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Mathematical Biosciences & Engineering

2017 , Volume 14 , Issue 5-6

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Daozhou Gao, Shigui Ruan and Jifa Jiang
2017, 14(5-6): i-ii doi: 10.3934/mbe.201705i +[Abstract](59) +[HTML](18) +[PDF](31.6KB)
Global stability of the steady states of an epidemic model incorporating intervention strategies
Yongli Cai, Yun Kang and Weiming Wang
2017, 14(5-6): 1071-1089 doi: 10.3934/mbe.2017056 +[Abstract](208) +[HTML](3) +[PDF](451.8KB)

In this paper, we investigate the global stability of the steady states of a general reaction-diffusion epidemiological model with infection force under intervention strategies in a spatially heterogeneous environment. We prove that the reproduction number \begin{document} $\mathcal{R}_0$ \end{document} can be played an essential role in determining whether the disease will extinct or persist: if \begin{document} $\mathcal{R}_0<1$ \end{document}, there is a unique disease-free equilibrium which is globally asymptotically stable; and if \begin{document} $\mathcal{R}_0>1$ \end{document}, there exists a unique endemic equilibrium which is globally asymptotically stable. Furthermore, we study the relation between \begin{document} $\mathcal{R}_0$ \end{document} with the diffusion and spatial heterogeneity and find that, it seems very necessary to create a low-risk habitat for the population to effectively control the spread of the epidemic disease. This may provide some potential applications in disease control.

Effect of seasonal changing temperature on the growth of phytoplankton
Ming Chen, Meng Fan, Xing Yuan and Huaiping Zhu
2017, 14(5-6): 1091-1117 doi: 10.3934/mbe.2017057 +[Abstract](121) +[HTML](6) +[PDF](659.3KB)

An non-autonomous nutrient-phytoplankton interacting model incorporating the effect of time-varying temperature is established. The impacts of temperature on metabolism of phytoplankton such as nutrient uptake, death rate, and nutrient releasing from particulate nutrient are investigated. The ecological reproductive index is formulated to present a threshold criteria and to characterize the dynamics of phytoplankton. The positive invariance, dissipativity, and the existence and stability of boundary and positive periodic solution are established. The analyses rely on the comparison principle, the coincidence degree theory and Lyapunov direct method. The effect of seasonal temperature and daily temperature on phytoplankton biomass are simulated numerically. Numerical simulation shows that the phytoplankton biomass is very robust to the variation of water temperature. The dynamics of the model and model predictions agree with the experimental data. Our model and analysis provide a possible explanation of triggering mechanism of phytoplankton blooms.

Effects of isolation and slaughter strategies in different species on emerging zoonoses
Jing-An Cui and Fangyuan Chen
2017, 14(5-6): 1119-1140 doi: 10.3934/mbe.2017058 +[Abstract](169) +[HTML](3) +[PDF](7057.7KB)

Zoonosis is the kind of infectious disease transmitting among different species by zoonotic pathogens. Different species play different roles in zoonoses. In this paper, we established a basic model to describe the zoonotic pathogen transmission from wildlife, to domestic animals, to humans. Then we put three strategies into the basic model to control the emerging zoonoses. Three strategies are corresponding to control measures of isolation, slaughter or similar in wildlife, domestic animals and humans respectively. We analyzed the effects of these three strategies on control reproductive numbers and equilibriums and we took avian influenza epidemic in China as an example to show the impacts of the strategies on emerging zoonoses in different areas at beginning.

An SIR epidemic model with vaccination in a patchy environment
Qianqian Cui, Zhipeng Qiu and Ling Ding
2017, 14(5-6): 1141-1157 doi: 10.3934/mbe.2017059 +[Abstract](283) +[HTML](3) +[PDF](668.0KB)

In this paper, an SIR patch model with vaccination is formulated to investigate the effect of vaccination coverage and the impact of human mobility on the spread of diseases among patches. The control reproduction number \begin{document} $\mathfrak{R}_v$ \end{document} is derived. It shows that the disease-free equilibrium is unique and is globally asymptotically stable if \begin{document} $\mathfrak{R}_v < 1$ \end{document}, and unstable if \begin{document} $\mathfrak{R}_v>1$ \end{document}. The sufficient condition for the local stability of boundary equilibria and the existence of equilibria are obtained for the case \begin{document} $n=2$ \end{document}. Numerical simulations indicate that vaccines can control and prevent the outbreak of infectious in all patches while migration may magnify the outbreak size in one patch and shrink the outbreak size in other patch.

