Function-valued adaptive dynamics and the calculus of variations

Adaptive dynamics has been widely used to study the evolution of scalar-valued, and occasionally vector-valued, strategies in ecologically realistic models. In many ecological situations, however, evolving strategies are best described as function-valued, and thus infinite-dimensional, traits. So far, such evolution has only been studied sporadically, mostly based on quantitative genetics models with limited ecological realism. In this article we show how to apply the calculus of variations to find evolutionarily singular strategies of function-valued adaptive dynamics: such a strategy has to satisfy Euler's equation with environmental feedback. We also demonstrate how second-order derivatives can be used to investigate whether or not a function-valued singular strategy is evolutionarily stable. We illustrate our approach by presenting several worked examples.     

React or wait: which optimal culling strategy to control infectious diseases in wildlife

We applied optimal control theory to an SI epidemic model to identify optimal culling strategies for diseases management in wildlife. We focused on different forms of the objective function, including linear control, quadratic control, and control with limited amount of resources. Moreover, we identified optimal solutions under different assumptions on disease-free host dynamics, namely: self-regulating logistic growth, Malthusian growth, and the case of negligible demography. We showed that the correct characterization of the disease-free host growth is crucial for defining optimal disease control strategies. By analytical investigations of the model with negligible demography, we demonstrated that the optimal strategy for the linear control can be either to cull at the maximum rate at the very beginning of the epidemic (reactive culling) when the culling cost is low, or never to cull, when culling cost is high. On the other hand, in the cases of quadratic control or limited resources, we demonstrated that the optimal strategy is always reactive. Numerical analyses for hosts with logistic growth showed that, in the case of linear control, the optimal strategy is always reactive when culling cost is low. In contrast, if the culling cost is high, the optimal strategy is to delay control, i.e. not to cull at the onset of the epidemic. Finally, we showed that for diseases with the same basic reproduction number delayed control can be optimal for acute infections, i.e. characterized by high disease-induced mortality and fast dynamics, while reactive control can be optimal for chronic ones.     

Stability analysis and optimal vaccination of an SIR epidemic model

Almost all mathematical models of diseases start from the same basic premise: the population can be subdivided into a set of distinct classes dependent upon experience with respect to the relevant disease. Most of these models classify individuals as either a susceptible individual S, infected individual I or recovered individual R. This is called the susceptible-infected-recovered (SIR) model. In this paper, we describe an SIR epidemic model with three components; S, I and R. We describe our study of stability analysis theory to find the equilibria for the model. Next in order to achieve control of the disease, we consider a control problem relative to the SIR model. A percentage of the susceptible populations is vaccinated in this model. We show that an optimal control exists for the control problem and describe numerical simulations using the Runge-Kutta fourth order procedure. Finally, we describe a real example showing the efficiency of this optimal control.     

Minimal time control of fed-batch bioreactor with product inhibition

This paper is devoted to the minimal time control problem for fed-batch bioreactors, in presence of an inhibitory product, which is released by the biomass proportionally to its growth. We first consider a growth rate with substrate saturation and product inhibition, and we prove that the optimal strategy is fill and wait (bang-bang). We then investigate the case of the Jin growth rate which takes into account substrate and product inhibition. For this type of growth function, we can prove the existence of singular arc paths defining singular strategies. Several configurations are addressed depending on the parameter set. For each case, we provide an optimal feedback control of the problem (of type bang-bang or bang-singular-bang). These results are obtained gathering the initial system into a planar one by using conservation laws. Thanks to Pontryagin maximum principle, Green’s theorem, and properties of the switching function, we obtain the optimal synthesis. A methodology is also proposed in order to implement the optimal feeding strategies.     

Optimal coordination and control of posture and locomotion.

This paper presents a theoretical model of stability and coordination of posture and locomotion, together with algorithms for continuous-time quadratic optimization of motion control. Explicit solutions to the Hamilton-Jacobi equation for optimal control of rigid-body motion are obtained by solving an algebraic matrix equation. The stability is investigated with Lyapunov function theory, and it is shown that global asymptotic stability holds. It is also shown how optimal control and adaptive control may act in concert in the case of unknown or uncertain system parameters. The solution describes motion strategies of minimum effort and variance. The proposed optimal control is formulated to be suitable as a posture and stance model for experimental validation and verification. The combination of adaptive and optimal control makes this algorithm a candidate for coordination and control of functional neuromuscular stimulation as well as of prostheses.     

