Almost all mathematical models of diseases start from the same basic premise: the population can be subdivided into a set of distinct classes dependent upon experience with respect to the relevant disease. Most of these models classify individuals as either a susceptible individual S, infected individual I or recovered individual R. This is called the susceptible-infected-recovered (SIR) model. In this paper, we describe an SIR epidemic model with three components; S, I and R. We describe our study of stability analysis theory to find the equilibria for the model. Next in order to achieve control of the disease, we consider a control problem relative to the SIR model. A percentage of the susceptible populations is vaccinated in this model. We show that an optimal control exists for the control problem and describe numerical simulations using the Runge-Kutta fourth order procedure. Finally, we describe a real example showing the efficiency of this optimal control.

Stability analysis and optimal vaccination of an SIR epidemic model

We formulate a dynamic mathematical model that describes the
interaction of the immune system with the human immunodeficiency
virus (HIV) and that permits drug ''cocktail'' therapies. We
derive HIV therapeutic strategies by formulating and analyzing an
optimal control problem using two types of dynamic treatments
representing reverse transcriptase (RT) inhibitors and protease
inhibitors (PIs). Continuous optimal therapies are found by
solving the corresponding optimality systems. In addition, using
ideas from dynamic programming, we formulate and derive suboptimal
structured treatment interruptions (STI) in antiviral therapy that
include drug-free periods of immune-mediated control of HIV. Our
numerical results support a scenario in which STI therapies can
lead to long-term control of HIV by the immune response system
after discontinuation of therapy.

https://www.aimsciences.org/article/exportPdf?id=581f3255-5440-4142-ba09-cc084fac3cc4

Dynamic multidrug therapies for hiv: optimal and sti control approaches.

HIV dynamics: Modeling, data analysis, and optimal treatment protocols

There is an urgent need to understand how the provision of information influences individual risk perception and how this in turn shapes the evolution of epidemics. Individuals are influenced by information in complex and unpredictable ways. Emerging infectious diseases, such as the recent swine flu epidemic, may be particular hotspots for a media-fueled rush to vaccination; conversely, seasonal diseases may receive little media attention, despite their high mortality rate, due to their perceived lack of newness. We formulate a deterministic transmission and vaccination model to investigate the effects of media coverage on the transmission dynamics of influenza. The population is subdivided into different classes according to their disease status. The compartmental model includes the effect of media coverage on reporting the number of infections as well as the number of individuals successfully vaccinated. A threshold parameter (the basic reproductive ratio) is analytically derived and used to discuss the local stability of the disease-free steady state. The impact of costs that can be incurred, which include vaccination, education, implementation and campaigns on media coverage, are also investigated using optimal control theory. A simplified version of the model with pulse vaccination shows that the media can trigger a vaccinating panic if the vaccine is imperfect and simplified messages result in the vaccinated mixing with the infectives without regard to disease risk. The effects of media on an outbreak are complex. Simplified understandings of disease epidemiology, propogated through media soundbites, may make the disease significantly worse.

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The impact of media coverage on the transmission dynamics of human influenza.

In this paper, we study the global properties of a class of human immunodeficiency virus (HIV) models. The basic model is a 5-dimensional nonlinear ODEs that describes the interaction of the HIV with two target cells, CD4+ T cells and macrophages. An HIV model with exposed state and a model with nonlinear incidence rate are also analyzed. Lyapunov functions are constructed to establish the global asymptotic stability of the uninfected and infected steady states. We have proven that if the basic reproduction number R 0 is less than unity, then the uninfected steady state is globally asymptotically stable. If R 0 > 1 (or if the infected steady state exists), then the infected steady state is globally asymptotically stable. In a control system framework, we have shown that the HIV model incorporating the effect of Highly Active AntiRetroviral Therapy (HAART) is globally asymptotically controllable to the uninfected steady state.

Global properties of a class of HIV models

A system of ordinary differential equations, which describes the interaction of HIV and T-cells in the immune system is utilized, and optimal controls representing drug treatment strategies of this model are explored. Two types of treatments are used, and existence and uniqueness results for the optimal control pair are established. The optimality system is derived and then solved numerically using an iterative method with a Runge–Kutta fourth order scheme. Copyright © 2002 John Wiley & Sons, Ltd.

Optimal control of an HIV immunology model

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Hem Raj Joshi

Papers

Optimal control of an HIV immunology model

Optimal Control Problems in PDE and ODE Systems