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.

Gompertz models with periodical treatment and applications to prostate cancer.

In this paper, Gompertz type models are proposed to understand the temporal tumor volume behavior of prostate cancer when a periodical treatment is provided. Existence, uniqueness, and stability of periodic solutions are established. The models are used to fit the data and to forecast the tumor growth behavior based on prostate cancer treatments using capsaicin and docetaxel anticancer drugs. Numerical simulations show that the combination of capsaicin and docetaxel is the most efficient treatment of prostate cancer.

Impact of climate change on vegetation patterns in Altay Prefecture, China.

Altay Prefecture, a typical arid region in northwestern China, has experienced the climate transition from warming-drying to warming-wetting since 1980s and has attracted widespread attention. Nonetheless, it is still unclear how climate change has influenced the distribution of vegetation in this region. In this paper, a reaction-diffusion model of the climate-vegetation system is proposed to study the impact of climate change (precipitation, temperature and carbon dioxide concentration) on vegetation patterns in Altay Prefecture. Our results indicate that the tendency of vegetation growth in Altay Prefecture improved gradually from 1985 to 2010. Under the current climate conditions, the increase of precipitation results in the change of vegetation pattern structures, and eventually vegetation coverage tends to be uniform. Moreover, we found that there exists an optimal temperature where the spot vegetation pattern structure remains stable. Furthermore, the increase in carbon dioxide concentration induces vegetation pattern transition. Based on four climate change scenarios of the Coupled Model Intercomparison Project Phase 6 (CMIP6), we used the power law range (PLR) to predict the optimal scenario for the sustainable development of the vegetation ecosystem in Altay Prefecture.

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.

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.

Linking mathematical models and trap data to infer the proliferation, abundance, and control of Aedes aegypti.

Aedes aegypti is one of the most dominant mosquito species in the urban areas of Miami-Dade County, Florida, and is responsible for the local arbovirus transmissions. Since August 2016, mosquito traps have been placed throughout the county to improve surveillance and guide mosquito control and arbovirus outbreak response. In this paper, we develop a deterministic mosquito population model, estimate model parameters by using local entomological and temperature data, and use the model to calibrate the mosquito trap data from 2017 to 2019. We further use the model to compare the Ae. aegypti population and evaluate the impact of rainfall intensity in different urban built environments. Our results show that rainfall affects the breeding sites and the abundance of Ae. aegypti more significantly in tourist areas than in residential places. In addition, we apply the model to quantitatively assess the effectiveness of vector control strategies in Miami-Dade County.

Age-Structured Population Dynamics with Nonlocal Diffusion.

Random diffusive age-structured population models have been studied by many researchers. Though nonlocal diffusion processes are more applicable to many biological and physical problems compared with random diffusion processes, there are very few theoretical results on age-structured population models with nonlocal diffusion. In this paper our objective is to develop basic theory for age-structured population dynamics with nonlocal diffusion. In particular, we study the semigroup of linear operators associated to an age-structured model with nonlocal diffusion and use the spectral properties of its infinitesimal generator to determine the stability of the zero steady state. It is shown that (i) the structure of the semigroup for the age-structured model with nonlocal diffusion is essentially determined by that of the semigroups for the age-structured model without diffusion and the nonlocal operator when both birth and death rates are independent of spatial variables; (ii) the asymptotic behavior can be determined by the sign of spectral bound of the infinitesimal generator when both birth and death rates are dependent on spatial variables; (iii) the weak solution and comparison principle can be established when both birth and death rates are dependent on spatial variables and time; and (iv) the above results can be generalized to an age-size structured model. In addition, we compare our results with the age-structured model with Laplacian diffusion in the first two cases (i) and (ii).

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.

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.

Modelling homosexual and heterosexual transmissions of hepatitis B virus in China.

Studies have shown that sexual transmission, both heterosexually and homosexually, is one of the main ways of HBV infection. Based on this fact, we propose a mathematical model to study the sexual transmission of HBV among adults by classifying adults into men and women and considering both same-sex and opposite-sex transmissions of HBV in adults. Firstly, we calculate the basic reproduction number R0 and the disease-free equilibrium point E0. Secondly, by analysing the sensitivity of R0 in terms of model parameters, we find that the infection rate among people who have same-sex partners, the frequency of homosexual contact and the immunity rate of adults play important roles in the transmission of HBV. Moreover, we use our model to fit the reported data in China and forecast the trend of hepatitis B. Our results demonstrate that popularizing the basic knowledge of HBV among residents, advocating healthy and reasonable sexual life style, reducing the number of adult carriers, and increasing the immunization rate of adults are effective measures to prevent and control hepatitis B.

Global Dynamics of a Susceptible-Infectious-Recovered Epidemic Model with a Generalized Nonmonotone Incidence Rate.

A susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone incidence rate kIS 1 + ß I + α I 2 ( ß > - 2 α such that 1 + ß I + α I 2 > 0 for all I ≥ 0 ) is considered in this paper. It is shown that the basic reproduction number R 0 does not act as a threshold value for the disease spread anymore, and there exists a sub-threshold value R ∗ ( < 1 ) such that: (i) if R 0 < R ∗ , then the disease-free equilibrium is globally asymptotically stable; (ii) if R 0 = R ∗ , then there is a unique endemic equilibrium which is a nilpotent cusp of codimension at most three; (iii) if R ∗ < R 0 < 1 , then there are two endemic equilibria, one is a weak focus of multiplicity at least three, the other is a saddle; (iv) if R 0 ≥ 1 , then there is again a unique endemic equilibrium which is a weak focus of multiplicity at least three. As parameters vary, the model undergoes saddle-node bifurcation, backward bifurcation, Bogdanov-Takens bifurcation of codimension three, Hopf bifurcation, and degenerate Hopf bifurcation of codimension three. Moreover, it is shown that there exists a critical value α 0 for the psychological effect α , a critical value k 0 for the infection rate k, and two critical values ß 0 , ß 1 ( ß 1 < ß 0 ) for ß that will determine whether the disease dies out or persists in the form of positive periodic coexistent oscillations or coexistent steady states under different initial populations. Numerical simulations are given to demonstrate the existence of one, two or three limit cycles.

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.

Optimal control of environmental cleaning and antibiotic prescription in an epidemiological model of methicillin-resistant Staphylococcus aureus infections in hospitals.

We consider a deterministic model of Methicillin-resistant Staphylococcus aureus infections in hospitals with seasonal oscillations of the antibiotic prescription rate. The model compartments consist of uncolonized patients with or without antibiotic exposure, colonized patients with or without antibiotic exposure, uncontaminated or contaminated healthcare workers, and free-living bacteria in the environment. We apply optimal control theory to this seven-compartment periodic system of ordinary differential equations to reduce the number of colonized patients and density of bacteria in the environment while minimizing the cost associated with environmental cleaning and antibiotic use in a particular time period. Characterizations of optimal control strategies are formulated and the ways hospitals should adjust these strategies for different scenarios are discussed. Numerical simulations strongly suggest that environmental cleaning is essential in the control of MRSA infections and antibiotic usage is suggested to be maintained at the least possible level. Screening, isolating, and shortening the extremely lengthened stays of colonized patients with antibiotic use history are all effective intervention strategies.

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.

Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate.

In this paper, we study a susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone and saturated incidence rate k I 2 S 1 + ß I + α I 2 , in which the infection function first increases to a maximum when a new infectious disease emerges, then decreases due to psychological effect, and eventually tends to a saturation level due to crowding effect. It is shown that there are a weak focus of multiplicity at most two and a cusp of codimension at most two for various parameter values, and the model undergoes saddle-node bifurcation, Bogdanov-Takens bifurcation of codimension two, Hopf bifurcation, and degenerate Hopf bifurcation of codimension two as the parameters vary. It is shown that there exists a critical value α = α 0 for the psychological effect, and two critical values k = k 0 , k 1 ( k 0 < k 1 ) for the infection rate such that: (i) when α > α 0 , or α ≤ α 0 and k ≤ k 0 , the disease will die out for all positive initial populations; (ii) when α = α 0 and k 0 < k ≤ k 1 , the disease will die out for almost all positive initial populations; (iii) when α = α 0 and k > k 1 , the disease will persist in the form of a positive coexistent steady state for some positive initial populations; and (iv) when α < α 0 and k > k 0 , the disease will persist in the form of multiple positive periodic coexistent oscillations and coexistent steady states for some positive initial populations. Numerical simulations, including the existence of one or two limit cycles and data-fitting of the influenza data in Mainland China, are presented to illustrate the theoretical results.

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.

Modeling the seasonality of Methicillin-resistant Staphylococcus aureus infections in hospitals with environmental contamination.

A deterministic mathematical model with periodic antibiotic prescribing rate is constructed to study the seasonality of Methicillin-resistant Staphylococcus aureus (MRSA) infections taking antibiotic exposure and environmental contamination into consideration. The basic reproduction number R0 for the periodic model is calculated under the assumption that there are only uncolonized patients with antibiotic exposure at admission. Sensitivity analysis of R0 with respect to some essential parameters is performed. It is shown that the infection would go to extinction if the basic reproduction number is less than unity and would persist if it is greater than unity. Numerical simulations indicate that environmental cleaning is the most important intervention to control the infection, which emphasizes the effect of environmental contamination in MRSA infections. It is also important to highlight the importance of effective antimicrobial stewardship programmes, increase active screening at admission and subsequent isolation of positive cases, and treat patients quickly and efficiently.