Price and Treatment Decisions in Epidemics: A Differential Game Approach

• Main Author: Andrea Di Liddo
• Format: Article
• Language: English
• Published: MDPI AG 2018-10-01
• Series: Mathematics
• Subjects: differential games; epidemics; cost functions; Stackelberg game; bargaining
• Online Access: http://www.mdpi.com/2227-7390/6/10/190

We consider a pharmaceutical company that sells a drug that is useful in the treatment of an infectious disease. A public authority buys the drug to heal at least a portion of the infected population. The authority has an overall budget for all health care costs in the country and can only allocate a (small) part of the budget to the purchase of the drug. The government chooses the amount of drug to be purchased in order to minimize both the number of infectious people and the perceived cost of the operation along a given time horizon. This cost can be modeled through a linear or quadratic function of the monetary cost (as generally happens in the literature) or through a specific function (blow-up) that makes the budget constraint endogenous. The pharmaceutical company chooses the price of the drug in order to maximize its profit and knowing the budget constraints of the buyer. The resulting differential game is studied by supposing the simplest possible dynamics for the population. Two different games are proposed and their solutions are discussed: a cooperative game in which the two players bargain for the price of the drug and the quantity is purchased with the aim of maximizing the overall payoff and a competitive game in which the seller announces a price strategy to the buyer and binds to it; the buyer reacts by choosing the quantity to be purchased. In the case of linear or quadratic costs, the solution provided (for budget levels is not high enough) that the government spends the entire budget to purchase the drug. This drawback does not occur when the blow-up cost function is used.