Hopf bifurcation in an age-structured predator-prey system with Beddington-DeAngelis functional response and constant harvesting.

In this paper, an age-structured predator-prey system with Beddington-DeAngelis (B-D) type functional response, prey refuge and harvesting is investigated, where the predator fertility function f(a) and the maturation function ß ( a ) are assumed to be piecewise functions related to their maturation period τ . Firstly, we rewrite the original system as a non-densely defined abstract Cauchy problem and show the existence of solutions. In particular, we discuss the existence and uniqueness of a positive equilibrium of the system. Secondly, we consider the maturation period τ as a bifurcation parameter and show the existence of Hopf bifurcation at the positive equilibrium by applying the integrated semigroup theory and Hopf bifurcation theorem. Moreover, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are studied by applying the center manifold theorem and normal form theory. Finally, some numerical simulations are given to illustrate of the theoretical results and a brief discussion is presented.

Effects of whaling and krill fishing on the whale-krill predation dynamics: bifurcations in a harvested predator-prey model with Holling type I functional response.

In the Antarctic, the whale population had been reduced dramatically due to the unregulated whaling. It was expected that Antarctic krill, the main prey of whales, would grow significantly as a consequence and exploratory krill fishing was practiced in some areas. However, it was found that there has been a substantial decline in abundance of krill since the end of whaling, which is the phenomenon of krill paradox. In this paper, to study the krill-whale interaction we revisit a harvested predator-prey model with Holling I functional response. We find that the model admits at most two positive equilibria. When the two positive equilibria are located in the region { ( N , P ) | 0 ≤ N < 2 N c , P ≥ 0 } , the model exhibits degenerate Bogdanov-Takens bifurcation with codimension up to 3 and Hopf bifurcation with codimension up to 2 by rigorous bifurcation analysis. When the two positive equilibria are located in the region { ( N , P ) | N > 2 N c , P ≥ 0 } , the model has no complex bifurcation phenomenon. When there is one positive equilibrium on each side of N = 2 N c , the model undergoes Hopf bifurcation with codimension up to 2. Moreover, numerical simulation reveals that the model not only can exhibit the krill paradox phenomenon but also has three limit cycles, with the outmost one crosses the line N = 2 N c under some specific parameter conditions.

Imperfect and Bogdanov-Takens bifurcations in biological models: from harvesting of species to isolation of infectives.

A natural biological system under human interventions may exhibit complex dynamical behaviors which could lead to either the collapse or stabilization of the system. The bifurcation theory plays an important role in understanding this evolution process by modeling and analyzing the biological system. In this paper, we examine two types of biological models that Fred Brauer made pioneer contributions: predator-prey models with stocking/harvesting and epidemic models with importation/isolation. First we consider the predator-prey model with Holling type II functional response whose dynamics and bifurcations are well-understood. By considering human interventions such as constant harvesting or stocking of predators, we show that the system under human interventions undergoes imperfect bifurcation and Bogdanov-Takens bifurcation, which induces much richer dynamical behaviors such as the existence of limit cycles or homoclinic loops. Then we consider an epidemic model with constant importation/isolation of infective individuals and observe similar imperfect and Bogdanov-Takens bifurcations when the constant importation/isolation rate varies.

A celebration of Fred Brauer's legacy in mathematical biology.

Fred Brauer (1932-2021), one of the pioneers of mathematical population biology, shaped generations of researchers through his lines of research, his books which have become key references in the field, and his mentoring of junior researchers. This dedication reviews some of his work in population harvesting and epidemiological modeling, highlighting how this special collection reflects the impact of his legacy through both his research accomplishments and the formation of new researchers.

Estimating asymptomatic, undetected and total cases for the COVID-19 outbreak in Wuhan: a mathematical modeling study.

