Optimal control of a social epidemic model with media coverage

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.     

An SIRVS epidemic model with pulse vaccination strategy

The aim of this paper is to analyze an SIRVS epidemic model in which pulse vaccination strategy (PVS) is included. We are interested in finding the basic reproductive number of the model which determine whether or not the disease dies out. The global attractivity of the disease-free periodic solution (DFPS for short) is obtained when the basic reproductive number is less than unity. The disease is permanent when the basic reproductive number is greater than unity, i.e., the epidemic will turn out to endemic. Our results indicate that the disease will go to extinction when the vaccination rate reaches some critical value.     

Optimal control of diffusion processes pertaining to an opioid epidemic dynamical model with random perturbations

Journal of Mathematical Biology - In this paper, we consider the problem of controlling a diffusion process pertaining to an opioid epidemic dynamical model with random perturbation so as to...     

Optimal timing of one-shot interventions for epidemic control

The interventions and outcomes in the ongoing COVID-19 pandemic are highly varied. The disease and the interventions both impose costs and harm on society. Some interventions with particularly high costs may only be implemented briefly. The design of optimal policy requires consideration of many intervention scenarios. In this paper we investigate the optimal timing of interventions that are not sustainable for a long period. Specifically, we look at at the impact of a single short-term non-repeated intervention (a “one-shot intervention”) on an epidemic and consider the impact of the intervention’s timing. To minimize the total number infected, the intervention should start close to the peak so that there is minimal rebound once the intervention is stopped. To minimise the peak prevalence, it should start earlier, leading to initial reduction and then having a rebound to the same prevalence as the pre-intervention peak rather than one very large peak. To delay infections as much as possible (as might be appropriate if we expect improved interventions or treatments to be developed), earlier interventions have clear benefit. In populations with distinct subgroups, synchronized interventions are less effective than targeting the interventions in each subcommunity separately.     

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.     

Stability properties of pulse vaccination strategy in SEIR epidemic model

The problem of the applicability of the pulse vaccination strategy (PVS) for the stable eradication of some relevant general class of infectious diseases is analyzed in terms of study of local asymptotic stability (LAS) and global asymptotic stability (GAS) of the periodic eradication solution for the SEIR epidemic model in which is included the PVS. Demographic variations due or not to diseased-related fatalities are also considered. Due to the non-triviality of the Floquet's matrix associate to the studied model, the LAS is studied numerically and in this way it is found a simple approximate (but analytical) sufficient criterion which is an extension of the LAS constraint for the stability of the trivial equilibrium in SEIR model without vaccination. The numerical simulations also seem to suggest that the PVS is slightly more efficient than the continuous vaccination strategy. Analytically, the GAS of the eradication solutions is studied and it is demonstrated that the above criteria for the LAS guarantee also the GAS.     

Optimal posture control of a musculo-skeletal arm model

 In this paper maximal performance posture control of the human arm is investigated by means of model simulations. Recent experiments (F.C.T. van der Helm, submitted, 2000) have shown that the reflexive feedback during postural control varies with the bandwidth of the applied force disturbances. This paper focusses on the influence of the frequency content of force disturbances on the reflexive feedback gains by means of optimization. The arm is modelled by a non-linear musculo-skeletal model with two degrees of freedom and six muscles. To facilitate the optimization of the model parameters, the arm model is linearized. A performance criterion is minimized for stochastic force disturbances in a two-step procedure: (1) optimization of static muscle activations using an additional energy criterion to obtain a unique and energy-efficient solution; and (2) optimization of reflex gains using an additional control effort criterion to obtain a unique solution. The optimization reveals that for the given task and posture, the shoulder muscles have the largest contribution, whereas the bi-articular muscles have a relatively small contribution to the behaviour. The dynamics at the endpoint level are estimated so that a comparison can be made with the experiments. Compared to the experiments, the intrinsic damping of the model is relatively large (about 150%), whereas the intrinsic stiffness is relatively small (about 60%). These differences can be attributed to unmodelled mechanical effects of cross-bridges in Hill-type muscle models. The optimized reflex gains show remarkable similarities with the values found in the experiments, implying that humans can adjust their reflexive feedback gains in an optimal way, weighting the performance and energy. The approach in this paper could be useful in the study of various posture tasks, for example in the prediction of the relation between the control parameters of various musculo-skeletal models and different experimental variables. Received: 24 January 2000 / Accepted in revised form: 7 July 2000     

Optimal intervention for an epidemic model under parameter uncertainty

We will be concerned with optimal intervention policies for a continuous-time stochastic SIR (susceptible-->infective-->removed) model for the spread of infection through a closed population. In previous work on such optimal policies, it is common to assume that model parameter values are known; in reality, uncertainty over parameter values exists. We shall consider the effect upon the optimal policy of changes in parameter estimates, and of explicitly taking into account parameter uncertainty via a Bayesian decision-theoretic framework. We consider policies allowing for (i) the isolation of any number of infectives, or (ii) the immunisation of all susceptibles (total immunisation). Numerical examples are given to illustrate our results.     