Global dynamics of a vector-host epidemic model with age of infection
Yan-Xia Dang, Zhi-Peng Qiu, Xue-Zhi Li and Maia Martcheva
2017, 14(5-6): 1159-1186 doi: 10.3934/mbe.2017060 +[Abstract](104) +[HTML](7) +[PDF](533.6KB)

In this paper, a partial differential equation (PDE) model is proposed to explore the transmission dynamics of vector-borne diseases. The model includes both incubation age of the exposed hosts and infection age of the infectious hosts which describe incubation-age dependent removal rates in the latent period and the variable infectiousness in the infectious period, respectively. The reproductive number \begin{document} $\mathcal R_0$ \end{document} is derived. By using the method of Lyapunov function, the global dynamics of the PDE model is further established, and the results show that the basic reproduction number \begin{document} $\mathcal R_0$ \end{document} determines the transmission dynamics of vector-borne diseases: the disease-free equilibrium is globally asymptotically stable if \begin{document} $\mathcal R_0≤ 1$ \end{document}, and the endemic equilibrium is globally asymptotically stable if \begin{document} $\mathcal{R}_0>1$ \end{document}. The results suggest that an effective strategy to contain vector-borne diseases is decreasing the basic reproduction number \begin{document} $\mathcal{R}_0$ \end{document} below one.

Invasion entire solutions in a time periodic Lotka-Volterra competition system with diffusion
Li-Jun Du, Wan-Tong Li and Jia-Bing Wang
2017, 14(5-6): 1187-1213 doi: 10.3934/mbe.2017061 +[Abstract](75) +[HTML](5) +[PDF](585.7KB)

This paper is concerned with invasion entire solutions of a monostable time periodic Lotka-Volterra competition-diffusion system. We first give the asymptotic behaviors of time periodic traveling wave solutions at infinity by a dynamical approach coupled with the two-sided Laplace transform. According to these asymptotic behaviors, we then obtain some key estimates which are crucial for the construction of an appropriate pair of sub-super solutions. Finally, using the sub-super solutions method and comparison principle, we establish the existence of invasion entire solutions which behave as two periodic traveling fronts with different speeds propagating from both sides of x-axis. In other words, we formulate a new invasion way of the superior species to the inferior one in a time periodic environment.

Structural calculations and propagation modeling of growing networks based on continuous degree
Junbo Jia, Zhen Jin, Lili Chang and Xinchu Fu
2017, 14(5-6): 1215-1232 doi: 10.3934/mbe.2017062 +[Abstract](93) +[HTML](8) +[PDF](575.4KB)

When a network reaches a certain size, its node degree can be considered as a continuous variable, which we will call continuous degree. Using continuous degree method (CDM), we analytically calculate certain structure of the network and study the spread of epidemics on a growing network. Firstly, using CDM we calculate the degree distributions of three different growing models, which are the BA growing model, the preferential attachment accelerating growing model and the random attachment growing model. We obtain the evolution equation for the cumulative distribution function \begin{document}$F(k,t)$\end{document}, and then obtain analytical results about \begin{document}$F(k,t)$\end{document} and the degree distribution \begin{document}$p(k,t)$\end{document}. Secondly, we calculate the joint degree distribution \begin{document}$p(k_1, k_2, t)$\end{document} of the BA model by using the same method, thereby obtain the conditional degree distribution \begin{document}$p (k_1|k_2) $\end{document}. We find that the BA model has no degree correlations. Finally, we consider the different states, susceptible and infected, according to the node health status. We establish the continuous degree SIS model on a static network and a growing network, respectively. We find that, in the case of growth, the new added health nodes can slightly reduce the ratio of infected nodes, but the final infected ratio will gradually tend to the final infected ratio of SIS model on static networks.