BACKGROUND: The COVID-19 outbreak in Wuhan started in December 2019 and was under control by the end of March 2020 with a total of 50,006 confirmed cases by the implementation of a series of nonpharmaceutical interventions (NPIs) including unprecedented lockdown of the city. This study analyzes the complete outbreak data from Wuhan, assesses the impact of these public health interventions, and estimates the asymptomatic, undetected and total cases for the COVID-19 outbreak in Wuhan. METHODS: By taking different stages of the outbreak into account, we developed a time-dependent compartmental model to describe the dynamics of disease transmission and case detection and reporting. Model coefficients were parameterized by using the reported cases and following key events and escalated control strategies. Then the model was used to calibrate the complete outbreak data by using the Monte Carlo Markov Chain (MCMC) method. Finally we used the model to estimate asymptomatic and undetected cases and approximate the overall antibody prevalence level. RESULTS: We found that the transmission rate between Jan 24 and Feb 1, 2020, was twice as large as that before the lockdown on Jan 23 and 67.6% (95% CI [0.584,0.759]) of detectable infections occurred during this period. Based on the reported estimates that around 20% of infections were asymptomatic and their transmission ability was about 70% of symptomatic ones, we estimated that there were about 14,448 asymptomatic and undetected cases (95% CI [12,364,23,254]), which yields an estimate of a total of 64,454 infected cases (95% CI [62,370,73,260]), and the overall antibody prevalence level in the population of Wuhan was 0.745% (95% CI [0.693%,0.814%]) by March 31, 2020. CONCLUSIONS: We conclude that the control of the COVID-19 outbreak in Wuhan was achieved via the enforcement of a combination of multiple NPIs: the lockdown on Jan 23, the stay-at-home order on Feb 2, the massive isolation of all symptomatic individuals via newly constructed special shelter hospitals on Feb 6, and the large scale screening process on Feb 18. Our results indicate that the population in Wuhan is far away from establishing herd immunity and provide insights for other affected countries and regions in designing control strategies and planing vaccination programs.

Mathematical analysis on an age-structured SIS epidemic model with nonlocal diffusion.

In this paper we propose an age-structured susceptible-infectious-susceptible epidemic model with nonlocal (convolution) diffusion to describe the geographic spread of infectious diseases via long-distance travel. We analyze the well-posedness of the model, investigate the existence and uniqueness of the nontrivial steady state corresponding to an endemic state, and study the local and global stability of this nontrivial steady state. Moreover, we discuss the asymptotic properties of the principal eigenvalue and nontrivial steady state with respect to the nonlocal diffusion rate. The analysis is carried out by using the theory of semigroups and the method of monotone and positive operators. The spectral radius of a positive linear operator associated to the solution flow of the model is identified as a threshold.

Spreading Speed in an Integrodifference Predator-Prey System without Comparison Principle.

In this paper, we study the spreading speed in an integrodifference system which models invasion of predators into the habitat of the prey. Without the requirement of comparison principle, we construct several auxiliary integrodifference equations and use the results of monotone scalar equations to estimate the spreading speed of the invading predators. We also present some numerical simulations to support our theoretical results and demonstrate that the integrodifference predator-prey system exhibits very complex dynamics. Our theory and numerical results imply that the invasion of predators may have a rough constant speed. Moreover, our numerical simulations indicate that the spatial contact of individuals and the overcompensatory phenomenon of the prey may be conducive to the persistence of nonmonotone biological systems and lead to instability of the predator-free state.

The effect of initial values on extinction or persistence in degenerate diffusion competition systems.

When the asymptotic spreading for classical monostable Lotka-Volterra competition diffusion systems is concerned, extinction or persistence of the two competitive species is completely determined by the dynamics of the corresponding kinetic systems, while the size of initial values does not affect the final states. The purpose of this paper is to demonstrate the rich dynamics induced by the initial values in a class of degenerate competition diffusion systems with weak Allee effect. We present various extinction or persistence results by selecting different initial values although the corresponding kinetic system is fixed, which also implies the existence of balance between degenerate nonlinear reaction and diffusion. For example, even if the positive steady state of the corresponding kinetic system is globally asymptotically stable, we observe four different spreading-vanishing phenomena by selecting different initial values. In addition, the interspecific competition of one species may be harmful to the persistence of the other species by taking proper initial values. Our results show that the superior competitor in the sense of the corresponding kinetic system is not always unbeatable, it can be wiped out by the inferior competitor in the sense of the corresponding kinetic system depending on the size of initial habitats as well as the intensity of Allee effect.