Optimal coordination and control of posture and movements

This paper presents a theoretical model of stability and coordination of posture and locomotion, together with algorithms for continuous-time quadratic optimization of motion control. Explicit solutions to the Hamilton–Jacobi equation for optimal control of rigid-body motion are obtained by solving an algebraic matrix equation. The stability is investigated with Lyapunov function theory and it is shown that global asymptotic stability holds. It is also shown how optimal control and adaptive control may act in concert in the case of unknown or uncertain system parameters. The solution describes motion strategies of minimum effort and variance. The proposed optimal control is formulated to be suitable as a posture and movement model for experimental validation and verification. The combination of adaptive and optimal control makes this algorithm a candidate for coordination and control of functional neuromuscular stimulation as well as of prostheses. Validation examples with experimental data are provided.     

Function-valued adaptive dynamics and optimal control theory

In this article we further develop the theory of adaptive dynamics of function-valued traits. Previous work has concentrated on models for which invasion fitness can be written as an integral in which the integrand for each argument value is a function of the strategy value at that argument value only. For this type of models of direct effect, singular strategies can be found using the calculus of variations, with singular strategies needing to satisfy Euler’s equation with environmental feedback. In a broader, more mechanistically oriented class of models, the function-valued strategy affects a process described by differential equations, and fitness can be expressed as an integral in which the integrand for each argument value depends both on the strategy and on process variables at that argument value. In general, the calculus of variations cannot help analyzing this much broader class of models. Here we explain how to find singular strategies in this class of process-mediated models using optimal control theory. In particular, we show that singular strategies need to satisfy Pontryagin’s maximum principle with environmental feedback. We demonstrate the utility of this approach by studying the evolution of strategies determining seasonal flowering schedules.     

Optimal risk management of human alveolar echinococcosis with vermifuge

In this study, we develop a bioeconomic model of human alveolar echinococcosis (HAE) and formulate the optimal strategies for managing the infection risks in humans by applying optimal control theory. The model has the following novel features: (i) the complex transmission cycle of HAE has been tractably incorporated into the framework of optimal control problems and (ii) the volume of vermifuge spreading to manage the risk is considered a control variable. With this model, we first obtain the stability conditions for the transmission dynamics under the condition of constant control. Second, we explicitly introduce a control variable of vermifuge spreading into the analysis by considering the associated control costs. In this optimal control problem, we have successfully derived a set of conditions for a bang-bang control and singular control, which are mainly characterized by the prevalence of infection in voles and foxes and the remaining time of control. The analytical results are demonstrated by numerical analysis and we discuss the effects of the parameter values on the optimal strategy and the transmission cycle. We find that when the prevalence of infection in foxes is low and the prevalence of infection in voles is sufficiently high, the optimal strategy is to expend no effort in vermifuge spreading.     

宏观生物经济系统线性二次型问题的稳定性分析

本文用控制方法研究宏观生物经济系统,对其进行了线性二次型最优控制设计,给出了该系统的线性二次型问题模型,并对其稳定性进行分析,利用李雅普诺夫稳定性定理,证明了该控制设计的最优控制系统是大范围稳定的,从而得到了该系统在线性二次型控制下保持生物产业经济稳定增长的结论,最后通过仿真验证了结论的正确性．     

Optimal control strategies and cost-effectiveness analysis of a malaria model

The aim of this paper is to investigate the effectiveness and cost-effectiveness of three malaria preventive measures (use of treated bednets, spray of insecticides and a possible treatment of infective humans that blocks transmission to mosquitoes). For this, we consider a mathematical model for the transmission dynamics of the disease that includes these measures. We first consider the constant control parameters’ case, we calculate the basic reproduction number and investigate the existence and stability of equilibria; the model is found to exhibit backward bifurcation. We then assess the relative impact of each of the constant control parameters measures by calculating the sensitivity index of the basic reproductive number to the model's parameters. In the time-dependent constant control case, we use Pontryagin's Maximum Principle to derive necessary conditions for the optimal control of the disease. We also calculate the Infection Averted Ratio (IAR) and the Incremental Cost-Effectiveness Ratio (ICER) to investigate the cost-effectiveness of all possible combinations of the three control measures. One of our findings is that the most cost-effective strategy for malaria control, is the combination of the spray of insecticides and treatment of infective individuals. This strategy requires a 100% effort in both treatment (for 20 days) and spray of insecticides (for 57 days). In practice, this will be extremely difficult, if not impossible to achieve. The second most cost-effective strategy which consists of a 100% use of treated bednets and 87% treatment of infective individuals for 42 and 100 days, respectively, is sustainable and therefore preferable.     

