. Definewhich is a Laplacian matrix whose column sums are zero [30]. Then, is irreducible because is irreducible. It follows from Lemma 1 in [29] that the solution space of the linear system has dimension 1 with a base with , where is the cofactor of the -th diagonal entry of .
DefinewhereDifferentiating , we obtainIt follows from [50] thatThen, using steady state equations (52), we havewhere with global maximum value . From the assumption (4), it is easy to validate thatMoreover, according to the property that arithmetic mean is not less than the associated geometric mean, and . Thus, we haveConsequently, we obtainIt follows from the equality that which is equivalent to . Then,and thus, for all . Following the graph-theoretic approach as proposed in [28–30], similar procedures as in [26] can be applied to verify thatThus, we have . Furthermore, it can be shown that the largest invariant set in is the singleton . Therefore, by the LaSalle’s invariance principle, is globally asymptotically stable.

6. Conclusions

In this paper, a multigroup SVIR model with diffusion and nonlinear incidence rate has been investigated. We defined the basic reproduction number for spatially heterogeneous environment, and we further proved that served as a threshold index which predicts the extinction and persistence of the disease. Particularly, by using comparison principle, we proved that the disease-free steady state is globally asymptotically stable when is less than one. If is great than one, then the disease will persist. Consequently, we obtained the existence of endemic steady state . Furthermore, we established the criteria on the global stability of the disease-free steady state and the endemic steady state of the model in a special case.

Although the existence of the endemic steady state is established, it is necessary to point that the uniqueness and stability of the endemic steady state remains an open problem. Moreover, the impact of the latent individuals should also be considered for some diseases during its spread, such as AIDS and COVID-19 (see, for example, [4–8, 51]), in which it has been validated that the latent individuals for these two diseases may also have probability of infection. Thus, an improved model with latent compartment should be investigated. We leave these problems for future investigation.

Data Availability

There are no data in the submission.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11701445, 11701451, and 11702214), Natural Science Basic Research Plan in Shaanxi Province of China (2018JQ1057 and 2020JQ-38), and Young Talent Fund of University Association for Science and Technology in Shaanxi, China (20180504).