Modeling the Transmission Dynamics of Rabies for Dog, Chinese Ferret Badger and Human Interactions in Zhejiang Province, China.

Human rabies is one of the major public health problems in China with an average of 1977 cases per year. It is estimated that 95% of these human rabies cases are due to dog bites. In recent years, the number of wildlife-associated human rabies cases has increased, particularly in the southeast and northeast regions of mainland China. Chinese ferret badgers (CFBs) are one of the most popular wildlife animals which are distributed mostly in the southeast region of China. Human cases caused by rabid CFB were first recorded in Huzhou, Zhejiang Province, in 1994. From 1996 to 2004, more than 30 human cases were caused by CFB bites in Zhejiang Province. In this paper, based on the reported data of the human rabies caused by both dogs and CFB in Zhejiang Province, we propose a multi-host zoonotic model for the dog-CFB-human transmission of rabies. We first evaluate the basic reproduction number [Formula: see text] discuss the stability of the disease-free equilibrium, and study persistence of the disease. Then we use our model to fit the reported data in Zhejiang Province from 2004 to 2017 and forecast the trend of human or livestock rabies. Finally by carrying out sensitivity analysis of the basic reproduction number in terms of parameters, we find that the transmission between dogs and CFB, the quantity of dogs, and the vaccination rate of dogs play important roles in the transmission of rabies. Our study suggests that rabies control and prevention strategies should include enhancing public education and awareness about rabies, increasing dog vaccination rate, reducing the dog and CFB interactions, and avoiding CFB bites or contact.

Modelling diapause in mosquito population growth.

Diapause, a period of arrested development caused by adverse environmental conditions, serves as a key survival mechanism for insects and other invertebrate organisms in temperate and subtropical areas. In this paper, a novel modelling framework, motivated by mosquito species, is proposed to investigate the effects of diapause on seasonal population growth, where the diapause period is taken as an independent growth process, during which the population dynamics are completely different from that in the normal developmental and post-diapause periods. More specifically, the annual growth period is divided into three intervals, and the population dynamics during each interval are described by different sets of equations. We formulate two models of delay differential equations (DDE) to explicitly describe mosquito population growth with a single diapausing stage, either immature or adult. These two models can be further unified into one DDE model, on which the well-posedness of the solutions and the global stability of the trivial and positive periodic solutions in terms of an index [Formula: see text] are analysed. The seasonal population abundances of two temperate mosquito species with different diapausing stages are simulated to identify the essential role on population persistence that diapause plays and predict that killing mosquitoes during the diapause period can lower but fail to prevent the occurrence of peak abundance in the following season. Instead, culling mosquitoes during the normal growth period is much more efficient to decrease the outbreak size. Our modelling framework may shed light on the diapause-induced variations in spatiotemporal distributions of different mosquito species.

Modeling the importation and local transmission of vector-borne diseases in Florida: The case of Zika outbreak in 2016.

Chikungunya, dengue, and Zika viruses are all transmitted by Aedes aegypti and Aedes albopictus mosquito species, had been imported to Florida and caused local outbreaks. We propose a deterministic model to study the importation and local transmission of these mosquito-borne diseases. The purpose is to model and mimic the importation of these viruses to Florida via travelers, local infections in domestic mosquitoes by imported travelers, and finally non-travel related transmissions to local humans by infected local mosquitoes. As a case study, the model will be used to simulate the accumulative Zika cases in Florida. Since the disease system is driven by a continuing input of infections from outside sources, orthodox analytic methods based on the calculation of the basic reproduction number are inadequate to describe and predict their behavior. Via steady-state analysis and sensitivity analysis, effective control and prevention measures for these mosquito-borne diseases are tested.

A conceptual model for optimizing vaccine coverage to reduce vector-borne infections in the presence of antibody-dependent enhancement.

