Showing papers in "Journal of Biological Dynamics in 2017"
TL;DR: An SIR-type model is developed which incorporates both the direct and indirect transmissions in such a manner that there is a provision of Ebola viruses, and it is proved that the full model has one (endemic) equilibrium which is locally asymptotically stable whereas, it is globally asymptonically stable in the absence of the Ebola virus shedding in the environment.
Abstract: We deal with the following question: Can the consumption of contaminated bush meat, the funeral practices and the environmental contamination explain the recurrence and persistence of Ebola virus disease outbreaks in Africa? We develop an SIR-type model which, incorporates both the direct and indirect transmissions in such a manner that there is a provision of Ebola viruses. We prove that the full model has one (endemic) equilibrium which is locally asymptotically stable whereas, it is globally asymptotically stable in the absence of the Ebola virus shedding in the environment. For the sub-model without the provision of Ebola viruses, the disease dies out or stabilizes globally at an endemic equilibrium. At the endemic level, the number of infectious is larger for the full model than for the sub-model without provision of Ebola viruses. We design a nonstandard finite difference scheme, which preserves the dynamics of the model. Numerical simulations are provided.
101 citations
TL;DR: This article presents the transmission dynamic of the acute and chronic hepatitis B epidemic problem and develops an optimal control strategy to control the spread of hepatitis B in a community by applying three control variables such as isolation of infected and non-infected individuals.
Abstract: In this article, we present the transmission dynamic of the acute and chronic hepatitis B epidemic problem and develop an optimal control strategy to control the spread of hepatitis B in a community. In order to do this, first we present the model formulation and find the basic reproduction number R0. We show that if R0≤1, then the disease-free equilibrium is both locally as well as globally asymptotically stable. Then, we prove that the model is locally and globally asymptotically stable, if R0>1. To control the spread of this infection, we develop a control strategy by applying three control variables such as isolation of infected and non-infected individuals, treatment and vaccination to minimize the number of acute infected, chronically infected with hepatitis B individuals and maximize the number of susceptible and recovered individuals. Finally, we present numerical simulation to illustrate the feasibility of the control strategy.
76 citations
TL;DR: An HIV latent infection model including both modes of transmission and time delays between viral entry and integration or viral production is developed, showing the existence, positivity and boundedness of the solution, and proving the local and global stability of the infection-free and infected steady states.
Abstract: HIV can infect cells via virus-to-cell infection or cell-to-cell viral transmission. These two infection modes may occur in a synergistic way and facilitate viral spread within an infected individual. In this paper, we developed an HIV latent infection model including both modes of transmission and time delays between viral entry and integration or viral production. We analysed the model by defining the basic reproductive number, showing the existence, positivity and boundedness of the solution, and proving the local and global stability of the infection-free and infected steady states. Numerical simulations have been performed to illustrate the theoretical results and evaluate the effects of time delays and fractions of infection leading to latency on the virus dynamics. The estimates of the relative contributions to the HIV latent reservoir and the virus population from the two modes of transmission have also been provided.
66 citations
TL;DR: It is shown that a smaller release rate and more frequent releases are more efficient in controlling the wild mosquito population for the periodic releases, but an early release of sterile mosquitoes is more effective for the state feedback releases.
Abstract: To study the impact of releasing sterile mosquitoes on mosquito-borne disease transmissions, we propose two mathematical models with impulsive releases of sterile mosquitoes. We consider periodic impulsive releases in the first model and obtain the existence, uniqueness, and globally stability of a wild-mosquito-eradication periodic solution. We also establish thresholds for the control of the wild mosquito population by selecting the release rate and the release period. In the second model, the impulsive releases are determined by the closely monitored wild mosquito density, or the state feedback. We prove the existence of an order one periodic solution and find a relatively small attraction region, which ensures the wild mosquito population is under control. We provide numerical analysis which shows that a smaller release rate and more frequent releases are more efficient in controlling the wild mosquito population for the periodic releases, but an early release of sterile mosquitoes is more effective for the state feedback releases.
43 citations
TL;DR: In this paper, a nonlinear mathematical model for the transmission dynamics of pneumonia disease in a population of varying size is proposed and analyzed using stability theory of differential equations and the effective reproduction number is obtained and also the asymptotic stability conditions for the disease free and as well as for the endemic equilibria are established.
