Stability analysis and optimal vaccination of an SIR epidemic model
Introduction
A major assumption of many mathematical models of epidemics is that the population can be divided into a set of distinct states. These states are defined with respect to disease status. The simplest model, which computes the theoretical number of people infected with a contagious illness in a closed population over time, described by Kermack and Mckendrick (1927), consists of three components: susceptible (S), infected (I) and recovered (R). The disease states are defined:
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Susceptible: Individuals that are susceptible have, in the case of the basic model, never been infected, and thus, they are able to catch the disease. Once they have it, they move into the infected state.
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Infected: Individuals who are infected can spread the disease to susceptible individuals. The time that individuals spend in the infected state is the infectious period; after, they enter the recovered state.
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Recovered: Individuals in the recovered state are assumed to be immune for life.
Mathematical models have become important tools in analyzing the spread and control of infectious diseases. The model formulation process clarifies assumptions, variables and parameters. There have been many studies that have mathematically analyzed infectious diseases. Several epidemic models and reviews on theoretical developments are described in Becker (1978), Dietz (1988), Herbert (2000), Wickwirem (1977) and Zhang et al. (2007). The Maximum Principle of Pontryagin and an embedded Newton algorithm are used to find an optimal control strategy in the model (Ogren and Martin, 2002). A system of difference equations for a measles epidemic with vaccination is derived in Linda (2007). Mickens studied the consequences of vaccination in a discrete-time model for the spread of periodic diseases (Mickens, 1992). For control in others, epidemic models, and optimal vaccination strategies, we refer the reader to Fister et al. (1998), Joshi (2002) and Muller (1999).
In this work, we first consider a general epidemic model (Herbert, 2000) and apply stability analysis theory to find the equilibria for the model (Brauer and Castillo-Chavez, 2001). Once the equilibria of the model without control terms are investigated, then we undertake a new model with a control (vaccination). The aim of this work is not to consider a special disease but to set up an optimal control problem relative to the model. To do this, we use a percentage of susceptible populations as a control in the model. This percentage is a function of time. Hence, the optimal control (vaccination) strategy is to minimize the infected and susceptible individuals, and to maximize the total number of recovered individuals by using possible minimal control variables and simultaneously, to investigate the sensitivity of the susceptible individuals by the control (vaccination). We illustrate how the optimal control theory and the percentage of the vaccination (control) variable can be applied to minimize the susceptible and infected individuals. Our second task is to derive the optimality system for the model with the percentage of a vaccinated individuals. Then we find an optimal vaccination regime for the control model and solve numerically this system by using an iterative Runge–Kutta fourth order procedure. Finally we describe an example of a real epidemic with authentic data where we show that optimal control (vaccination) strategies reduce susceptible and infected individuals and increase the total number of recovered individuals. Hence, the method presented here gives a justification for realistic situations and models: smoking, alcoholics, measles, chickenpox, scarlet fever, mumps, etc.
This paper is organized as follows. In Section 2, we investigate stability analysis for the epidemic model. Using this analysis, we determine the equilibria of the system on the model. A control system for the optimality and its existence, and the optimal control are derived in Section 3. In Section 4, utilizing the representation of the optimal control, we describe a numerical solution of the optimality system consisting of the original state system, the adjoint system, and their boundary conditions. In Section 5, we describe, in detail, a real application of our optimal control theory. Finally, we conclude in Section 6.
Section snippets
Equilibria in the Model
Stability criteria for first-order systems or for higher order difference equations depend on the behavior of the system. The classic endemic model is the model with vital dynamics (births and deaths) given byhere is the death rate (and equally, the birth rate), is the recover rate from the infection and is a transmission coefficient. Susceptible individuals acquire the infection
The Optimal Vaccination
Optimal control techniques are of great use in developing optimal strategies to control various kinds of diseases. To solve the challenges of obtaining an optimal vaccination strategy, we use optimal control theory (Morton and Nancy, 2000). We use the model presented in Herbert (2000) to reduce the numbers of susceptible and infected individuals and increase the number of recovered individuals.
In the system (1), we have three state variables , and . For the optimal control problem,
Numerical Method and Discussion
In this section, we solve the optimality system using an iterative method with a Runge–Kutta fourth order procedure. We solve the state system with an initial guess forward in time and then we solve the adjoint system backward in time. First, starting with an initial guess for the adjoint variables, we solve the state equations by a forward Runge–Kutta fourth order procedure in time. Then, those state values are used to solve the adjoint equations by a backward Runge–Kutta fourth order
Numerical Simulations of the Smoking Epidemic
In this section, we apply the above optimal control theory with consideration of its applicability to real life situations. Our optimal control techniques can be applied to real epidemics such as measles, chickenpox, smoking addiction and smallpox. We consider the giving up smoking model which treats smoking as a disease (Brauer and Castillo-Chavez, 2001). There are significant amounts of mathematical theory on the concept of diseases and epidemics. The basic ideas in these theories are that
Conclusion
In this paper, we proposed an epidemic model. We discussed stability analysis and demonstrated that the model was asymptotically stable. In fact, the analysis we introduced in this work played an important role in analyzing whether a system was stable or unstable. Our theoretical studies for the optimal control problem and its numerical simulations were described where certain values of the parameters considered were appropriate to make the model asymptotically stable.
In the case of a
Acknowledgements
We would like to thank Ms. A. Kellerman and referees for their careful reading of the original manuscript and their many valuable comments and suggestions that greatly improve a presentation of this work.
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2005-003-C00018).
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