BACKGROUND: Many vector-borne diseases co-circulate, as the viruses from the same family are also transmitted by the same vector species. For example, Zika and dengue viruses belong to the same Flavivirus family and are primarily transmitted by a common mosquito species Aedes aegypti. Zika outbreaks have also commonly occurred in dengue-endemic areas, and co-circulation and co-infection of both viruses have been reported. As recent immunological cross-reactivity studies have confirmed that convalescent plasma following dengue infection can enhance Zika infection, and as global efforts of developing dengue and Zika vaccines are intensified, it is important to examine whether and how vaccination against one disease in a large population may affect infection dynamics of another disease due to antibody-dependent enhancement. METHODS: Through a conceptual co-infection dynamics model parametrized by reported dengue and Zika epidemic and immunological cross-reactivity characteristics, we evaluate impact of a hypothetical dengue vaccination program on Zika infection dynamics in a single season when only one particular dengue serotype is involved. RESULTS: We show that an appropriately designed and optimized dengue vaccination program can not only help control the dengue spread but also, counter-intuitively, reduce Zika infections. We identify optimal dengue vaccination coverages for controlling dengue and simultaneously reducing Zika infections, as well as the critical coverages exceeding which dengue vaccination will increase Zika infections. CONCLUSION: This study based on a conceptual model shows the promise of an integrative vector-borne disease control strategy involving optimal vaccination programs, in regions where different viruses or different serotypes of the same virus co-circulate, and convalescent plasma following infection from one virus (serotype) can enhance infection against another virus (serotype). The conceptual model provides a first step towards well-designed regional and global vector-borne disease immunization programs.

Analysis of a Dengue Model with Vertical Transmission and Application to the 2014 Dengue Outbreak in Guangdong Province, China.

There is evidence showing that vertical transmission of dengue virus exists in Aedes mosquitoes. In this paper, we propose a deterministic dengue model with vertical transmission in mosquitoes by including aquatic mosquitoes (eggs, larvae and pupae), adult mosquitoes (susceptible, exposed and infectious) and human hosts (susceptible, exposed, infectious and recovered). We first analyze the existence and stability of disease-free equilibria, calculate the basic reproduction number and discuss the existence of the disease-endemic equilibrium. Then, we study the impact of vertical transmission of the virus in mosquitoes on the spread dynamics of dengue. We also use the model to simulate the reported infected human data from the 2014 dengue outbreak in Guangdong Province, China, carry out sensitivity analysis of the basic reproduction number in terms of the model parameters, and seek for effective control measures for the transmission of dengue virus.

Traveling wave solutions in a two-group SIR epidemic model with constant recruitment.

Host heterogeneity can be modeled by using multi-group structures in the population. In this paper we investigate the existence and nonexistence of traveling waves of a two-group SIR epidemic model with time delay and constant recruitment and show that the existence of traveling waves is determined by the basic reproduction number [Formula: see text] More specifically, we prove that (i) when the basic reproduction number [Formula: see text] there exists a minimal wave speed [Formula: see text] such that for each [Formula: see text] the system admits a nontrivial traveling wave solution with wave speed c and for [Formula: see text] there exists no nontrivial traveling wave satisfying the system; (ii) when [Formula: see text] the system admits no nontrivial traveling waves. Finally, we present some numerical simulations to show the existence of traveling waves of the system.

Linear and Weakly Nonlinear Stability Analyses of Turing Patterns for Diffusive Predator-Prey Systems in Freshwater Marsh Landscapes.

We study a diffusive predator-prey model describing the interactions of small fishes and their resource base (small invertebrates) in the fluctuating freshwater marsh landscapes of the Florida Everglades. The spatial model is described by a reaction-diffusion system with Beddington-DeAngelis functional response. Uniform bound, local and global asymptotic stability of the steady state of the PDE model under the no-flux boundary conditions are discussed in details. Sufficient conditions on the Turing (diffusion-driven) instability which induces spatial patterns in the model are derived via linear analysis. Existence of one-dimensional and two-dimensional spatial Turing patterns, including rhombic and hexagonal patterns, are established by weakly nonlinear analyses. These results provide theoretical explanations and numerical simulations of spatial dynamical behaviors of the wetland ecosystems of the Florida Everglades.

Global properties of vector-host disease models with time delays.