Abstract: We propose and analyse a nonlinear mathematical model for the transmission dynamics of pneumonia disease in a population of varying size. The deterministic compartmental model is studied using stability theory of differential equations. The effective reproduction number is obtained and also the asymptotic stability conditions for the disease free and as well as for the endemic equilibria are established. The possibility of bifurcation of the model and the sensitivity indices of the basic reproduction number to the key parameters are determined. Using Pontryagin's maximum principle, the optimal control problem is formulated with three control strategies: namely disease prevention through education, treatment and screening. The cost-effectiveness analysis of the adopted control strategies revealed that the combination of prevention and treatment is the most cost-effective intervention strategies to combat the pneumonia pandemic. Numerical simulation is performed and pertinent results are displayed g...
42 citations
TL;DR: An ordinary differential equation model is developed and analysed to investigate the transmission dynamics of releasing Wolbachia-infected mosquitoes to establish an endemic infection in a population of wild uninfected mosquitoes and it is observed that the most effective approach to establish the infection in the wild is based on reducing mosquitoes in both the adult and aquatic stages.
Abstract: We develop and analyse an ordinary differential equation model to investigate the transmission dynamics of releasing Wolbachia-infected mosquitoes to establish an endemic infection in a population of wild uninfected mosquitoes. Wolbachia is a genus of endosymbiotic bacteria that can infect mosquitoes and reduce their ability to transmit some viral mosquito-transmitted diseases, including dengue fever, chikungunya, and Zika. Although the bacterium is transmitted vertically from infected mothers to their offspring, it can be difficult to establish an endemic infection in a wild mosquito population. Our transmission model for the adult and aquatic-stage mosquitoes takes into account Wolbachia-induced fitness change and cytoplasmic incompatibility. We show that, for a wide range of realistic parameter values, the basic reproduction number, [Formula: see text], is less than one. Hence, the epidemic will die out if only a few Wolbachia-infected mosquitoes are introduced into the wild population. Even though the basic reproduction number is less than one, an endemic Wolbachia infection can be established if a sufficient number of infected mosquitoes are released. This threshold effect is created by a backward bifurcation with three coexisting equilibria: a stable zero-infection equilibrium, an intermediate-infection unstable endemic equilibrium, and a high-infection stable endemic equilibrium. We analyse the impact of reducing the wild mosquito population before introducing the infected mosquitoes and observed that the most effective approach to establish the infection in the wild is based on reducing mosquitoes in both the adult and aquatic stages.
41 citations
TL;DR: Model analysis of the temporal model suggests that virus burst size and virus infection rate determine the success of the virotherapy treatment, whereas travelling wave solutions to the spatiotemporal model show that tumour diffusivity and growth rate are critical during chemovirother therapy.
Abstract: Chemovirotherapy is a combination therapy with chemotherapy and oncolytic viruses It is gaining more interest and attracting more attention in the clinical setting due to its effective therapy and potential synergistic interactions against cancer In this paper, we develop and analyse a mathematical model in the form of parabolic non-linear partial differential equations to investigate the spatiotemporal dynamics of tumour cells under chemovirotherapy treatment The proposed model consists of uninfected and infected tumour cells, a free virus, and a chemotherapeutic drug The analysis of the model is carried out for both the temporal and spatiotemporal cases Travelling wave solutions to the spatiotemporal model are used to determine the minimum wave speed of tumour invasion A sensitivity analysis is performed on the model parameters to establish the key parameters that promote cancer remission during chemovirotherapy treatment Model analysis of the temporal model suggests that virus burst size
40 citations
TL;DR: This categorization framework is shown to be applicable in categorizing other types of multiscale models of infectious diseases beyond HL-IEMs through modifying the initialategorization framework presented in this study.
Abstract: Modelling of infectious disease systems has entered a new era in which disease modellers are increasingly turning to multiscale modelling to extend traditional modelling frameworks into new application areas and to achieve higher levels of detail and accuracy in characterizing infectious disease systems. In this paper we present a categorization framework for categorizing multiscale models of infectious disease systems. The categorization framework consists of five integration frameworks and five criteria. We use the categorization framework to give a complete categorization of host-level immuno-epidemiological models (HL-IEMs). This categorization framework is also shown to be applicable in categorizing other types of multiscale models of infectious diseases beyond HL-IEMs through modifying the initial categorization framework presented in this study. Categorization of multiscale models of infectious disease systems in this way is useful in bringing some order to the discussion on the structure of these multiscale models.
40 citations
TL;DR: This work formulate stage-structured continuous-time mathematical models, based on systems of differential equations, for the interactive dynamics of the wild and sterile mosquitoes, and investigates the model dynamics, including the existence of positive equilibria and their stability.