Global dynamics of a delay virus model with recruitment and saturation effects of immune responses
Cuicui Jiang, Kaifa Wang and Lijuan Song
2017, 14(5-6): 1233-1246 doi: 10.3934/mbe.2017063 +[Abstract](184) +[HTML](4) +[PDF](732.4KB)

In this paper, we formulate a virus dynamics model with the recruitment of immune responses, saturation effects and an intracellular time delay. With the help of uniform persistence theory and Lyapunov method, we show that the global stability of the model is totally determined by the basic reproductive number $R_0$. Furthermore, we analyze the effects of the recruitment of immune responses on virus infection by numerical simulation. The results show ignoring the recruitment of immune responses will result in overestimation of the basic reproductive number and the severity of viral infection.

Theoretical investigation on models of circadian rhythms based on dimerization and proteolysis of PER and TIM
Jifa Jiang, Qiang Liu and Lei Niu
2017, 14(5-6): 1247-1259 doi: 10.3934/mbe.2017064 +[Abstract](320) +[HTML](4) +[PDF](477.5KB)

Circadian rhythms of physiology and behavior are widespread\break mechanisms in many organisms. The internal biological rhythms are driven by molecular clocks, which oscillate with a period nearly but not exactly \begin{document}$24$\end{document} hours. Many classic models of circadian rhythms are based on a time-delayed negative feedback, suggested by the protein products inhibiting transcription of their own genes. In 1999, based on stabilization of PER upon dimerization, Tyson et al. [J. J. Tyson, C. I. Hong, C. D. Thron, B. Novak, Biophys. J. 77 (1999) 2411-2417] proposed a crucial positive feedback to the circadian oscillator. This idea was mathematically expressed in a three-dimensional model. By imposing assumptions that the dimerization reactions were fast and dimeric proteins were in rapid equilibrium, they reduced the model to a pair of nonlinear ordinary differential equations of mRNA and total protein concentrations. Then they used phase plane analysis tools to investigate circadian rhythms. In this paper, the original three-dimensional model is studied. We explore the existence of oscillations and their periods. Much attention is paid to investigate how the periods depend on model parameters. The numerical simulations are in good agreement with their reduced work.

Modulation of first-passage time for bursty gene expression via random signals
Qiuying Li, Lifang Huang and Jianshe Yu
2017, 14(5-6): 1261-1277 doi: 10.3934/mbe.2017065 +[Abstract](240) +[HTML](4) +[PDF](786.0KB)

The stochastic nature of cell-specific signal molecules (such as transcription factor, ribosome, etc.) and the intrinsic stochastic nature of gene expression process result in cell-to-cell variations at protein levels. Increasing experimental evidences suggest that cell phenotypic variations often depend on the accumulation of some special proteins. Hence, a natural and fundamental question is: How does input signal affect the timing of protein count up to a given threshold? To this end, we study effects of input signal on the first-passage time (FPT), the time at which the number of proteins crosses a given threshold. Input signal is distinguished into two types: constant input signal and random input signal, regulating only burst frequency (or burst size) of gene expression. Firstly, we derive analytical formulae for FPT moments in each case of constant signal regulation and random signal regulation. Then, we find that random input signal tends to increases the mean and noise of FPT compared with constant input signal. Finally, we observe that different regulation ways of random signal have different effects on FPT, that is, burst size modulation tends to decrease the mean of FPT and increase the noise of FPT compared with burst frequency modulation. Our findings imply a fundamental mechanism that random fluctuating environment may prolong FPT. This can provide theoretical guidance for studies of some cellular key events such as latency of HIV and lysis time of bacteriophage $λ.$ In conclusion, our results reveal impacts of external signal on FPT and aid understanding the regulation mechanism of gene expression.

A mathematical model for the seasonal transmission of schistosomiasis in the lake and marshland regions of China
Yingke Li, Zhidong Teng, Shigui Ruan, Mingtao Li and Xiaomei Feng
2017, 14(5-6): 1279-1299 doi: 10.3934/mbe.2017066 +[Abstract](90) +[HTML](3) +[PDF](699.1KB)

Schistosomiasis, a parasitic disease caused by Schistosoma Japonicum, is still one of the most serious parasitic diseases in China and remains endemic in seven provinces, including Hubei, Anhui, Hunan, Jiangsu, Jiangxi, Sichuan, and Yunnan. The monthly data of human schistosomiasis cases in Hubei, Hunan, and Anhui provinces (lake and marshland regions) released by the Chinese Center for Disease Control and Prevention (China CDC) display a periodic pattern with more cases in late summer and early autumn. Based on this observation, we construct a deterministic model with periodic transmission rates to study the seasonal transmission dynamics of schistosomiasis in these lake and marshland regions in China. We calculate the basic reproduction number $R_{0}$, discuss the dynamical behavior of solutions to the model, and use the model to fit the monthly data of human schistosomiasis cases in Hubei. We also perform some sensitivity analysis of the basic reproduction number $R_{0}$ in terms of model parameters. Our results indicate that treatment of at-risk population groups, improving sanitation, hygiene education, and snail control are effective measures in controlling human schistosomiasis in these lakes and marshland regions.