Since there exist extrinsic and intrinsic incubation periods of pathogens in the feedback interactions between the vectors and hosts, it is necessary to consider the incubation delays in vector-host disease transmission dynamics. In this paper, we propose vector-host disease models with two time delays, one describing the incubation period in the vector population and another representing the incubation period in the host population. Both distributed and discrete delays are used. By constructing suitable Liapunov functions, we obtain sufficient conditions for the global stability of the endemic equilibria of these models. The analytic results reveal that the global dynamics of such vector-host disease models with time delays are completely determined by the basic reproduction number. Some specific cases with discrete delay are studied and the corresponding results are improved.

On the sexual transmission dynamics of hepatitis B virus in China.

In a previous study we noticed that there might be co-infections of HBV and HIV by comparing incidence rates of these two diseases in China. The comparisons between the incidence data of HBV and sexually transmitted diseases (including AIDS, HIV, syphilis and gonorrhea) in China demonstrate that sexual transmission is an important route of spread of HBV in China. On the basis of this fact, in this paper we propose a compartmental model including under-aged children, male adults, and female adults. The effect of sexual transmission on the spread and prevalence of HBV in China is studied. The model is employed to simulate the HBV incidence data reported by the Chinese Center for Disease Control and Prevention for under-aged children, adult males, and adult females, respectively. The sensitivity analysis of the basic reproduction number indicates that it is important and crucial to increase the immunization rate for both under-aged children and adults in order to control the transmission of HBV in China. Our study suggests that effective control measures for hepatitis B in China include enhancing public education and awareness about hepatitis B virus, particularly about the fact that hepatitis B is a sexually transmitted disease, and increasing the immunization rate for both under-aged children and adults, especially for those groups of high risk.

A modeling approach to investigate epizootic outbreaks and enzootic maintenance of Rift Valley fever virus.

We propose a mathematical model to investigate the transmission dynamics of Rift Valley fever (RVF) virus among ruminants. Our findings indicate that in endemic areas RVF virus maintains at a very low level among ruminants after outbreaks and subsequent outbreaks may occur when new susceptible ruminants are recruited into endemic areas or abundant numbers of mosquitoes emerge when herd immunity decreases. Many factors have been shown to have impacts on the severity of RVF outbreaks; a higher probability of death due to RVF among ruminants, a higher mosquito:ruminant ratio, or a shorter lifespan of animals can amplify the magnitude of the outbreaks; vaccination helps to reduce the magnitude of RVF outbreaks and the loss of animals efficiently, and the maximum vaccination effort (a high vaccination rate and a larger number of vaccinated animals) is recommended before the commencement of an outbreak but can be reduced later during the enzootic.

Periodic and chaotic oscillations in a tumor and immune system interaction model with three delays.

In this paper, a tumor and immune system interaction model consisted of two differential equations with three time delays is considered in which the delays describe the proliferation of tumor cells, the process of effector cells growth stimulated by tumor cells, and the differentiation of immune effector cells, respectively. Conditions for the asymptotic stability of equilibria and existence of Hopf bifurcations are obtained by analyzing the roots of a second degree exponential polynomial characteristic equation with delay dependent coefficients. It is shown that the positive equilibrium is asymptotically stable if all three delays are less than their corresponding critical values and Hopf bifurcations occur if any one of these delays passes through its critical value. Numerical simulations are carried out to illustrate the rich dynamical behavior of the model with different delay values including the existence of regular and irregular long periodic oscillations.

A PERIODIC ROSS-MACDONALD MODEL IN A PATCHY ENVIRONMENT.

Based on the classical Ross-Macdonald model, in this paper we propose a periodic malaria model to incorporate the effects of temporal and spatial heterogeneity on disease transmission. The temporal heterogeneity is described by assuming that some model coefficients are time-periodic, while the spatial heterogeneity is modeled by using a multi-patch structure and assuming that individuals travel among patches. We calculate the basic reproduction number [Formula: see text] and show that either the disease-free periodic solution is globally asymptotically stable if [Formula: see text] or the positive periodic solution is globally asymptotically stable if [Formula: see text]. Numerical simulations are conducted to confirm the analytical results and explore the effect of travel control on the disease prevalence.