Evolutionary branching and evolutionarily stable coexistence of predator species: Critical function analysis

On the ecological timescale, two predator species with linear functional responses can stably coexist on two competing prey species. In this paper, with the methods of adaptive dynamics and critical function analysis, we investigate under what conditions such a coexistence is also evolutionarily stable, and whether the two predator species may evolve from a single ancestor via evolutionary branching. We assume that predator strategies differ in capture rates and a predator with a high capture rate for one prey has a low capture rate for the other and vice versa. First, by using the method of critical function analysis, we identify the general properties of trade-off functions that allow for evolutionary branching in the predator strategy. It is found that if the trade-off curve is weakly convex in the vicinity of the singular strategy and the interspecific prey competition is not strong, then this singular strategy is an evolutionary branching point, near which the resident and mutant predator populations can coexist and diverge in their strategies. Second, we find that after branching has occurred in the predator phenotype, if the trade-off curve is globally convex, the predator population will eventually branch into two extreme specialists, each completely specializing on a particular prey species. However, in the case of smoothed step function-like trade-off, an interior dimorphic singular coalition becomes possible, the predator population will eventually evolve into two generalist species, each feeding on both of the two prey species. The algebraical analysis reveals that an evolutionarily stable dimorphism will always be attractive and that no further branching is possible under this model.     

Optimal treatment of an SIR epidemic model with time delay

In this paper the optimal control strategies of an SIR (susceptible–infected–recovered) epidemic model with time delay are introduced. In order to do this, we consider an optimally controlled SIR epidemic model with time delay where a control means treatment for infectious hosts. We use optimal control approach to minimize the probability that the infected individuals spread and to maximize the total number of susceptible and recovered individuals. We first derive the basic reproduction number and investigate the dynamical behavior of the controlled SIR epidemic model. We also show the existence of an optimal control for the control system and present numerical simulations on real data regarding the course of Ebola virus in Congo. Our results indicate that a small contact rate(probability of infection) is suitable for eradication of the disease (Ebola virus) and this is one way of optimal treatment strategies for infectious hosts.     

Optimal isolation control strategies and cost-effectiveness analysis of a two-strain avian influenza model

The most important and effective measures against disease outbreaks in the absence of valid medicines or vaccine are quarantine and isolation strategies. In this paper optimal control theory is applied to a system of ordinary differential equation describing a two-strain avian influenza transmission via the Pontryagin's Maximum Principle. To this end, a pair of control variables representing the isolation strategies for individuals with avian and mutant strains were incorporated into the transmission model. The infection averted ratio (IAR) and the incremental cost-effectiveness ratio (ICER) were calculated to investigate the cost-effectiveness of all possible combinations of the control strategies. The simulation results show that the implementation of the combination strategy during the epidemic is the most cost-effective strategy for avian influenza transmission. This is followed by the control strategy involving isolation of individuals with the mutant strain. Also observed was the fact that low mutating and more virulent virus results in an increased control effort of isolating individuals with the avian strain; and high mutating with more virulent virus results in increased efforts in isolating individuals with the mutant strain.     

Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control

In this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace spanned by specially chosen observable functions. The Koopman operator is an infinite-dimensional linear operator that evolves functions of the state of a dynamical system. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear measurements of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems. Choosing the right nonlinear observable functions to form an invariant subspace where it is possible to obtain linear reduced-order models, especially those that are useful for control, is an open challenge. Here, we investigate the choice of observable functions for Koopman analysis that enable the use of optimal linear control techniques on nonlinear problems. First, to include a cost on the state of the system, as in linear quadratic regulator (LQR) control, it is helpful to include these states in the observable subspace, as in DMD. However, we find that this is only possible when there is a single isolated fixed point, as systems with multiple fixed points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis by leveraging a new algorithm to determine relevant terms in a dynamical system by ℓ₁-regularized regression of the data in a nonlinear function space; we also show how this algorithm is related to DMD. Finally, we demonstrate the usefulness of nonlinear observable subspaces in the design of Koopman operator optimal control laws for fully nonlinear systems using techniques from linear optimal control.     

