This paper is concerned with the mathematical analysis of an age-structured multi-group heroin epidemic model, which can be used to describe the spread of heroin habituation and addiction in ...heterogeneous environment. Under general assumptions on the different level of susceptibility and the relapse to frequent heroin use, we establish sharp criteria for heroin spreading and vanishing. We rigorously investigate the well-posedness of the model, the existence of equilibria, the asymptotic smoothness of solution orbits, and the global stability of equilibria. Specifically, we rigorously show that the drug-free equilibrium is globally asymptotically stable if a threshold value ℜ0 is less than one, and the unique drug-endemic equilibrium is globally attractive if ℜ0 is greater than one. In the proofs of global stability of equilibria, we construct suitable Lyapunov functions by using a graph-theoretic method.
Compared with the existing HIV/AIDS host model that considers only age-since-infection or only spacial diffusion, we propose a new age–space structured model that incorporating both two infection ...ages, spacial diffusion, and HAART (highly active antiretroviral therapy) to analyze the global dynamics of HIV/AIDS epidemic and study the incident of its transmission among MSM (men who have sex with men) group. We first obtain the well-posedness and positivity of the solution for the model. The generation operator R is obtained by the process of renewal, followed by the basic reproduction number R0. Then we prove that the disease-free equilibrium is globally asymptotically stable (g.a.s) when R0<1, the space-independent steady state is g.a.s, and the disease will be present when R0>1. Numerical simulations are conducted to reinforce these analytical results. Finally, numerical results show that key model parameters influence the value of R0 effectively, and indicate that the spatial factor plays a crucial role as a control measure and cannot be ignored during the intervention for HIV transmission among MSM group. Moreover, our work suggests that treatment adherence should be improved during the HAART for HIV infected individuals among MSM.
We study a two-stage structured population model for which the juveniles diffuse purely by random walk while the adults exhibit long range dispersal. Questions on the persistence or extinction of the ...species are examined. It is shown that the population eventually dies out if the principal spectrum point λp of the linearized system at the trivial solution is nonpositive. However, the species persists if λp>0. Moreover, at least one positive steady state exists when λp>0. The uniqueness and global stability of the positive steady-state solution is obtained under some special cases. We also establish a sup/inf characterization of λp.
In the predator-prey system, predators can affect the prey population (1) by direct killing and (2) by inducing predation fear, which ultimately force preys to adopt some anti-predator strategies. ...However, the anti-predator strategy is not the same for all individual preys of different life stages. Also, anti-predator behavior has both cost and benefit, but most of the mathematical models observed the dynamics by incorporating its cost only. In the present study, we formulate a predator-prey model dividing the prey population into two stages: juvenile and adult. We assume that adult preys are only adapting group defense as an anti-predator strategy when they are sensitive to predation. Group defense plays a positive role for adult prey by reducing their predation, but, on the negative side, it simultaneously decreases their reproductive potential. A parameter, anti-predator sensitivity is introduced to interlink both the benefit and cost of group defense. Our result shows that when adult preys are not showing anti-predator behavior, with an increase of maturation rate, the system exhibits a population cycle of abruptly increasing amplitude, which may drive all species of the system to extinction. Anti-predator sensitivity may exclude oscillation through homoclinic bifurcation and avert the prey population for any possible random extinction. Anti-predator sensitivity also decreases the predator population density and produces bistable dynamics. Higher values of anti-predator sensitivity may lead to the extinction of the predator population and benefit adult preys to persist with large population density. Below a threshold value of anti-predator sensitivity, it may possible to retain the predator population in the system by increasing the fear level of the predator. We also observe our fear-induced stage-structured model exhibits interesting and rich dynamical behaviors, various types of bistabilities in different bi-parameter planes. Finally, we discuss the potential impact of our findings.
In the late stages of an epidemic, infections are often sporadic and geographically distributed. Spatially structured stochastic models can capture these important features of disease dynamics, ...thereby allowing a broader exploration of interventions. Here we develop a stochastic model of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) transmission among an interconnected group of population centers representing counties, municipalities, and districts (collectively, “counties”). The model is parameterized with demographic, epidemiological, testing, and travel data from Ontario, Canada. We explore the effects of different control strategies after the epidemic curve has been flattened. We compare a local strategy of reopening (and reclosing, as needed) schools and workplaces county by county, according to triggers for county-specific infection prevalence, to a global strategy of province-wide reopening and reclosing, according to triggers for province-wide infection prevalence. For trigger levels that result in the same number of COVID-19 cases between the two strategies, the local strategy causes significantly fewer person-days of closure, even under high intercounty travel scenarios. However, both cases and person-days lost to closure rise when county triggers are not coordinated and when testing rates vary among counties. Finally, we show that local strategies can also do better in the early epidemic stage, but only if testing rates are high and the trigger prevalence is low. Our results suggest that pandemic planning for the far side of the COVID-19 epidemic curve should consider local strategies for reopening and reclosing.