Abstract: To study the impact of the sterile insect technique and effects of the mosquitoes' metamorphic stage structure on the transmission dynamics of mosquito-borne diseases, we formulate stage-structured continuous-time mathematical models, based on systems of differential equations, for the interactive dynamics of the wild and sterile mosquitoes. We incorporate different strategies for the releases of sterile mosquitoes in the models and investigate the model dynamics, including the existence of positive equilibria and their stability. Numerical examples are provided to demonstrate the dynamical features of the models.
40 citations
TL;DR: Revised, new, simple models for interactive wild and sterile mosquitoes which are better approximations to real biological situations but mathematically more tractable are formulated.
Abstract: Based on previous research, we formulate revised, new, simple models for interactive wild and sterile mosquitoes which are better approximations to real biological situations but mathematically more tractable. We give basic investigations of the dynamical features of these simple models such as the existence of equilibria and their stability. Numerical examples to demonstrate our findings and brief discussions are also provided.
39 citations
TL;DR: A deterministic compartmental eco- epidemiological model with optimal control of Newcastle disease in Tanzania is proposed and analysed and it is observed that combination of chicken vaccination and human education campaign strategy is the best strategy to implement in limited resources.
Abstract: In this paper, a deterministic compartmental eco- epidemiological model with optimal control of Newcastle disease (ND) in Tanzania is proposed and analysed. Necessary conditions of optimal control problem were rigorously analysed using Pontryagin's maximum principle and the numerical values of model parameters were estimated using maximum likelihood estimator. Three control strategies were incorporated such as chicken vaccination (preventive), human education campaign and treatment of infected human (curative) and its' impact were graphically observed. The incremental cost effectiveness analysis technique used to determine the most cost effectiveness strategy and we observe that combination of chicken vaccination and human education campaign strategy is the best strategy to implement in limited resources. Therefore, ND can be controlled if the farmers will apply chicken vaccination properly and well in time.
TL;DR: The basic reproductive number was defined and the local and global stability of the steady states were shown and the stability crossing curves for the model were shown to be a series of open-ended curves.
Abstract: Time delays can affect the dynamics of HIV infection predicted by mathematical models. In this paper, we studied two mathematical models each with two time delays. In the first model with HIV latency, one delay is the time between viral entry into a cell and the establishment of HIV latency, and the other delay is the time between cell infection and viral production. We defined the basic reproductive number and showed the local and global stability of the steady states. Numerical simulations were performed to evaluate the influence of time delays on the dynamics. In the second model with HIV immune response, one delay is the time between cell infection and viral production, and the other delay is the time needed for the adaptive immune response to emerge to control viral replication. With two positive delays, we obtained the stability crossing curves for the model, which were shown to be a series of open-ended curves.
TL;DR: A new modelling framework is proposed to study the within-host and between-host dynamics of cholera, a severe intestinal infection caused by the bacterium Vibrio cholerae, and the result regarding the backward bifurcation highlights the challenges in the prevention and control of Cholera.
Abstract: A new modelling framework is proposed to study the within-host and between-host dynamics of cholera, a severe intestinal infection caused by the bacterium Vibrio cholerae. The within-host dynamics are characterized by the growth of highly infectious vibrios inside the human body. These vibrios shed from humans contribute to the environmental bacterial growth and the transmission of the disease among humans, providing a link from the within-host dynamics at the individual level to the between-host dynamics at the population and environmental level. A fast-slow analysis is conducted based on the two different time scales in our model. In particular, a bifurcation study is performed, and sufficient and necessary conditions are derived that lead to a backward bifurcation in cholera epidemics. Our result regarding the backward bifurcation highlights the challenges in the prevention and control of cholera.
TL;DR: An epidemic model in which all disease transmission is through shedding of virus by infectives and acquisition by susceptibles, rather than by direct contact leads to an susceptible-infectious-virus-removed (SIVR) model for which the basic reproduction number and the final size relation are determined.
Abstract: We consider an epidemic model in which all disease transmission is through shedding of virus by infectives and acquisition by susceptibles, rather than by direct contact. This leads to an susceptible-infectious-virus-removed (SIVR) model for which we can determine the basic reproduction number and the final size relation. We extend the model to an age of infection model with virus shedding a function of the age of infection.
TL;DR: A more realistic binge drinking model with time delay is introduced and it is obtained that the alcohol-free equilibrium is globally asymptotically stable under certain conditions.
