Optimal Control of a Vaccinating Game toward Increasing Overall Coverage

Abstract

In this paper, we study an asymmetric game that characterizes the intentions of players to adopt a vaccine. The game describes a decision-making process of two players differentiated by income level and perceived treatment cost, who consider a vaccination against an infectious disease. The process is a noncooperative game since their vaccination decision has a direct impact on vaccine coverage in the population. We introduce a replicator dynamics (RD) to investigate the players' optimal strategy selections over time. The dynamics reveal the long-term stability of the unique Nash-Pareto equilibrium strategy of this game, which is an extension of the notion of an evolutionarily stable strategy pair for asymmetric games. This Nash-Pareto pair is dependent on perceived costs to each player type, on perceived loss upon getting infected, and on the probability of getting infected from an infected person. Last but not least, we introduce a payoff parameter that plays the role of cost-incentive towards vaccination. We use an optimal control problem associated with the RD system to show that the Nash-Pareto pair can be controlled to evolve towards vaccination strategies that lead to a higher overall expected vaccine coverage.

Classical Continuous Boundary Optimal Control Vector Problem for Triple Nonlinear Parabolic System

“…, with 𝑐(𝑡, 𝑦 , 𝑦 ) ≥ 𝛼 ̅ ‖𝑦 ‖ 1 2 . Where ‖𝜐‖ 0 ,and ‖𝜐‖ 1 are denote to the norms in the spaces L 2 (Ω), 𝐻 1 (Ω) resp. and ‖𝜐 ‖ 1 2 = ∑ ‖𝜐 𝑖 ‖ 1 , 𝛼 𝑖 , 𝛼 ̅ 𝑖 , 𝛽 𝑖 , 𝛽 ̅ 𝑖 ( ∀ 𝑖 = 1,2,3), ∈ 𝑖 (∀ 𝑖 = 1,2,3,4,5,6) and 𝛼 ̅ are real positive constants.…”

“…Then there exists a CCBOCV. Proof: From the assumptions on 𝑈 𝑙 (𝑙 = 1,2,3), (56) 〈𝑦 3𝑘𝑡 , 𝑣 3 〉 + 𝑎 3 (𝑡, 𝑦 3𝑘 , 𝑣 3 ) + (𝑏 3 (𝑡)𝑦 3𝑘 , 𝑣 3 ) Ω + (𝑏 5 (𝑡)𝑦 1𝑘 , 𝑣 3 ) − (𝑏 6 (𝑡)𝑦 2𝑘 , 𝑣 3 ) Ω Ω = (𝑓 3 (𝑥, 𝑡, 𝑦 3𝑘 ), 𝑣 3 ) Ω + (𝑢 3𝑘 , 𝑣 3 ) Γ (57) M.B.S of ((55)-(57)) by 𝜑 𝑙 (𝑡) ∈ 𝐶 1 [𝐼] resp, with 𝜑 𝑙 (𝑇) = 0 , ∀𝑙 = 1,2 ,3 and then I.B.S w.r.t. 𝑡 from 0 to 𝑇, and using I.B.P for the 1 𝑠𝑡 terms in the L.H.S.…”

“…Optimal control problems (OCPs) play an important role in many practical applications, such as in medicine [1], aircraft [2], economics [3], robotics [4], weather conditions [5] and many other scientific fields. They are two types of OCPs; the classical and the relax type, each one of these two types is dominated either by nonlinear ODEs [6] or by nonlinear PDEs (NLPDEs) [7].…”

Controlling infection in predator-prey systems with transmission dynamics

“…A numerical method to solve this problem is the steepest descent (SD) method whose algorithm is presented in (Kirk, 2012) and (Wang, 2009). We used this method in similar work to solve an optimal control of a social norm game in (Jaber & Cojocaru, 2018) and to solve the optimal control of a preventive treatment game in (Cojocaru & Jaber, 2018) to maximize the number of players who choose to vaccinate in an epidemiological model.…”

The Classical Continuous Optimal Control for Quaternary Nonlinear Parabolic Boundary Value Problems

“…The problem seeks to optimize the objective function subject to the constraints construed by the model describing the evolution of the underlying system 1 . Optimal control problems (OCPs) play an important role in many practical applications, such as in weather conditions 2 , economics 3 , robotics 4 , aircraft 5 , medicine 6 , and many other scientific fields. They are two types of OCPs; the classical and the relax type, each one of these two types is dominated either by nonlinear ODEs 7 or by nonlinear PDEs (NLPDEs) 8 .…”