We present an overview of some concepts and methodologies we believe useful in modeling HIV pathogenesis. After a brief discussion of motivation for and previous efforts in the development of mathematical models for progression of HIV infection and treatment, we discuss mathematical and statistical ideas relevant to Structured Treatment Interruptions (STI). Among these are model development and validation procedures including parameter estimation, data reduction and representation, and optimal control relative to STI. Results from initial attempts in each of these areas by an interdisciplinary team of applied mathematicians, statisticians and clinicians are presented.

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 inhibitors (RTIs) 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.

In this paper we study the dynamics of a vector-transmitted disease using two deterministic models. First, we look at time dependent prevention and treatment efforts, where optimal control theory is applied. Using analytical and numerical techniques, it is shown that there are cost effective control efforts for treatment of hosts and prevention of host-vector contacts. Then, we considered the autonomous counter part of the mode and we established global stability results based on the reproductive number. The model is applied to study the effects of prevention and treatment controls on a malaria disease while keeping the implementation cost at a minimum. Numerical results indicate the effects of the two controls (prevention and treatment) in lowering exposed and infected members of each of the populations. The study also highlights the effects of some model parameters on the results.

We consider optimal dynamic multidrug therapies for human immunodeficiency virus (HIV) type 1 infection. In this context, we describe an optimal tracking problem attempting to drive the states of the system to a stationary state in which the viral load is low and the immune response is strong. We consider optimal feedback control with full-state as well as with partial-state measurements. In the case of partialstate measurement, a state estimator is constructed based on viral load and T-cell count measurements. We demonstrate by numerical simulations that by anticipation of and response to the disease progression, the dynamic multidrug strategy reduces the viral load, increases the CD4+ T-cell count and improves the immune response.

Since the 1980s, there has been a worldwide re-emergence of vector-borne diseases including Malaria, Dengue, Yellow fever or, more recently, chikungunya. These viruses are arthropod-borne viruses (arboviruses) transmitted by arthropods like mosquitoes of Aedes genus. The nature of these arboviruses is complex since it conjugates human, environmental, biological and geographical factors. Recent researchs have suggested, in particular during the Reunion Island epidemic in 2006, that the transmission by Aedes albopictus (an Aedes genus specie) has been facilitated by genetic mutations of the virus and the vector capacity to adapt to non tropical regions. In this paper we formulate an optimal control problem, based on biological observations. Three main efforts are considered in order to limit the virus transmission. Indeed, there is no vaccine nor specific treatment against chikungunya, that is why the main measures to limit the impact of such epidemic have to be considered. Therefore, we look at time dependent breeding sites destruction, prevention and treatment efforts, for which optimal control theory is applied. Using analytical and numerical techniques, it is shown that there exist cost effective control efforts.

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We consider the increasingly important and highly complex immunological control problem: control of the dynamics of immunosuppression for organ transplant recipients. The goal in this problem is to maintain the delicate balance between over-suppression (where opportunistic latent viruses threaten the patient) and under-suppression (where rejection of the transplanted organ is probable). First, a mathematical model is formulated to describe the immune response to both viral infection and introduction of a donor kidney in a renal transplant recipient. Some numerical results are given to qualitatively validate and demonstrate that this initial model exhibits appropriate characteristics of primary infection and reactivation for immunosuppressed transplant recipients. In addition, we develop a computational framework for designing adaptive optimal treatment regimes with partial observations and low frequency sampling, where the state estimates are obtained by solving a second deterministic optimal tracking problem. Numerical results are given to illustrate the feasibility of this method in obtaining optimal treatment regimes with a balance between under-suppression and over-suppression of the immune system.

Countries around the world have taken control measures to mitigate the spread of COVID-19, including Korea. Social distancing is considered an essential strategy to reduce transmission in the absence of vaccination or treatment. While interventions have been successful in controlling COVID-19 in Korea, maintaining the current restrictions incurs great social costs. Thus, it is important to analyze the impact of different polices on the spread of the epidemic. To model the COVID-19 outbreak, we use an extended age-structured SEIR model with quarantine and isolation compartments. The model is calibrated to age-specific cumulative confirmed cases provided by the Korea Disease Control and Prevention Agency (KDCA). Four control measures—school closure, social distancing, quarantine, and isolation—are investigated. Because the infectiousness of the exposed has been controversial, we study two major scenarios, considering contributions to infection of the exposed, the quarantined, and the isolated. Assuming the transmission rate would increase more than 1.7 times after the end of social distancing, a second outbreak is expected in the first scenario. The epidemic threshold for increase of contacts between teenagers after school reopening is 3.3 times, which brings the net reproduction number to 1. The threshold values are higher in the second scenario. If the average time taken until isolation and quarantine reduces from three days to two, cumulative cases are reduced by 60% and 47% in the first scenario, respectively. Meanwhile, the reduction is 33% and 41%, respectively, for rapid isolation and quarantine in the second scenario. Without social distancing, a second wave is possible, irrespective of whether we assume risk of infection by the exposed. In the non-infectivity of the exposed scenario, early detection and isolation are significantly more effective than quarantine. Furthermore, quarantining the exposed is as important as isolating the infectious when we assume that the exposed also contribute to infection.