Computation of $\mathcal R $ in age-structured epidemiological models with maternal and temporary immunity
Zhilan Feng Qing Han Zhipeng Qiu Andrew N. Hill John W. Glasser
For infectious diseases such as pertussis, susceptibility is determined by immunity, which is chronological age-dependent. We consider an age-structured epidemiological model that accounts for both passively acquired maternal antibodies that decay and active immunity that wanes, permitting re-infection. The model is a 6-dimensional system of partial differential equations (PDE). By assuming constant rates within each age-group, the PDE system can be reduced to an ordinary differential equation (ODE) system with aging from one age-group to the next. We derive formulae for the effective reproduction number ${\mathcal R}$ and provide their biological interpretation in some special cases. We show that the disease-free equilibrium is stable when ${\mathcal R}<1$ and unstable if ${\mathcal R}>1$.
keywords: multiple infections. Age-structured epidemiological model partial immunity reproduction numbers
Complex dynamics of a nutrient-plankton system with nonlinear phytoplankton mortality and allelopathy
Zhipeng Qiu Huaiping Zhu
Understanding the plankton dynamics can help us take effective measures to settle the critical issue on how to keep plankton ecosystem balance. In this paper, a nutrient-phytoplankton-zooplankton (NPZ) model is formulated to understand the mechanism of plankton dynamics. To account for the harmful effect of the phytoplankton allelopathy, a prototype for a non-monotone response function is used to model zooplankton grazing, and nonlinear phytoplankton mortality is also included in the NPZ model. Using the model, we will focus on understanding how the phytoplankton allelopathy and nonlinear phytoplankton mortality affect the plankton population dynamics. We first examine the existence of multiple equilibria and provide a detailed classification for the equilibria, then stability and local bifurcation analysis are also studied. Sufficient conditions for Hopf bifurcation and zero-Hopf bifurcation are given respectively. Numerical simulations are finally conducted to confirm and extend the analytic results. Both theoretical and numerical findings imply that the phytoplankton allelopathy and nonlinear phytoplankton mortality may lead to a rich variety of complex dynamics of the nutrient-plankton system. The results of this study suggest that the effects of the phytoplankton allelopathy and nonlinear phytoplankton mortality should receive more attention to understand the plankton dynamics.
keywords: bifurcation Nutrient-phytoplankton-zooplankton model nonlinear phytoplankton mortality allelopathy complex dynamics.
Optimal control of a vector-host epidemics model
Qingkai Kong Zhipeng Qiu Zi Sang Yun Zou
In this paper, a vector-host epidemic model with control measures is considered to assess the impact of control measures on the prevalence of the vector-host diseases. We incorporated mosquito-reduction strategy and host medical treatment into the model. For the basic vector-host model, we provide sufficient conditions for the local stability of the disease free equilibrium (DFE) and the sensitivity analysis for the reproduction number with respect to the model parameters. Using the optimal control theory, the optimal levels of the two controls are characterized, and then the existence and uniqueness for the optimal control pair are established. Numerical simulations are further conducted to confirm and extend the analytical results. Numerical results suggest that optimal multi-control strategy is a more beneficial choice in fighting the outbreak of vector-host diseases. For the vector-host epidemics, vector control measures should be taken prior to other measures.
keywords: Epidemic model sensitivity indices reproduction number. optimal control
Assessing the effect of non-pharmaceutical interventions on containing an emerging disease
Zi Sang Zhipeng Qiu Xiefei Yan Yun Zou
Non-pharmaceutical interventions, such as quarantine, isolation and entry screening, are usually the primary public health measures to control the spread of an emerging infectious disease through a human population. This paper proposes a multi-regional deterministic compartmental model to assess the effectiveness and implications of non-pharmaceutical interventions. The reproduction number is determined as the spectral radius of a nonnegative matrix product. Comparisons are made using the reproduction number, epidemic peaks and cumulative number of infections and mortality as indexes. Simulation results show that quarantine of suspected cases and isolation of cases with symptom are effective in reducing disease burden for multiple regions. Using entry screening strategy leads to a moderate time delay for epidemic peaks, but is of no help for preventing an epidemic breaking out. The study further shows that isolation strategy is always the best choice in the presence or absence of stringent hygiene precautions and should be given priority in combating an emerging epidemic.
keywords: isolation metapopualtion epidemic model. Quarantine entry screening
Multi-host transmission dynamics of schistosomiasis and its optimal control
Chunxiao Ding Zhipeng Qiu Huaiping Zhu