Modeling co-infection of Ixodes tick-borne pathogens
Yijun Lou, Li Liu and Daozhou Gao
2017, 14(5-6): 1301-1316 doi: 10.3934/mbe.2017067 +[Abstract](113) +[HTML](5) +[PDF](1920.3KB)

Ticks, including the Ixodes ricinus and Ixodes scapularis hard tick species, are regarded as the most common arthropod vectors of both human and animal diseases in Europe and the United States capable of transmitting a large number of bacteria, viruses and parasites. Since ticks in larval and nymphal stages share the same host community which can harbor multiple pathogens, they may be co-infected with two or more pathogens, with a subsequent high likelihood of co-transmission to humans or animals. This paper is devoted to the modeling of co-infection of tick-borne pathogens, with special focus on the co-infection of Borrelia burgdorferi (agent of Lyme disease) and Babesia microti (agent of human babesiosis). Considering the effect of co-infection, we illustrate that co-infection with B. burgdorferi increases the likelihood of B. microti transmission, by increasing the basic reproduction number of B. microti below the threshold smaller than one to be possibly above the threshold for persistence. The study confirms a mechanism of the ecological fitness paradox, the establishment of B. microti which has weak fitness (basic reproduction number less than one). Furthermore, co-infection could facilitate range expansion of both pathogens.

An SEI infection model incorporating media impact
Xuejuan Lu, Shaokai Wang, Shengqiang Liu and Jia Li
2017, 14(5-6): 1317-1335 doi: 10.3934/mbe.2017068 +[Abstract](81) +[HTML](3) +[PDF](798.9KB)

To study the impact of media coverage on spread and control of infectious diseases, we use a susceptible-exposed-infective (SEI) model, including individuals' behavior changes in their contacts due to the influences of media coverage, and fully investigate the model dynamics. We define the basic reproductive number \begin{document}$\Re_0$\end{document} for the model, and show that the modeled disease dies out regardless of initial infections when \begin{document}$\Re_0 < 1$\end{document}, and becomes uniformly persistently endemic if \begin{document}$\Re_0>1$\end{document}. When the disease is endemic and the influence of the media coverage is less than or equal to a critical number, there exists a unique endemic equilibrium which is asymptotical stable provided \begin{document}$\Re_0 $\end{document} is greater than and near one. However, if \begin{document}$\Re_0 $\end{document} is larger than a critical number, the model can undergo Hopf bifurcation such that multiple endemic equilibria are bifurcated from the unique endemic equilibrium as the influence of the media coverage is increased to a threshold value. Using numerical simulations we obtain results on the effects of media coverage on the endemic that the media coverage may decrease the peak value of the infectives or the average number of the infectives in different cases. We show, however, that given larger \begin{document}$\Re_0$\end{document}, the influence of the media coverage may as well result in increasing the average number of the infectives, which brings challenges to the control and prevention of infectious diseases.

Global stability in a tuberculosis model of imperfect treatment with age-dependent latency and relapse
Shanjing Ren
2017, 14(5-6): 1337-1360 doi: 10.3934/mbe.2017069 +[Abstract](94) +[HTML](4) +[PDF](603.4KB)

In this paper, an \begin{document}$SEIR$\end{document} epidemic model for an imperfect treatment disease with age-dependent latency and relapse is proposed. The model is well-suited to model tuberculosis. The basic reproduction number \begin{document}$R_{0}$\end{document} is calculated. We obtain the global behavior of the model in terms of \begin{document}$R_{0}$\end{document}. If \begin{document}$R_{0} < 1$\end{document}, the disease-free equilibrium is globally asymptotically stable, whereas if \begin{document}$R_{0}>1$\end{document}, a Lyapunov functional is used to show that the endemic equilibrium is globally stable amongst solutions for which the disease is present.