Abstract: A more realistic binge drinking model with time delay is introduced. Time delay is used to represent the time lag of the immunity against drinking in our model. For the model without the time delay, using Routh–Hurwitz criterion, we obtain that the alcohol-free equilibrium is locally asymptotically stable if R0 1. For the model with time delay, the local stability of all the equilibria is derived. Furthermore, regardless of the time delay length, using comparison theorem and iteration technique, we obtain that the alcohol-free equilibrium is globally asymptotically stable under certain conditions. Numerical simulations are also conducted to illustrate and extend our analytic results.
TL;DR: A discrete time, structured population dynamic model that is motivated by recent field observations concerning certain life history strategies of colonial-nesting gulls, specifically the glaucous-winged gull, is studied.
Abstract: We study a discrete time, structured population dynamic model that is motivated by recent field observations concerning certain life history strategies of colonial-nesting gulls, specifically the glaucous-winged gull (Larus glaucescens). The model focuses on mechanisms hypothesized to play key roles in a population's response to degraded environment resources, namely, increased cannibalism and adjustments in reproductive timing. We explore the dynamic consequences of these mechanics using a juvenile-adult structure model. Mathematically, the model is unusual in that it involves a high co-dimension bifurcation at [Formula: see text] which, in turn, leads to a dynamic dichotomy between equilibrium states and synchronized oscillatory states. We give diagnostic criteria that determine which dynamic is stable. We also explore strong Allee effects caused by positive feedback mechanisms in the model and the possible consequence that a cannibalistic population can survive when a non-cannibalistic population cannot.
TL;DR: The results show that media coverage is an effective measure to quit drinking.
Abstract: A new social epidemic model to depict alcoholism with media coverage is proposed in this paper. Some fundamental properties of the model including existence and positivity as well as boundedness of equilibria are investigated. Stability of all equilibria are studied. The existence of the optimal control pair and mathematical expressions of optimal control are obtained by Pontryagin's maximum principle. Numerical simulations are also performed to illustrate our results. Our results show that media coverage is an effective measure to quit drinking.
TL;DR: This paper study the nonlinear age-structured model of a polycyclic two-phase population dynamics including delayed effect of population density growth on the mortality, modelled as a system of initial boundary values problem for semi-linear transport equation with delay and initial problem for nonlinear delay ODE.
Abstract: In this paper we study the nonlinear age-structured model of a polycyclic two-phase population dynamics including delayed effect of population density growth on the mortality. Both phases are modelled as a system of initial boundary values problem for semi-linear transport equation with delay and initial problem for nonlinear delay ODE. The obtained system is studied both theoretically and numerically. Three different regimes of population dynamics for asymptotically stable states of autonomous systems are obtained in numerical experiments for the different initial values of population density. The quasi-periodical travelling wave solutions are studied numerically for the autonomous system with the different values of time delays and for the system with oscillating death rate and birth modulus. In both cases it is observed three types of travelling wave solutions: harmonic oscillations, pulse sequence and single pulse.
TL;DR: It is proved that all solutions of this model are bounded, and there exist some values of the parameters such that the model has a global attractor.
Abstract: In this paper, we study the global dynamics and bifurcations of a two-dimensional discrete time host–parasitoid model with strong Allee effect. The existence of fixed points and their stability are analysed in all allowed parametric region. The bifurcation analysis shows that the model can undergo fold bifurcation and Neimark–Sacker bifurcation. As the parameters vary in a small neighbourhood of the Neimark–Sacker bifurcation condition, the unique positive fixed point changes its stability and an invariant closed circle bifurcates from the positive fixed point. From the viewpoint of biology, the invariant closed curve corresponds to the periodic or quasi-periodic oscillations between host and parasitoid populations. Furthermore, it is proved that all solutions of this model are bounded, and there exist some values of the parameters such that the model has a global attractor. These theoretical results reveal the complex dynamics of the present model.
TL;DR: The results reveal that an appropriate immune activation time delay plays a significant role in control of tumour growth and the dual role of this delay can induce stability switches exhibiting destabilization as well as stabilization of the tumour-presence equilibrium.
