Continuous Population Models

Brauer, Fred; Castillo‐Chavez, Carlos

doi:10.1007/978-1-4614-1686-9_1

Cited by 10 publications

(9 citation statements)

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“…Without a predator, the prey species grows exponentially, while the predator has a constant per capita death rate. Besides that, the presence of the prey increases the predator population, while the presence of the predator species reduces the prey population [4].…”

Section: Introductionmentioning

confidence: 99%

“…In the work of Lotka [18], this model was developed to describe certain chemical reactions applying the chemical law for mass action. In epidemiology, the Lotka-Volterra system has been used to model the evolution of certain microorganisms, such as the experiments Gause made in 1934 [4]. In this work, we will try to study the most general case, in which many different applications could be included.…”

Section: Introductionmentioning

confidence: 99%

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Optimal control of the Lotka–Volterra system: turnpike property and numerical simulations

Ibañez

2016

Journal of Biological Dynamics

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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. ARTICLE HISTORY

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Optimal control of the Lotka–Volterra system: turnpike property and numerical simulations

Ibañez

2016

Journal of Biological Dynamics

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“…We see that with these calculations we can analyse the asymptotic behaviour of the aggregated variables of system (4). Using the function r(θ ), we can determine existence and uniqueness of an endemic steady-state solution and using the set-membership estimation, we can conclude that this steady state is globally asymptotically stable for all initial data u(·) ∈ U.…”

Section: Numerical Analysismentioning

confidence: 96%

“…The model involves a system of first-order PDEs (of the type of the so-called size-structured systems), which is similar to (but different from) [26]. It could be interpreted in terms of an influenza infection, but similar models may be appropriate to simulate sexually transmitted diseases [4]. In [26], it is argued that this framework can also be used to model microparasite infections.…”

Section: Introductionmentioning

confidence: 99%

Modelling and estimation of infectious diseases in a population with heterogeneous dynamic immunity

Veliov

Widder

2016

Journal of Biological Dynamics

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The paper presents a model for the evolution of an infectious disease in a population with individual-specific immunity. The immune state of an individual varies with time according to its own dynamics, depending on whether the individual is infected or not. The model involves a system of size-structured (first-order) PDEs that capture both the dynamics of the immune states and the transition between compartments consisting of infected, susceptible, etc. individuals. Due to the unavailability of precise data about the immune states of the individuals, the main focus in the paper is on developing a technique for set-membership estimations of aggregated quantities of interest. The technique involves solving specific optimization problems for the underlying PDE system and is developed up to a numerical method. Results of numerical simulations are presented for a benchmark model of SIS-type, potentially applicable to diseases like influenza and to various sexually transmitted diseases. ARTICLE HISTORY

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“…SRNs are employed to describe the time evolution of biochemical reactions, epidemic processes [11,12], and transcription and translation in genomics and virus kinetics [27,42], among other important applications. Let X be an SRN taking values in Z d + and defined in the time-interval [0, T ], where T > 0 is a user-selected final time.…”

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confidence: 99%

Multilevel hybrid split-step implicit tau-leap

2016

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Abstract. In biochemically reactive systems with small copy numbers of one or more reactant molecules, the dynamics is dominated by stochastic effects. To approximate those systems, discrete state-space and stochastic simulation approaches have been shown to be more relevant than continuous state-space and deterministic ones. In systems characterized by having simultaneously fast and slow timescales, existing discrete space-state stochastic path simulation methods, such as the stochastic simulation algorithm (SSA) and the explicit tau-leap (explicit-TL ) method, can be very slow. Implicit approximations have been developed to improve numerical stability and provide efficient simulation algorithms for those systems. Here, we propose an efficient Multilevel Monte Carlo (MLMC) method in the spirit of the work by Anderson and Higham (2012). This method uses split-step implicit tau-leap (SSI-TL ) at levels where the explicit-TL method is not applicable due to numerical stability issues. We present numerical examples that illustrate the performance of the proposed method.

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Continuous Population Models

Cited by 10 publications

References 0 publications

Optimal control of the Lotka–Volterra system: turnpike property and numerical simulations

Optimal control of the Lotka–Volterra system: turnpike property and numerical simulations

Modelling and estimation of infectious diseases in a population with heterogeneous dynamic immunity

Multilevel hybrid split-step implicit tau-leap

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