A bacteriophage model based on CRISPR/Cas immune system in a chemostat
Mengshi Shu, Rui Fu and Wendi Wang
2017, 14(5-6): 1361-1377 doi: 10.3934/mbe.2017070 +[Abstract](91) +[HTML](5) +[PDF](489.7KB)

Clustered regularly interspaced short palindromic repeats (CRISPRs) along with Cas proteins are a widespread immune system across bacteria and archaea. In this paper, a mathematical model in a chemostat is proposed to investigate the effect of CRISPR/Cas on the bacteriophage dynamics. It is shown that the introduction of CRISPR/Cas can induce a backward bifurcation and transcritical bifurcation. Numerical simulations reveal the coexistence of a stable infection-free equilibrium with an infection equilibrium, or a stable infection-free equilibrium with a stable periodic solution.

A mathematical model of stem cell regeneration with epigenetic state transitions
Qiaojun Situ and Jinzhi Lei
2017, 14(5-6): 1379-1397 doi: 10.3934/mbe.2017071 +[Abstract](210) +[HTML](8) +[PDF](500.9KB)

In this paper, we study a mathematical model of stem cell regeneration with epigenetic state transitions. In the model, the heterogeneity of stem cells is considered through the epigenetic state of each cell, and each epigenetic state defines a subpopulation of stem cells. The dynamics of the subpopulations are modeled by a set of ordinary differential equations in which epigenetic state transition in cell division is given by the transition probability. We present analysis for the existence and linear stability of the equilibrium state. As an example, we apply the model to study the dynamics of state transition in breast cancer stem cells.

On the continuity of the function describing the times of meeting impulsive set and its application
Sanyi Tang and Wenhong Pang
2017, 14(5-6): 1399-1406 doi: 10.3934/mbe.2017072 +[Abstract](90) +[HTML](3) +[PDF](363.9KB)

The properties of the limit sets of orbits of planar impulsive semi-dynamic system strictly depend on the continuity of the function, which describes the times of meeting impulsive sets. In this note, we will show a more realistic counter example on the continuity of this function which has been proven and widely used in impulsive dynamical system and applied in life sciences including population dynamics and disease control. Further, what extra condition should be added to guarantee the continuity of the function has been addressed generally, and then the applications and shortcomings have been discussed when using the properties of this function.

Dynamics of epidemic models with asymptomatic infection and seasonal succession
Yilei Tang, Dongmei Xiao, Weinian Zhang and Di Zhu
2017, 14(5-6): 1407-1424 doi: 10.3934/mbe.2017073 +[Abstract](96) +[HTML](5) +[PDF](476.4KB)

In this paper, we consider a compartmental SIRS epidemic model with asymptomatic infection and seasonal succession, which is a periodic discontinuous differential system. The basic reproduction number \begin{document}$\mathcal{R}_0$\end{document} is defined and evaluated directly for this model, and uniform persistence of the disease and threshold dynamics are obtained. Specially, global dynamics of the model without seasonal force are studied. It is shown that the model has only a disease-free equilibrium which is globally stable if \begin{document}$\mathcal{R}_0≤ 1$\end{document}, and as \begin{document}$\mathcal{R}_0>1$\end{document} the disease-free equilibrium is unstable and there is an endemic equilibrium, which is globally stable if the recovering rates of asymptomatic infectives and symptomatic infectives are close. These theoretical results provide an intuitive basis for understanding that the asymptomatically infective individuals and the seasonal disease transmission promote the evolution of the epidemic, which allow us to predict the outcomes of control strategies during the course of the epidemic.

Dynamical behaviors of an Echinococcosis epidemic model with distributed delays
Kai Wang, Zhidong Teng and Xueliang Zhang
2017, 14(5-6): 1425-1445 doi: 10.3934/mbe.2017074 +[Abstract](84) +[HTML](7) +[PDF](1121.7KB)