Abstract: In this paper, a previous tumour–immune interaction model is simplified by neglecting a relatively weak direct immune activation by the tumour cells, which can still keep the essential dynamics properties of the original model. As the immune activation process is not instantaneous, we now incorporate one delay for the activation of the effector cells (ECs) by helper T cells (HTCs) into the model. Furthermore, we investigate the stability and instability regions of the tumour-presence equilibrium state of the delay-induced system with respect to two parameters, the activation rate of ECs by HTCs and the HTCs stimulation rate by the presence of identified tumour antigens. We show the dual role of this delay that can induce stability switches exhibiting destabilization as well as stabilization of the tumour-presence equilibrium. Besides, our results reveal that an appropriate immune activation time delay plays a significant role in control of tumour growth.
TL;DR: An susceptible-infective-removed epidemic model incorporating media coverage with time delay with results that show that the time delay in media coverage can not affect the stability of the disease-free equilibrium when the basic reproduction number is less than unity.
Abstract: An susceptible-infective-removed epidemic model incorporating media coverage with time delay is proposed. The stability of the disease-free equilibrium and endemic equilibrium is studied. And then, the conditions which guarantee the existence of local Hopf bifurcation are given. Furthermore, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. The obtained results show that the time delay in media coverage can not affect the stability of the disease-free equilibrium when the basic reproduction number is less than unity. However, the time delay affects the stability of the endemic equilibrium and produces limit cycle oscillations while the basic reproduction number is greater than unity. Finally, some examples for numerical simulations are included to support the theoretical prediction.
TL;DR: An optimal control problem with malaria treatment and insecticide-treated nets as control functions is formulated and numerical simulations suggest the possibility of eliminating malaria and reducing HIV prevalence significantly, within a short time horizon.
Abstract: We propose and study a mathematical model for malaria-HIV co-infection transmission and control, in which malaria treatment and insecticide-treated nets are incorporated. The existence of a backward bifurcation is established analytically, and the occurrence of such backward bifurcation is influenced by disease-induced mortality, insecticide-treated bed-net coverage and malaria treatment parameters. To further assess the impact of malaria treatment and insecticide-treated bed-net coverage, we formulate an optimal control problem with malaria treatment and insecticide-treated nets as control functions. Using reasonable parameter values, numerical simulations of the optimal control suggest the possibility of eliminating malaria and reducing HIV prevalence significantly, within a short time horizon.
TL;DR: The results indicate that the combined effect of two stressors, both of which can be tolerated by the colony individually, might lead to colony failure, suggesting multi-factorial causes behind losses of honey bee colonies.
Abstract: We present a mathematical model (a) for the infection of a honey bee colony with Nosema ceranae. This is a system of five ordinary differential equations for the dependent variables healthy and infected worker bees in the hive, healthy and infected forager bees, and disease potential deposited in the hive. The model is then (b) extended to account for increased forager losses, e.g. caused by exposure to external stressors. The model is non-autonomous with periodic coefficient functions. Algebraic complexity prevents a rigorous mathematical analysis. Therefore, we resort to computer simulations in addition to some analytical results in the constant coefficient case. We investigate each of the two stressors (a) and (b) individually and jointly. Our results indicate that the combined effect of two stressors, both of which can be tolerated by the colony individually, might lead to colony failure, suggesting multi-factorial causes behind losses of honey bee colonies.
TL;DR: This model analysis shows that early media alert and strong media effects are preferable to decrease the numbers of infected cases at endemic equilibria and global stability is obtained when the model admits a unique endemic equilibrium.
Abstract: In general, media coverage would not be implemented unless the number of infected cases reaches some critical number. To reflect this feature, we incorporate the media effect and a critical number of infected cases into the disease transmission rate and consider an susceptible-infected-susceptible epidemic model with logistic growth. Our model analysis shows that early media alert and strong media effects are preferable to decrease the numbers of infected cases at endemic equilibria. Furthermore, we noticed that the model may have up to three endemic equilibria and bi-stability can occur in a threshold interval for the critical number. Note that the interval depends on parameters for the focal disease and the media effect. It is possible to roughly estimate the interval for re-emerging diseases in a given region. Therefore, the result could be useful to health policymakers. Global stability is also obtained when the model admits a unique endemic equilibrium.
TL;DR: Novel deterministic and stochastic models are proposed in this paper for the within-host dynamics of cholera, with a focus on the bacterial–viral interaction, and the result indicates that there is a sharp disease threshold characterized by the basic reproduction number.