In this paper we formulate a dynamical model to study the transmission dynamics of schistosomiasis in humans and snails. We also incorporate bovines in the model to study their impact on transmission and controlling the spread of Schistosoma japonicum in humans in China. The dynamics of the model is rigorously analyzed by using the theory of dynamical systems. The theoretical results show that the disease free equilibrium is globally asymptotically stable if $\mathcal R_0<1$, and if $\mathcal R_0>1$ the system has only one positive equilibrium. The local stability of the unique positive equilibrium is investigated and sufficient conditions are also provided for the global stability of the positive equilibrium. The optimal control theory are further applied to the model to study the corresponding optimal control problem. Both analytical and numerical results suggest that: (a) the infected bovines play an important role in the spread of schistosomiasis among humans, and killing the infected bovines will be useful to prevent transmission of schistosomiasis among humans; (b) optimal control strategy performs better than the constant controls in reducing the prevalence of the infected human and the cost for implementing optimal control is much less than that for constant controls; and (c) improving the treatment rate of infected humans, the killing rate of the infected bovines and the fishing rate of snails in the early stage of spread of schistosomiasis are very helpful to contain the prevalence of infected human case as well as minimize the total cost.
keywords: optimal control. Multi-hosts transmission dynamics stability schistosomiasis
Competitive exclusion in an infection-age structured vector-host epidemic model
Yanxia Dang Zhipeng Qiu Xuezhi Li
The competitive exclusion principle means that the strain with the largest reproduction number persists while eliminating all other strains with suboptimal reproduction numbers. In this paper, we extend the competitive exclusion principle to a multi-strain vector-borne epidemic model with age-since-infection. The model includes both incubation age of the exposed hosts and infection age of the infectious hosts, both of which describe the different removal rates in the latent period and the variable infectiousness in the infectious period, respectively. The formulas for the reproduction numbers $\mathcal R^j_0$ of strain $j,j=1,2,···, n$, are obtained from the biological meanings of the model. The strain $j$ can not invade the system if $\mathcal R^j_0<1$, and the disease free equilibrium is globally asymptotically stable if $\max_j\{\mathcal R^j_0\}<1$. If $\mathcal R^{j_0}_0>1$, then a single-strain equilibrium $\mathcal{E}_{j_0}$ exists, and the single strain equilibrium is locally asymptotically stable when $\mathcal R^{j_0}_0>1$ and $\mathcal R^{j_0}_0>\mathcal R^{j}_0,j≠ j_0$. Finally, by using a Lyapunov function, sufficient conditions are further established for the global asymptotical stability of the single-strain equilibrium corresponding to strain $j_0$, which means strain $j_0$ eliminates all other stains as long as $\mathcal R^{j}_0/\mathcal R^{j_0}_0<b_j/b_{j_0}<1,j≠ j_0$, where $b_j$ denotes the probability of a given susceptible vector being transmitted by an infected host with strain $j$.
keywords: Age-structure competitive exclusion global stability vector-borne disease Lyapunov function
The asymptotic behavior of a chemostat model
Zhipeng Qiu Jun Yu Yun Zou
In this paper, the Chemostat model with stage-structure and the Beddington-DeAngelies functional responses is studied. Sufficient conditions for uniform persistence of this model with delay are obtained via uniform persistence of infinite dimensional dynamical systems; and for the model without delay, sufficient conditions for the global asymptotic stability of the positive equilibrium are presented.
keywords: global asymptotic stability uniform persistence stage-structure Chemostat mutual interference.
Global stability of an age-structured cholera model
Jianxin Yang Zhipeng Qiu Xue-Zhi Li
In this paper, an age-structured epidemic model is formulated to describe the transmission dynamics of cholera. The PDE model incorporates direct and indirect transmission pathways, infection-age-dependent infectivity and variable periods of infectiousness. Under some suitable assumptions, the PDE model can be reduced to the multi-stage models investigated in the literature. By using the method of Lyapunov function, we established the dynamical properties of the PDE model, and the results show that the global dynamics of the model is completely determined by the basic reproduction number $\mathcal R_0$: if $\mathcal R_0 < 1$ the cholera dies out, and if $\mathcal R_0 >1$ the disease will persist at the endemic equilibrium. Then the global results obtained for multi-stage models are extended to the general continuous age model.
keywords: Age of infection global stability Lyapunov function. cholera
An SIR epidemic model with vaccination in a patchy environment
Qianqian Cui Zhipeng Qiu Ling Ding

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 $\mathfrak{R}_v$ is derived. It shows that the disease-free equilibrium is unique and is globally asymptotically stable if $\mathfrak{R}_v < 1$, and unstable if $\mathfrak{R}_v>1$. The sufficient condition for the local stability of boundary equilibria and the existence of equilibria are obtained for the case $n=2$. 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.

keywords: SIR model vaccination patchy environment transmission dynamics reproduction number

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