In this paper, a novel spreading dynamical model for Echinococcosis with distributed time delays is proposed. For the model, we firstly give the basic reproduction number \begin{document}$\mathcal{R}_0$\end{document} and the existence of a unique endemic equilibrium when \begin{document}$\mathcal{R}_0>1$\end{document}. Furthermore, we analyze the dynamical behaviors of the model. The results show that the dynamical properties of the model is completely determined by \begin{document}$\mathcal{R}_0$\end{document}. That is, if \begin{document}$\mathcal{R}_0<1$\end{document}, the disease-free equilibrium is globally asymptotically stable, and if \begin{document}$\mathcal{R}_0>1$\end{document}, the model is permanent and the endemic equilibrium is globally asymptotically stable. According to human Echinococcosis cases from January 2004 to December 2011 in Xinjiang, China, we estimate the parameters of the model and study the transmission trend of the disease in Xinjiang, China. The model provides an approximate estimate of the basic reproduction number \begin{document}$\mathcal{R}_0=1.23$\end{document} in Xinjiang, China. From theoretic results, we further find that Echinococcosis is endemic in Xinjiang, China. Finally, we perform some sensitivity analysis of several model parameters and give some useful measures on controlling the transmission of Echinococcosis.

Modeling transcriptional co-regulation of mammalian circadian clock
Yanqin Wang, Xin Ni, Jie Yan and Ling Yang
2017, 14(5-6): 1447-1462 doi: 10.3934/mbe.2017075 +[Abstract](125) +[HTML](10) +[PDF](543.1KB)

The circadian clock is a self-sustaining oscillator that has a period of about 24 hours at the molecular level. The oscillator is a transcription-translation feedback loop system composed of several genes. In this paper, a scalar nonlinear differential equation with two delays, modeling the transcriptional co-regulation in mammalian circadian clock, is proposed and analyzed. Sufficient conditions are established for the asymptotic stability of the unique nontrivial positive equilibrium point of the model by studying an exponential polynomial characteristic equation with delay-dependent coefficients. The existence of the Hopf bifurcations can be also obtained. Numerical simulations of the model with proper parameter values coincide with the theoretical result.

Transition of interaction outcomes in a facilitation-competition system of two species
Yuanshi Wang and Hong Wu
2017, 14(5-6): 1463-1475 doi: 10.3934/mbe.2017076 +[Abstract](91) +[HTML](2) +[PDF](573.9KB)

A facilitation-competition system of two species is considered, where one species has a facilitation effect on the other but there is spatial competition between them. Our aim is to show mechanism by which the facilitation promotes coexistence of the species. A lattice gas model describing the facilitation-competition system is analyzed, in which nonexistence of periodic solution is shown and previous results are extended. Global dynamics of the model demonstrate essential features of the facilitation-competition system. When a species cannot survive alone, a strong facilitation from the other would lead to its survival. However, if the facilitation is extremely strong, both species go extinct. When a species can survive alone and its mortality rate is not larger than that of the other species, it would drive the other one into extinction. When a species can survive alone and its mortality rate is larger than that of the other species, it would be driven into extinction if the facilitation from the other is weak, while it would coexist with the other if the facilitation is strong. Moreover, an extremely strong facilitation from the other would lead to extinction of species. Bifurcation diagram of the system exhibits that interaction outcome between the species can transition between competition, amensalism, neutralism and parasitism in a smooth fashion. A novel result of this paper is the rigorous and thorough analysis, which displays transparency of dynamics in the system. Numerical simulations validate the results.

Asymptotic behavior of a delayed stochastic logistic model with impulsive perturbations
Sanling Yuan, Xuehui Ji and Huaiping Zhu
2017, 14(5-6): 1477-1498 doi: 10.3934/mbe.2017077 +[Abstract](74) +[HTML](4) +[PDF](692.4KB)

In this paper, we investigate the dynamics of a delayed logistic model with both impulsive and stochastic perturbations. The impulse is introduced at fixed moments and the stochastic perturbation is of white noise type which is assumed to be proportional to the population density. We start with the existence and uniqueness of the positive solution of the model, then establish sufficient conditions ensuring its global attractivity. By using the theory of integral Markov semigroups, we further derive sufficient conditions for the existence of the stationary distribution of the system. Finally, we perform the extinction analysis of the model. Numerical simulations illustrate the obtained theoretical results.

Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network
Qiuyan Zhang, Lingling Liu and Weinian Zhang
2017, 14(5-6): 1499-1514 doi: 10.3934/mbe.2017078 +[Abstract](85) +[HTML](3) +[PDF](594.3KB)

There have been some results on bifurcations of codimension one (such as saddle-node, transcritical, pitchfork) and degenerate Hopf bifurcations for an enzyme-catalyzed reaction system comprising a branched network but no further discussion for bifurcations at its cusp. In this paper we give conditions for the existence of a cusp and compute the parameter curves for the Bogdanov-Takens bifurcation, which induces the appearance of homoclinic orbits and periodic orbits, indicating the tendency to steady-states or a rise of periodic oscillations for the concentrations of the substrate and the product.