Optimal control of a vectored plant disease model for a crop with continuous replanting

ABSTRACT

Vector-transmitted diseases of plants have had devastating effects on agricultural production worldwide, resulting in drastic reductions in yield for crops such as cotton, soybean, tomato, and cassava. Plant-vector-virus models with continuous replanting are investigated in terms of the effects of selection of cuttings, roguing, and insecticide use on disease prevalence in plants. Previous models are extended to include two replanting strategies: frequencyreplanting and abundance-replanting. In frequency-replanting, replanting of infected cuttings depends on the selection frequency parameter ε, whereas in abundance-replanting, replanting depends on plant abundance via a selection rate parameter also denoted as ε. The two models are analysed and new thresholds for disease elimination are defined for each model. Parameter values for cassava, whiteflies, and African cassava mosaic virus serve as a case study. A numerical sensitivity analysis illustrates how the equilibrium densities of healthy and infected plants vary with parameter values. Optimal control theory is used to investigate the effects of roguing and insecticide use with a goal of maximizing the healthy plants that are harvested. Differences in the control strategies in the two models are seen for large values of ε. Also, the combined strategy of roguing and insecticide use performs better than a single control.     

Stability analysis and optimal control of an SIR epidemic model with vaccination

This paper focuses on the study of a nonlinear mathematical SIR epidemic model with a vaccination program. We have discussed the existence and the stability of both the disease free and endemic equilibrium. Vaccine induced reproduction number is determined and the impact of vaccination in reducing the vaccine induced reproduction number is discussed. Then to achieve control of the disease, a control problem is formulated and it is shown that an optimal control exists for our model. The optimality system is derived and solved numerically using the Runge-Kutta fourth order procedure.     

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.     

Statistical inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case study

A stochastic discrete-time susceptible-exposed-infectious-recovered (SEIR) model for infectious diseases is developed with the aim of estimating parameters from daily incidence and mortality time series for an outbreak of Ebola in the Democratic Republic of Congo in 1995. The incidence time series exhibit many low integers as well as zero counts requiring an intrinsically stochastic modeling approach. In order to capture the stochastic nature of the transitions between the compartmental populations in such a model we specify appropriate conditional binomial distributions. In addition, a relatively simple temporally varying transmission rate function is introduced that allows for the effect of control interventions. We develop Markov chain Monte Carlo methods for inference that are used to explore the posterior distribution of the parameters. The algorithm is further extended to integrate numerically over state variables of the model, which are unobserved. This provides a realistic stochastic model that can be used by epidemiologists to study the dynamics of the disease and the effect of control interventions.     

A simple model for the spatial spread and control of rabies

A simple mathematical model for the spatial spread of rabies is presented. It models the dynamics of the front of an epizootic wave. We show how the model can be used to estimate the minimum width (in kilometers) of a break, that is, a region in which a control scheme is employed in order to stop the spatial progression of the rabies wave front. A simple expression is derived for the surviving fox population, after the passage of the epizootic, in terms of measurable parameters of the model.     

Optimal temperature control for a structured model of plant cell culture

This article calculates optimal open-loop temperature trajectories that maximize the average rate of product synthesis of a plant cell culture. It uses a previously published five-state mathematical model which describes the growth and product synthesis of a batch plant cell suspension culture of Catharanthus roseus under temperature control. The optimal open-loop temperatures maximize the final product concentration for predefined fermentation periods. A single switch in temperature is shown by computer simulation to be near optimal, with a 22% increase in final product yield over that obtained at the optimal constant temperature. Examination of the achieved final product yield as a function of fermentation period allows this period also to be chosen optimally. This time is reduced from 16 days in the constant temperature case to 12 days in the switched temperature case.     

On a simple epidemic model with a heterogeneous infective population

This paper is concerned with a generalization of the simple epidemic model in which the infective population is partitioned intom classes, each of specific infectiousness. Attention is restricted, however, to the case where all the meeting rates between two individuals are equal to each other. Both deterministic and stochastic versions are examined. In either case the development in time of the epidemic process is investigated by exploiting a connection with the standard simple epidemic model. Finally, it is shown that the technique used also applies to a similar model for the spread of information.     

A simple SIS epidemic model with a backward bifurcation

It is shown that an SIS epidemic model with a non-constant contact rate may have multiple stable equilibria, a backward bifurcation and hysteresis. The consequences for disease control are discussed. The model is based on a Volterra integral equation and allows for a distributed infective period. The analysis includes both local and global stability of equilibria.     

Quarantine in a multi-species epidemic model with spatial dynamics

Motivation is provided for the development of infectious disease models that incorporate the movement of individuals over a range of spatial scales. A general model is formulated for a disease that can be transmitted between different species and multiple patches, and the behavior of the system is investigated in the case in which the spatial component consists of a ring of patches. The influence of various parameters on the spatial and temporal spread of the disease is studied numerically, with particular focus on the role of quarantine in the form of travel restriction.     

Threshold dynamics in a time-delayed epidemic model with dispersal

The global dynamics of a time-delayed model with population dispersal between two patches is investigated. For a general class of birth functions, persistence theory is applied to prove that a disease is persistent when the basic reproduction number is greater than one. It is also shown that the disease will die out if the basic reproduction number is less than one, provided that the initial size of the infected population is relatively small. Numerical simulations are presented using some typical birth functions from biological literature to illustrate the main ideas and the relevance of dispersal.