Abstract: Novel deterministic and stochastic models are proposed in this paper for the within-host dynamics of cholera, with a focus on the bacterial–viral interaction. The deterministic model is a system of differential equations describing the interaction among the two types of vibrios and the viruses. The stochastic model is a system of Markov jump processes that is derived based on the dynamics of the deterministic model. The multitype branching process approximation is applied to estimate the extinction probability of bacteria and viruses within a human host during the early stage of the bacterial–viral infection. Accordingly, a closed-form expression is derived for the disease extinction probability, and analytic estimates are validated with numerical simulations. The local and global dynamics of the bacterial–viral interaction are analysed using the deterministic model, and the result indicates that there is a sharp disease threshold characterized by the basic reproduction number R0: if R0<1, vibrios...
TL;DR: A model of HIV infection including age-structured latently infected cells is developed and mathematically analyse the model and numerical simulations with different activation functions show that the model can explain the persistence of low-level viremia and the latent reservoir stability in patients on therapy.
Abstract: HIV latency remains a major obstacle to viral elimination. The activation rate of latently infected cells may depend on the age of latent infection. In this paper, we develop a model of HIV infection including age-structured latently infected cells. We mathematically analyse the model and use numerical simulations with different activation functions to show that the model can explain the persistence of low-level viremia and the latent reservoir stability in patients on therapy. Sensitivity tests suggest that the model is robust to the changes of most parameters but is sensitive to the relative magnitude of the net generation rate and the long-term activation rate of latently infected cells. To reduce the sensitivity, we extend the model to include homeostatic proliferation of latently infected cells. The new model is robust in reproducing the long-term dynamics of the virus and latently infected cells observed in patients receiving prolonged combination therapy.
TL;DR: The results show that alcohol immigrants increase the difficulty of the temperance work of the region.
Abstract: A drinking model with immigration is constructed. For the model with problem drinking immigration, the model admits only one problem drinking equilibrium. For the model without problem drinking immigration, the model has two equilibria, one is problem drinking-free equilibrium and the other is problem drinking equilibrium. By employing the method of Lyapunov function, stability of all kinds of equilibria is obtained. Numerical simulations are also provided to illustrate our analytical results. Our results show that alcohol immigrants increase the difficulty of the temperance work of the region.
TL;DR: Optimal control methods are applied to a deterministic mathematical model to characterize the factors contributing to the replacement of hospital-acquired methicillin-resistant Staphylococcus aureus (HA-MRSA) with community- Acquired methiillin- resistant Staphy coli (CA- MRSA), and quantify the effectiveness of three interventions aimed at limiting the spread of CA-MRsa in healthcare settings.
Abstract: Optimal control methods are applied to a deterministic mathematical model to characterize the factors contributing to the replacement of hospital-acquired methicillin-resistant Staphylococcus aureus (HA-MRSA) with community-acquired methicillin-resistant Staphylococcus aureus (CA-MRSA), and quantify the effectiveness of three interventions aimed at limiting the spread of CA-MRSA in healthcare settings. Characterizations of the optimal control strategies are established, and numerical simulations are provided to illustrate the results.
TL;DR: A new artificial nucleic acid biochemical reaction network is described, and its capacity to exhibit oscillatory solutions is demonstrated, and it is suggested that oscillations occur in a realistic range of reaction rates and concentrations.
Abstract: Oscillators are essential to fuel autonomous behaviours in molecular systems. Artificial oscillators built with programmable biological molecules such as DNA and RNA are generally easy to build and tune, and can serve as timers for biological computation and regulation. We describe a new artificial nucleic acid biochemical reaction network, and we demonstrate its capacity to exhibit oscillatory solutions. This network can be built in vitro using nucleic acids and three bacteriophage enzymes, and has the potential to be implemented in cells. Numerical simulations suggest that oscillations occur in a realistic range of reaction rates and concentrations.
TL;DR: This work analyses the optimal hunting problem paying special attention to the nature of the optimal state and control trajectories in long time intervals, and implements a variant of the single shooting method to solve the previous optimisation problem, taking the middle of the time interval as starting point.
Abstract: The Lotka-Volterra model is a differential system of two coupled equations representing the interaction of two species: a prey one and a predator one. We formulate an optimal control problem adding the effect of hunting both species as the control variable. We analyse the optimal hunting problem paying special attention to the nature of the optimal state and control trajectories in long time intervals. To do that, we apply recent theoretical results on the frame to show that, when the time horizon is large enough, optimal strategies are nearly steady-state. Such path is known as turnpike property. Some experiments are performed to observe such turnpike phenomenon in the hunting problem. Based on the turnpike property, we implement a variant of the single shooting method to solve the previous optimisation problem, taking the middle of the time interval as starting point.