Onset and termination of oscillation of disease spread through contaminated environment
Xue Zhang, Shuni Song and Jianhong Wu
2017, 14(5-6): 1515-1533 doi: 10.3934/mbe.2017079 +[Abstract](67) +[HTML](4) +[PDF](498.7KB)

We consider a reaction diffusion equation with a delayed nonlocal nonlinearity and subject to Dirichlet boundary condition. The model equation is motivated by infection dynamics of disease spread (avian influenza, for example) through environment contamination, and the nonlinearity takes into account of distribution of limited resources for rapid and slow interventions to clean contaminated environment. We determine conditions under which an equilibrium with positive value in the interior of the domain (disease equilibrium) emerges and determine conditions under which Hope bifurcation occurs. For a fixed pair of rapid and slow response delay, we show that nonlinear oscillations can be avoided by distributing resources for both fast or slow interventions.

Threshold dynamics of a time periodic and two--group epidemic model with distributed delay
Lin Zhao, Zhi-Cheng Wang and Liang Zhang
2017, 14(5-6): 1535-1563 doi: 10.3934/mbe.2017080 +[Abstract](191) +[HTML](15) +[PDF](534.7KB)

In this paper, a time periodic and two–group reaction–diffusion epidemic model with distributed delay is proposed and investigated. We firstly introduce the basic reproduction number \begin{document}$R_0$\end{document} for the model via the next generation operator method. We then establish the threshold dynamics of the model in terms of \begin{document}$R_0$\end{document}, that is, the disease is uniformly persistent if \begin{document}$R_0 > 1$\end{document}, while the disease goes to extinction if \begin{document}$R_0 < 1$\end{document}. Finally, we study the global dynamics for the model in a special case when all the coefficients are independent of spatio–temporal variables.

The risk index for an SIR epidemic model and spatial spreading of the infectious disease
Min Zhu, Xiaofei Guo and Zhigui Lin
2017, 14(5-6): 1565-1583 doi: 10.3934/mbe.2017081 +[Abstract](85) +[HTML](4) +[PDF](609.7KB)

In this paper, a reaction-diffusion-advection SIR model for the transmission of the infectious disease is proposed and analyzed. The free boundaries are introduced to describe the spreading fronts of the disease. By exhibiting the basic reproduction number \begin{document}$R_0^{DA}$\end{document} for an associated model with Dirichlet boundary condition, we introduce the risk index \begin{document}$R^F_0(t)$\end{document} for the free boundary problem, which depends on the advection coefficient and time. Sufficient conditions for the disease to prevail or not are obtained. Our results suggest that the disease must spread if \begin{document}$R^F_0(t_0)≥q 1$\end{document} for some \begin{document}$t_0$\end{document} and the disease is vanishing if \begin{document}$R^F_0(∞)<1$\end{document}, while if \begin{document}$R^F_0(0)<1$\end{document}, the spreading or vanishing of the disease depends on the initial state of infected individuals as well as the expanding capability of the free boundary. We also illustrate the impacts of the expanding capability on the spreading fronts via the numerical simulations.

Modeling and analyzing the transmission dynamics of visceral leishmaniasis
Lan Zou, Jing Chen and Shigui Ruan
2017, 14(5-6): 1585-1604 doi: 10.3934/mbe.2017082 +[Abstract](98) +[HTML](6) +[PDF](906.2KB)

In this paper, we develop a mathematical model to study the transmission dynamics of visceral leishmaniasis. Three populations: dogs, sandflies and humans, are considered in the model. Based on recent studies, we include vertical transmission of dogs in the spread of the disease. We also investigate the impact of asymptomatic humans and dogs as secondary reservoirs of the parasites. The basic reproduction number and sensitivity analysis show that the control of dog-sandfly transmission is more important for the elimination of the disease. Vaccination of susceptible dogs, treatment of infective dogs, as well as control of vertical transmission in dogs are effective prevention and control measures for visceral leishmaniasis.

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