Treatments consisting of mixtures of pharmacological agents have been shown to have superior effects to treatments involving single compounds. Given the vast amount of possible combinations involving multiple drugs and the restrictions in time and resources required to test all such combinations in vitro, mathematical methods are essential to model the interactive behavior of the drug mixture and the target, ultimately allowing one to better predict the outcome of the combination. In this review, we investigate various mathematical methods that model combination therapies. This survey includes the methods that focus on predicting the outcome of drug combinations with respect to synergism and antagonism, as well as the methods that explore the dynamics of combination therapy and its role in combating drug resistance. This comprehensive investigation of the mathematical methods includes models that employ pharmacodynamics equations, those that rely on signaling and how the underlying chemical networks are affected by the topological structure of the target proteins, and models that are based on stochastic models for evolutionary dynamics. Additionally, this article reviews computational methods including mathematical algorithms, machine learning, and search algorithms that can identify promising combinations of drug compounds. A description of existing data and software resources is provided that can support investigations in drug combination therapies. Finally, the article concludes with a summary of future directions for investigation by the research community.

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Drug resistance in cancer treatments is a frequent problem that, when it arises, leads to failure in therapeutic efforts. Tumor heterogeneity is the primary reason for resistance emergence and a precise treatment design that takes heterogeneity into account is required to postpone the rise of resistant subpopulations in the tumor environment. In this paper, we present a mathematical framework involving clonal evolution modeling of drug-sensitive and drug-resistant clones. Using our framework, we examine delaying the rise of resistance in heterogeneous tumors during control phase of therapy in a containment treatment approach. We apply pharmacokinetic/pharmacodynamic (PKPD) modeling and show that dosage strategies can be designed to control the resistant subpopulation. Our results show that the drug dosage and schedule determine the relative dynamics of sensitive and resistant clones. We present an optimal control problem that finds the dosing strategy that maximizes the delay in resistance emergence for a given period of containment treatment.

The course of the COVID-19 pandemic has given rise to many disease trends at various population scales, ranging from local to global. Understanding these trends and the epidemiological phenomena that lead to the changing dynamics associated with disease progression is critical for public health officials and the global community to rein in further spread of this and other virulent diseases. Classic epidemiological modeling based on dynamical systems are powerful tools used for modeling and understanding diseases, but often necessitate modifications to the classic compartmental models to reflect empirical observations. In this paper, we present a collection of extensions to the classic SIRS model to support public health decisions associated with viral pandemics. Specifically, we present models that reflect different levels of disease severity among infected individuals, capture the effect of vaccination on different population groups, capture the effect of different vaccines with different levels of effectiveness, and model the impact of a vaccine with varying number of doses. Further, our mathematical models support the investigation of a pandemic’s trend under the emergence of new variants and the associated reduction in vaccine effectiveness. Our models are supported through numerical simulations, which we use to illustrate phenomena that have been observed in the COVID-19 pandemic. Our findings also confirm observations that the mild infectious group accounts for the majority of infected individuals, and that prompt immunization results in weaker pandemic waves across all levels of infection as well as a lower number of disease-caused deaths. Finally, using our models, we demonstrate that, when dealing with a single variant and having access to a highly effective vaccine, a three-dose vaccine has a strong ability to reduce the infectious population. However, when a new variant with higher transmissibility and lower vaccine efficiency emerges, it becomes the dominant circulating variant, as was observed in the recent emergence of the Omicron variant.

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