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The representation of the cost of a therapy is a key element in the formulation of the optimal control problem for the treatment of infectious diseases. The cost of the treatment is usually modeled by a function of the price and quantity of drugs administered; this function should be the cost as subjectively perceived by the decision-maker. Neverth...
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... Figure 1 illustrates the different optimal policies with respect to the price p and to the effectiveness α. ...
Context 2
... total infected population rapidly tends to zero while the percentage of the total budget used for the treatment starts at 45% and approximates zero at the end, reaching a maximum of around 50% (see Figure 8). Time % of consumed budget budget=1; p=1; alpha=1; I(0)=1 Figure 10. Optimal treatment of an infectious disease with chronic stage. ...
Citations
... The quadratic terms in (13) permit to make the Lagrangian convex with respect to the control variables ( ), = 1, 2, 3, 4, to guarantee the existence of a unique solution. Di Liddo in [51] gives more explanations about the choice of costs type in the objective functional. ...
This study proposes a non-linear mathematical model for analysing the effect of COVID-19 dynamics on the student population in higher education institutions. The theory of positivity and boundedness of solution is used to investigate the well-posedness of the model. The disease-free equilibrium solution is examined analytically. The next-generation operator method calculates the basic reproduction number (R0). Sensitivity analyses are carried out to determine the relative importance of the model parameters in spreading COVID-19. In light of the sensitivity analysis results, the model is further extended to an optimal control problem by introducing four time-dependent control variables: personal protective measures, quarantine (or self-isolation), treatment, and management measures to mitigate the community spread of COVID-19 in the population. Simulations evaluate the effects of different combinations of the control variables in minimizing COVID-19 infection. Moreover, a cost-effectiveness analysis is conducted to ascertain the most effective and least expensive strategy for preventing and controlling the spread of COVID-19 with limited resources in the student population.
... Optimal control models have received credit from the scientific community for their contribution in determining essential measures for pushing down cases of infections and the spike of new variants [26][27][28][29][30][31][32][33]. In December 2019, when WHO ascertained the existence of the pandemic, intriguing optimal control studies have been carried out that helped to wrestle the epidemic from spreading. ...
... The newly translated model Equation (29) with the bifurcation parameter η 1 = η * 1 has at least one eigenvalue zero, and it affirms the existence of bifurcation in the transformed model (29). As expounded by Castillo-Chavez and Song ( [54], refer to Theorem 5.1.), ...
... are the principal determinants of the direction of bifurcation (whether backwards or forward bifurcation) in the transformed system (29). When a > 0 and b > 0, then the bifurcation is backward and when a < 0 and b > 0, the bifurcation is forward. ...
The world is on its path from the post-COVID period, but a fresh wave of the coronavirus infection engulfing most European countries makes the pandemic catastrophic. Mathematical models are of significant importance in unveiling strategies that could stem the spread of the disease. In this paper, a deterministic mathematical model of COVID-19 is studied to characterize a range of feasible control strategies to mitigate the disease. We carried out an analytical investigation of the model’s dynamic behaviour at its equilibria and observed that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number, R 0 is less than unity. The endemic equilibrium is also shown to be globally asymptotically stable when R 0 > 1 . Further, we showed that the model exhibits forward bifurcation around R 0 = 1 . Sensitivity analysis was carried out to determine the impact of various factors on the basic reproduction number R 0 and consequently, the spread of the disease. An optimal control problem was formulated from the sensitivity analysis. Cost-effectiveness analysis is conducted to determine the most cost-effective strategy that can be adopted to control the spread of COVID-19. The investigation revealed that combining self-protection and environmental control is the most cost-effective control strategy among the enlisted strategies.
... In addition, W1 and W2 are weight for cattle vaccination and cattle culling. The control efforts in equation (20) are assumed to be nonlinear-quadratic, since a quadratic structure in the control has mathematical advantages, such as: if the control set is compact and convex it follows that the Hamiltonian attains its minimum over the control set at a unique point [43,44,45,46,47,48]. Further, W1u 2 1 (t), and W2u 2 2 (t) describe the costs associated with vaccination and culling, respectively. ...