References

A. G. McKendrick, “Applications of mathematics to medical problems,” Proceedings of the Edinburgh Mathematical Society, vol. 44, pp. 98–130, 1925.
View at: Google Scholar
O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, Wiley, Chichester, UK, 2000.
X. Liu, Y. Takeuchi, and S. Iwami, “SVIR epidemic models with vaccination strategies,” Journal of Theoretical Biology, vol. 253, no. 1, pp. 1–11, 2008.
View at: Publisher Site | Google Scholar
L. Wang, J. Wang, J. Wang et al., “Modelling and assessing the effects of medical resources on transmission of novel coronavirus (COVID-19) in Wuhan, China,” Mathematical Biosciences and Engineering, vol. 17, no. 4, pp. 2936–2949, 2020.
View at: Publisher Site | Google Scholar
J. Jiao, Z. Liu, and S. Cai, “Dynamics of an SEIR model with infectivity in incubation period and homestead-isolation on the susceptible,” Applied Mathematics Letters, vol. 107, Article ID 106442, 2020.
View at: Publisher Site | Google Scholar
W. Zhou, A. Wang, F. Xia et al., “Effects of media reporting on mitigating spread of COVID-19 in the early phase of the outbreak,” Math. Biosci. Eng., vol. 17, no. 3, pp. 2683–2707, 2020.
View at: Google Scholar
B. Tang, F. Xia, S. Tang et al., “The effectiveness of quarantine and isolation determine the trend of the COVID-19 epidemics in the final phase of the current outbreak in China,” International Journal of Infectious Diseases, vol. 95, pp. 288–293, 2020.
View at: Publisher Site | Google Scholar
B. Tang, F. Xia, S. Tang et al., “The evolution of quarantined and suspected cases determines the final trend of the 2019-nCoV epidemics based on multi-source data analyses,” 2020, https://ssrn.com/abstract=3537099.
View at: Google Scholar
L. Peng, W. Yang, D. Zhang et al., “Epidemic analysis of COVID-19 in China by dynamical modeling,” 2020, http://arxiv.org/abs/2002.06563.
View at: Google Scholar
B. Tang, X. Wang, Q. Li et al., “Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions,” Journal of Clinical Medicine, vol. 9, no. 2, 2020.
View at: Publisher Site | Google Scholar
J. Wang and J. Wang, “Analysis of a reaction-diffusion cholera model with distinct dispersal rates in the human population,” Journal of Dynamics and Differential Equations, pp. 1–27, 2020.
View at: Google Scholar
R. Peng and X.-Q. Zhao, “A reaction-diffusion SIS epidemic model in a time-periodic environment,” Nonlinearity, vol. 25, no. 5, pp. 1451–1471, 2012.
View at: Publisher Site | Google Scholar
Y. Lou and X.-Q. Zhao, “A reaction-diffusion malaria model with incubation period in the vector population,” Journal of Mathematical Biology, vol. 62, no. 4, pp. 543–568, 2011.
View at: Publisher Site | Google Scholar
W. Wang and X.-Q. Zhao, “Basic reproduction numbers for reaction-diffusion epidemic models,” SIAM Journal on Applied Dynamical Systems, vol. 11, no. 4, pp. 1652–1673, 2012.
View at: Publisher Site | Google Scholar
Y. Wu and X. Zou, “Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism,” Journal of Differential Equations, vol. 261, no. 8, pp. 4424–4447, 2016.
View at: Publisher Site | Google Scholar
T. Kuniya and J. Wang, “Global dynamics of an SIR epidemic model with nonlocal diffusion,” Nonlinear Analysis: Real World Applications, vol. 43, pp. 262–282, 2018.
View at: Publisher Site | Google Scholar
Y. Cai, Y. Kang, M. Banerjee, and W. Wang, “Complex dynamics of a host-parasite model with both horizontal and vertical transmissions in a spatial heterogeneous environment,” Nonlinear Analysis: Real World Applications, vol. 40, pp. 444–465, 2018.
View at: Publisher Site | Google Scholar
Z. Bai, R. Peng, and X.-Q. Zhao, “A reaction-diffusion malaria model with seasonality and incubation period,” Journal of Mathematical Biology, vol. 77, no. 1, pp. 201–228, 2018.
View at: Publisher Site | Google Scholar
X. Ren, Y. Tian, L. Liu, and X. Liu, “A reaction-diffusion within-host HIV model with cell-to-cell transmission,” Journal of Mathematical Biology, vol. 76, no. 7, pp. 1831–1872, 2018.
View at: Publisher Site | Google Scholar
Y.-C. Shyu, R.-N. Chien, and F.-B. Wang, “Global dynamics of a West Nile virus model in a spatially variable habitat,” Nonlinear Analysis: Real World Applications, vol. 41, pp. 313–333, 2018.
View at: Publisher Site | Google Scholar
Z. Xu, Y. Xu, and Y. Huang, “Stability and traveling waves of a vaccination model with nonlinear incidence,” Computers & Mathematics with Applications, vol. 75, no. 2, pp. 561–581, 2018.
View at: Publisher Site | Google Scholar
R. Zhang, S. Liu, and S. Liu, “Traveling waves for SVIR epidemic model with nonlocal dispersal,” Mathematical Biosciences and Engineering, vol. 16, no. 3, pp. 1654–1682, 2019.
View at: Publisher Site | Google Scholar
A. Lajmanovich and J. A. Yorke, “A deterministic model for gonorrhea in a nonhomogeneous population,” Math. Biosci., vol. 28, no. 3-4, pp. 221–236, 1976.
View at: Publisher Site | Google Scholar
Y. Muroya, Y. Enatsu, and T. Kuniya, “Global stability for a multi-group SIRS epidemic model with varying population sizes,” Nonlinear Analysis: Real World Applications, vol. 14, no. 3, pp. 1693–1704, 2013.
View at: Publisher Site | Google Scholar
T. Kuniya, “Global stability of a multi-group SVIR epidemic model,” Nonlinear Analysis: Real World Applications, vol. 14, no. 2, pp. 1135–1143, 2013.