Every community experiences crime, therefore being aware of it is essential. It is natural to characterize individuals based on age when we consider the modelling of crime, as we do not expect a ...child to commit a crime. Also, a particular age group may be more prone to commit crimes. Keeping this in mind, we consider an age-structured model of the criminal and noncriminal-minded population in a society. The criminal-minded population is further divided into three categories. We develop a model and study the interaction between them. One crucial factor which we study is the effect of law enforcement. The global existence of steady state is shown using the method of induction on the real line. We check the stability of steady states by constructing appropriate Lyapunov functions. We have established the stability of the crime-free equilibrium. This equilibrium is more important as every society wants to achieve it. We also show the existence of forward bifurcation. It is interesting to see that the outcomes are very natural and enhance our understanding of such classes. Numerical simulations are also given for better illustration.
Immunotherapy is a targeted therapy that can be applied to cervical cancer patients to prevent DNA damage caused by human papillomavirus (HPV). The HPV infects normal cervical cells withing a ...specific cell age interval, i.e., between the $ G_1 $ to $ S $ phase of the cell cycle. In this study, we developed a new mathematical model of age-dependent immunotherapy for cervical cancer. The model is a four-dimensional first-order partial differential equation with time- and age-independent variables. The cell population is divided into four sub-populations, i.e., susceptible cells, cells infected by HPV, precancerous cells, and cancer cells. The immunotherapy term has been added to precancerous cells since these cells can experience regression if appointed by proper treatments. The immunotherapy process is closely related to the rate of T-cell division. The treatment works in the same cell cycle that stimulates and inhibits the immune system. In our model, immunotherapy is represented as a periodic function with a small amplitude. It is based on the fluctuating interaction between T-cells and precancerous cells. We have found that there are two types of steady-state conditions, i.e., infection-free and endemic. The local and global stability of an infection-free steady-state has been analyzed based on basic reproduction numbers. We have solved the Riccati differential equation to show the existence of an endemic steady-state. The stability analysis of the endemic steady-state has been determined by using the perturbation approach and solving integral equations. Some numerical simulations are also presented in this paper to illustrate the behavior of the solutions.
Fisheries have reduced the abundances of large piscivores-such as gadids (cod, pollock, etc.) and tunas-in ecosystems around the world. Fisheries also target smaller species-such as herring, capelin, ...and sprat-that are important parts of the piscivores' diets. It has been suggested that harvesting of these so-called forage fish will harm piscivores. Multispecies models used for fisheries assessments typically ignore important facets of fish community dynamics, such as individual-level bioenergetics and/or size structure. We test the effects of fishing for both forage fish and piscivores using a dynamic, multitrophic, size-structured, bioenergetics model of the Baltic Sea. In addition, we analyze historical patterns in piscivore-biomass declines and fishing mortalities of piscivores and forage fish using global fish-stock assessment data. Our community-dynamics model shows that piscivores benefit from harvesting of their forage fish when piscivore fishing mortality is high. With substantial harvesting of forage fish, the piscivores can withstand higher fishing mortality. On the other hand, when piscivore fishing mortality is low, piscivore biomass decreases with more fishing of the forage fish. In accordance with these predictions, our statistical analysis of global fisheries data shows a positive interaction between the fishing mortalities of forage-fish stocks and piscivore stocks on the strength of piscivore-biomass declines. While overfishing of forage fish must be prevented, our study shows that reducing fishing pressures on forage fish may have unwanted negative side effects on piscivores. In some cases, decreasing forage-fish exploitation could cause declines, or even collapses, of piscivore stocks.
In this paper, we study the global stability of a class of HIV viral infection models with continuous age-structure using the direct Lyapunov method. In each of the cases where the incidence rates ...are given by nonlinear infection rate F(T)G(V), Holling type II functional response and Crowley–Martin functional response, we define the basic reproduction number and prove that it is a sharp threshold determining whether the infection dies out or not. We give a rigorous mathematical analysis on technical materials and necessary arguments, including relative compactness of the orbit and uniform persistence of system, by reformulating the system as a system of Volterra integral equations. We further investigate global behaviors of HIV viral infection models with Holling type II functional response and Crowley–Martin functional response through numerical simulations.
We study sparse group Lasso for high-dimensional double sparse linear regression, where the parameter of interest is simultaneously element-wise and group-wise sparse. This problem is an important ...instance of the simultaneously structured model - an actively studied topic in statistics and machine learning. In the noiseless case, matching upper and lower bounds on sample complexity are established for the exact recovery of sparse vectors and for stable estimation of approximately sparse vectors, respectively. In the noisy case, upper and matching minimax lower bounds for estimation error are obtained. We also consider the debiased sparse group Lasso and investigate its asymptotic property for the purpose of statistical inference. Finally, numerical studies are provided to support the theoretical results.
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