Brucellosis is one of the most common zoonotic infections globally. It affects humans, domestic animals and wildlife. In this paper, we conduct an intrinsic analysis of human brucellosis dynamics in non-periodic and periodic environments. As such we propose and study two mathematical models for human brucellosis transmission and control, in which humans acquire infection from cattle and wildlife. The first model is an autonomous dynamical system and the second is a non-autonomous dynamical system in which the seasonal transmission of brucellosis is incorporated. Disease intervention strategies incorporated in this study are cattle vaccination, culling of infectious cattle and human treatment. For both models we conduct both epidemic and endemic analysis, with a focus on the threshold dynamics characterized by the basic reproduction numbers. Using sensitivity analysis we established that R0 is most sensitive to the rate of brucellosis transmission from buffalos to cattle, the result suggest that in order to control human brucellosis there is a need to control cattle infection. Based on our models, we also formulate an optimal control problem with cattle vaccination and culling of infectious cattle as control functions. Using reasonable parameter values, numerical simulations of the optimal control demonstrate the possibility of reducing brucellosis incidence in humans, wildlife and cattle, within a finite time horizon, for both periodic and non-periodic environments.
... It is important to note that unlike some previous works (see for example [69,26,88] and the references therein), we combined linear state-dependent costs (LSD) with quadratic state-independent costs (QSI). Linear state-dependent costs (LSD) depend on the number of people treated [89] while quadratic stateindependent costs are used to take into account nonlinear costs potentially arising at high levels of intervention [90,91] . Also, these quadratic terms permit to make of the problem convex and thus, guarantee that a unique solution exists [92,69,93] . ...
... Also, these quadratic terms permit to make of the problem convex and thus, guarantee that a unique solution exists [92,69,93] . More explanations about the choice of costs type in the objective function can be found in [89] . ...
In this work, we extend existing models of vector-borne diseases by including density-dependent rates and some existing control mechanisms to decrease the disease burden in the human population. We begin by analyzing the vector model dynamics and by determining the offspring reproductive number denoted by N as well as the trivial and nontrivial equilibria. Using theory of cooperative systems and the general theory of Lyapunov, we prove that, although there is a possibility that the trivial equilibrium coexists with a positive equilibrium, it remains globally asymptotically stable whenever N≤1. The fact that the non-trivial equilibrium is globally asymptotically stable permits us to reduce the study of the full model to the study of a reduced model whenever N>1. Thus, we analyze the reduced model by computing the basic reproduction number R0, equilibrium points as well as asymptotic stability of each equilibrium point. We also explore the nature of the bifurcation for the disease-free equilibrium from R0=1. By the application of the centre manifold theory, we prove that the backward bifurcation phenomenon can occur in our model, which means that the necessary condition R0<1 is not sufficient to guarantee the final extinction of the disease in human populations. To calibrate our model, we estimate model parameters on clinical data from the last Chikungunya epidemic which occurred in Chad, using the non-linear least-square method. We find out that R0=1.8519, which means that we are in an endemic state since R0>1. To determine model parameters that are responsible for disease spread in the human community, we perform sensitivity analysis (SA) using a global method. It follows that the density-dependent death rate of mosquitoes and the average number of mosquito bites are key parameters in the disease dynamics. Following this, we thus formulate an optimal control model by including in the autonomous model, four time-dependent control functions to fight the disease spread. Pontryagin’s maximum principle is used to characterize our optimal controls. Numerical simulations, using parameter values of Chikungunya transmission dynamics, and efficiency analysis, are conducted to determine the better control strategy which guaranteed the final extinction of the disease in human populations.
... -The development of treatment capability of the disease is contingent both upon treatment efforts and the current state of health infrastructures, which itself, in turn, is endogenous. This is in contrast to the existing literature that usually assumes that the treatment capabilities for a disease are exogenously given as bounded control variables (e.g., Behncke, 2000;Joshi, 2002;Hansen and Day, 2010;Lee et al., 2011;Di Liddo, 2016), or as a budget constraint (e.g., Rowthorn et al., 2009). We choose to assume that the treatment capabilities are built up based on the joint deployment of health infrastructures and treatment efforts while the epidemic ...