View at: Publisher Site | Google Scholar
Y. Geng and J. Xu, “Stability preserving NSFD scheme for a multi-group SVIR epidemic model,” Mathematical Methods in the Applied Sciences, vol. 40, no. 13, pp. 4917–4927, 2017.
View at: Google Scholar
H. Shu, D. Fan, and J. Wei, “Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission,” Nonlinear Analysis: Real World Applications, vol. 13, no. 4, pp. 1581–1592, 2012.
View at: Publisher Site | Google Scholar
H. Guo, M. Y. Li, and Z. Shuai, “A graph-theoretic approach to the method of global Lyapunov functions,” Proceedings of the American Mathematical Society, vol. 136, no. 08, pp. 2793–2802, 2008.
View at: Publisher Site | Google Scholar
H. Guo, M. Y. Li, and Z. Shuai, “Global stability of the endemic equilibrium of multigroup SIR epidemic models,” Canadian Applied Mathematics Quarterly, vol. 14, pp. 259–284, 2006.
View at: Google Scholar
M. Y. Li and Z. Shuai, “Global-stability problem for coupled systems of differential equations on networks,” Journal of Differential Equations, vol. 248, no. 1, pp. 1–20, 2010.
View at: Publisher Site | Google Scholar
S.-L. Wu and G. Chen, “Uniqueness and exponential stability of traveling wave fronts for a multi-type SIS nonlocal epidemic model,” Nonlinear Analysis: Real World Applications, vol. 36, pp. 267–277, 2017.
View at: Publisher Site | Google Scholar
S.-L. Wu and P. Weng, “Entire solutions for a multi-type SIS nonlocal epidemic model inRorZ , ,” Journal of Mathematical Analysis and Applications, vol. 394, no. 2, pp. 603–615, 2012.
View at: Publisher Site | Google Scholar
S.-L. Wu, P. Li, and H. Cao, “Dynamics of a nonlocal multi-type SIS epidemic model with seasonality,” Journal of Mathematical Analysis and Applications, vol. 463, no. 1, pp. 111–133, 2018.
View at: Publisher Site | Google Scholar
Y. Luo, S. Tang, Z. Teng, and L. Zhang, “Global dynamics in a reaction-diffusion multi-group SIR epidemic model with nonlinear incidence,” Nonlinear Analysis: Real World Applications, vol. 50, pp. 365–385, 2019.
View at: Publisher Site | Google Scholar
G. Huang, Y. Takeuchi, W. Ma, and D. Wei, “Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate,” Bulletin of Mathematical Biology, vol. 72, no. 5, pp. 1192–1207, 2010.
View at: Publisher Site | Google Scholar
A. Korobeinikov and P. K. Maini, “Non-linear incidence and stability of infectious disease models,” Mathematical Medicine and Biology: A Journal of the IMA, vol. 22, no. 2, pp. 113–128, 2005.
View at: Publisher Site | Google Scholar
Y. Yang, J. Zhou, and C.-H. Hsu, “Threshold dynamics of a diffusive SIRI model with nonlinear incidence rate,” Journal of Mathematical Analysis and Applications, vol. 478, no. 2, pp. 874–896, 2019.
View at: Publisher Site | Google Scholar
H. L. Smith, “Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems,” in Mathematical Surveys and Monographs, vol. 41, American Mathematical Society, Providence, RI, 1995.
View at: Google Scholar
R. H. Martin and H. L. Smith, “Abstract functional differential equations and reaction-diffusion systems,” Transactions of the American Mathematical Society, vol. 321, no. 1, pp. 1–44, 1990.
View at: Publisher Site | Google Scholar
Z. Guo, F.-B. Wang, and X. Zou, “Threshold dynamics of an infective disease model with a fixed latent period and non-local infections,” Journal of Mathematical Biology, vol. 65, no. 6-7, pp. 1387–1410, 2012.
View at: Publisher Site | Google Scholar
Y. Du, “Order structure and topological methods in nonlinear partial differential equations,” in Maximum Principles and Applications, vol. 1, World Scientific, New Jersey, NJ, USA, 2006.
View at: Google Scholar
R. B. Guenther and J. W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Dover. Public. Inc., Mineola, NY, USA, 1996.
M. Wang, Nonlinear Elliptic Equations, Science Public., Beijing, China, 2010.
J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, NY, USA, 1996.
J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, USA, 1988.
H. R. Thieme, “Convergence results and a Poincar -Bendixson trichotomy for asymptotically autonomous differential equations,” J. Math. Biol., vol. 30, pp. 755–763, 1992.
View at: Publisher Site | Google Scholar
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, NY, USA, 1984.
H. L. Smith and X.-Q. Zhao, “Robust persistence for semidynamical systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 9, pp. 6169–6179, 2001.
View at: Publisher Site | Google Scholar
P. Magal and X.-Q. Zhao, “Global attractors and steady states for uniformly persistent dynamical systems,” SIAM Journal on Mathematical Analysis, vol. 37, no. 1, pp. 251–275, 2005.
View at: Publisher Site | Google Scholar
J. Groeger, “Divergence theorems and the supersphere,” Journal of Geometry and Physics, vol. 77, pp. 13–29, 2014.
View at: Publisher Site | Google Scholar
H.-F. Huo and L.-X. Feng, “Global stability for an HIV/AIDS epidemic model with different latent stages and treatment,” Applied Mathematical Modelling, vol. 37, no. 3, pp. 1480–1489, 2013.
View at: Publisher Site | Google Scholar

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Copyright © 2020 Jinhu Xu and Yan Geng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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