On the impossibility of coexistence of infinitely many strategies

We investigate the possibility of coexistence of pure, inherited strategies belonging to a large set of potential strategies. We prove that under biologically relevant conditions every model allowing for coexistence of infinitely many strategies is structurally unstable. In particular, this is the case when the "interaction operator" which determines how the growth rate of a strategy depends on the strategy distribution of the population is compact. The interaction operator is not assumed to be linear. We investigate a Lotka-Volterra competition model with a linear interaction operator of convolution type separately because the convolution operator is not compact. For this model, we exclude the possibility of robust coexistence supported on the whole real line, or even on a set containing a limit point. Moreover, we exclude coexistence of an infinite set of equidistant strategies when the total population size is finite. On the other hand, for infinite populations it is possible to have robust coexistence in this case. These results are in line with the ecological concept of "limiting similarity" of coexisting species. We conclude that the mathematical structure of the ecological coexistence problem itself dictates the discreteness of the species.     

Optimization of control strategies for epidemics in heterogeneous populations with symmetric and asymmetric transmission

There is growing interest in incorporating economic factors into epidemiological models in order to identify optimal strategies for disease control when resources are limited. In this paper we consider how to optimize the control of a pathogen that is capable of infecting multiple hosts with different rates of transmission within and between species. Our objective is to find control strategies that maximize the discounted number of healthy individuals. We consider two classes of host-pathogen system, comprising two host species and a common pathogen, one with asymmetrical and the other with symmetrical transmission rates, applicable to a wide range of SI (susceptible-infected) epidemics of plant and animal pathogens. We motivate the analyses with an example of sudden oak death in California coastal forests, caused by Phytophthora ramorum, in communities dominated by bay laurel (Umbellularia californica) and tanoak (Lithocarpus densiflorus). We show for the asymmetric case that it is optimal to give priority in treating disease to the more infectious species, and to treat the other species only when there are resources left over. For the symmetric case, we show that although a switching strategy is an optimum, in which preference is first given to the species with the lower level of susceptibles and then to the species with the higher level of susceptibles, a simpler strategy that favors treatment of infected hosts for the more susceptible species is a robust alternative for practical application when the optimal switching time is unknown. Finally, since transmission rates are notoriously difficult to estimate, we analyze the robustness of the strategies when the true state with respect to symmetry or otherwise is unknown but one or other is assumed.     

The structure of optimal time- and age-dependent harvesting in the Lotka-McKendrik population model

The paper analyzes optimal harvesting of age-structured populations described by the Lotka-McKendrik model. It is shown that the optimal time- and age-dependent harvesting control involves only one age at natural conditions. This result leads to a new optimization problem with the time-dependent harvesting age as an unknown control. The integral Lotka model is employed to explicitly describe the time-varying age of harvesting. It is proven that in the case of the exponential discounting and infinite horizon the optimal strategy is a stationary solution with a constant harvesting age. A numeric example on optimal forest management illustrates the theoretical findings. Discussion and interpretation of the results are provided.     

Linear selection indices for non-linear profit functions

Summary Two methods of deriving linear selection indices for non-linear profit functions have been proposed. One is by linear approximation of profit, and another is the graphical method of Moav and Hill (1966). When profit is defined as the function of population means, the graphical method is optimal. In this paper, profit is defined as the function of the phenotypic values of individual animals; it is then shown that the graphical method is not generally optimal. We propose new methods for constructing selection indices. First, a numerical method equivalent to the graphical method is proposed. Furthermore, we propose two other methods using quadratic approximation of profit: one is based on Taylor series about means before selection, and the other is based on Tayler series about means after selection. Among these different methods, it is shown that the method using quadratic approximation based on Taylor series about means after selection is the most efficient.     

On invasion boundaries and the unprotected coexistence of two strategies

In this paper we present, in terms of invasion fitness functions, a sufficient condition for a coexistence of two strategies which are not protected from extinction when rare. In addition, we connect the result to the local characterization of singular strategies in the theory of adaptive dynamics. We conclude with some illustrative examples.