... -The economic and social losses incurred for each disease-related death are accounted for in a tradeoff that seeks to determine an optimal policy against the epidemic disease. This departs from most of the existing literature that accounts for the cost induced by all (e.g., Yan and Zu, 2008;Blayneh et al., 2009) or part (e.g., Lee et al., 2010;Hansen and Day, 2011;Lee et al., 2011;Tchuenche et al., 2011;Lee et al., 2012;Ullah et al., 2012;Di Liddo, 2016;Collins and Duffy, 2018) of the infected categories except deaths. Following Caulkins et al. (2020) and Charpentier et al. (2020), wchoose here to assume that disease-related deaths cannot be welfare-neutral. ...
... and affecting the decisions made as well as the evolution of the disease over time. Most of the literature assumes popular neutrality regarding disease-related social costs, which implies that the policymaker's decisions are not directly under time pressure (e.g., Behncke, 2000;Joshi, 2002;Ya and Zou, 2008;Blayneh et al., 2009;Lee et al., 2011;Hansen and Day, 2011;Tchuenche et al., 2011;Lee et al., 2012;Ullah et al., 2012;Di Liddo, 2016;Collins and Duffy, 2018;Caulkins et al., 2020). A few studies assume growing popular complacency through a discounting function of the overall social cost, (e.g., Sethi, 1974;Sethi, 1978;Sethi and Staats, 1978;Rowthorn et al., 2009;Charpentier et al., 2020), which implies that future decisions and related outcomes are less costly than current decisions and outcomes. ...
... A particular issue that is a very common source of constant controversy in biomedical applications is a form of the dependency of the cost function on the considered controls (for example, see discussion and the bibliography in [22,4]). The problem is that, in the majority of biomedical applications, the actual form of the costs and the dependency on these on the controls are usually unclear and can hardly be defined with a necessary degree of accuracy. ...
... For instance, Ledzewicz et al. [13] argued that the L 1 -type functionals, that is, the cost functionals with a linear dependence on the control, may capture the cost of an intervention policy more accurately than the L 2 -type functionals. Di Liddo [4] remarked that, if u(t) is the fraction of a population N(t) treated by some drug and p is the unitary cost of the drugs, then a natural way to define the cost C of the treatment is C = pu(t)N(t). Moreover, as an ultimate way to exclude the controversy of the intervention cost, Grigorieva et al. [11,9,7,8,6] suggested to completely disregard the cost of the intervention under the assumption that it is "tolerable" and small compared to the losses that can be inflicted otherwise. ...
... The above simplification allows us to focus on the impact of the cost function σ i (u, I) in the form of the optimal control solution. Following the ideas in-troduced in [4], we consider the following forms for the cost function σ i (u, I): ...
In applications of the optimal control theory to problems in medicine and biology, the dependency of the objective functional on the control itself is often a matter of controversy. In this paper, we explore the impact of the dependency using reasonably simple \emph{SIR} and \emph{SEIRS} epidemic models. To qualitatively compare the outcomes for different objective functionals, we apply the cost-effectiveness analysis. Our result shows that, at least for the comparatively inexpensive controls, the variation of the power at the controls in a biologically feasible range does not significantly affect the forms of the optimal controls and the corresponding optimal state solutions. Moreover, the costs and effectiveness are affected even less. At the same time, the dependency of the cost on the state variables can be very significant.
... [23][24][25][26][27] and references therein. The by far largest part of these works deals with optimal control of epidemics through vaccination and immunization [28][29][30][31], medical treatment [32,33] and combinations thereof [34][35][36][37][38][39]. Significantly fewer papers are concerned with the optimal control of transmission dynamics and the mitigation of epidemics through social distancing measures. ...
When effective medical treatment and vaccination are not available, non-pharmaceutical interventions such as social distancing, home quarantine and far-reaching shutdown of public life are the only available strategies to prevent the spread of epidemics. Based on an extended SEIR (susceptible-exposed-infectious-recovered) model and continuous-time optimal control theory, we compute the optimal non-pharmaceutical intervention strategy for the case that a vaccine is never found and complete containment (eradication of the epidemic) is impossible. In this case, the optimal control must meet competing requirements: First, the minimization of disease-related deaths, and, second, the establishment of a sufficient degree of natural immunity at the end of the measures, in order to exclude a second wave. Moreover, the socioeconomic costs of the intervention shall be kept at a minimum. The numerically computed optimal control strategy is a single-intervention scenario that goes beyond heuristically motivated interventions and simple "flattening of the curve". Careful analysis of the computed control strategy reveals, however, that the obtained solution is in fact a tightrope walk close to the stability boundary of the system, where socioeconomic costs and the risk of a new outbreak must be constantly balanced against one another. The model system is calibrated to reproduce the initial exponential growth phase of the COVID-19 pandemic in Germany.
... e subject of optimal control problem (OCP) plays a basic role in many real life problems in different branches of sciences; for example, in, medicine [1], engineering and social sciences [2], biology [3], ecology [4], electric power [5], aerospace [6], and many other branches. ...
... We denote by (υ, υ), (υ, υ) 1 ...
The focus of this work lies on proving the existence theorem of a unique state vector solution (Stvs) of the triple nonlinear hyperbolic boundary value problem (TNHBVP) when the classical continuous control vector (CCCVE) is fixed by using the Galerkin method (Galm), proving the existence theorem of a unique constraints classical continuous optimal control vector (CCCOCVE) with vector state constraints (equality EQVC and inequality INEQVC). Also, it consists of studying for the existence and uniqueness adjoint vector solution (Advs) of the triple adjoint vector equations (TAEqs) associated with the considered triple state equations (Tsteqs). The Fréchet Derivative (Frde.) of the Hamiltonian (HAM) is found. At the end, the theorems for the necessary conditions and the sufficient conditions of optimality (Necoop and Sucoop) are achieved.
... The optimal control problems play an important role in many fields in the real life problems, for examples in robotics [1], in an electric power [2], in civil engineering [3], in Aeronautics and Astronautics [4], in medicine [5], in economic [6], in heat conduction [7], in biology [8] and many others fields. This importance of optimal control problems encouraged many researchers interested to study the optimal control problems of systems are governed either by nonlinear ordinary differential equations as in [9] and [10] or by linear partial differential equations as in [11] or are governed by nonlinear partial differential equations either of a hyperbolic type as in [12] or of a parabolic type as in [13] or by an elliptic type as in [14], or optimal control problem are governed either by a couple of nonlinear partial differential equations of a hyperbolic type as in [15] or of a parabolic type as in [16] or by an elliptic type as in [17], or of an elliptic type but involve a boundary control as in [18]. ...
This paper is concerned with, the proof of the existence and the uniqueness theorem for the solution of the state vector of a couple of nonlinear elliptic partial differential equations using the Minty-Browder theorem, where the continuous classical boundary control vector is given. Also the existence theorem of a continuous classical boundary optimal control vector governing by the couple of nonlinear elliptic partial differential equation with equality and inequality constraints is proved. The existence of the uniqueness solution of the couple of adjoint equations associated with the considered couple of the state equations with equality and inequality constraints is studied. The necessary conditions theorem and the sufficient conditions theorem for optimality of the couple of nonlinear elliptic equations with equality and inequality constraints are proved using the Kuhn-Tucker-Lagrange multipliers theorems
... In the control of infection spread, on the other hand, the objective is to derive the transition probabilities such that a prescribed performance index, which is a function of implementation cost and the number of infected nodes, is minimized [14][15][16][17][18]. This problem is computationally difficult to solve, in general. ...
... For example, in [14][15][16] it is assumed that the network dynamics and cost are linear in control action (that is immunization or curing rates), which leads to bang-bang control strategy. The interested reader is referred to [17,18] for more details on optimal resource allocation methods. This paper studies the optimal control of a network consisting of an arbitrary number of nodes that are influenced (coupled) by the empirical distribution of infected nodes (such couplings are not necessarily linear). ...
