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In recent years, mathematical models have become an indispensable tool in the planning, evaluation, and implementation of public health interventions. Models must often provide detailed information for many levels of population stratification. Such detail comes at a price: in addition to the computational costs, the number of considered input parameters can be large, making effective study design difficult. To address these difficulties, we propose a novel technique to reduce the dimension of the model input space to simplify model-informed intervention planning. The method works by first applying a dimension reduction technique on the model output space. We then develop a method which allows us to map each reduced output to a corresponding vector in the input space, thereby reducing its dimension. We apply the method to the HIV Optimization and Prevention Economics (HOPE) model, to validate the approach and establish proof of concept.

Partial differential equation (PDE) models for infectious disease have received renewed interest in recent years. Most models of this type extend classical compartmental formulations with additional terms accounting for spatial dynamics, with Fickian diffusion being the most common such term. However, while diffusion may be appropriate for modeling vector-borne diseases, or diseases among plants or wildlife, the spatial propagation of airborne diseases in human populations is heavily dependent on human contact and mobility patterns, which are not necessarily well-described by diffusion. By including an additional chemotaxis-inspired term, in which the infection is propagated along the positive gradient of the susceptible population (from regions of low- to high-density of susceptibles), one may provide a more suitable description of these dynamics. This article introduces and analyzes a mathematical model of infectious disease incorporating a modified chemotaxis-type term. The model is analyzed mathematically and the well-posedness of the resulting PDE system is demonstrated. A series of numerical simulations are provided, demonstrating the ability of the model to naturally capture important phenomena not easily observed in standard diffusion models, including propagation over long spatial distances over short time scales and the emergence of localized infection hotspots

The OraQuick In-Home HIV self-test represents a fast, inexpensive, and convenient method for users to assess their HIV status. If integrated thoughtfully into existing testing practices, accompanied by efficient pathways to formal diagnosis, self-testing could both enhance HIV awareness and reduce HIV incidence. However, currently available self-tests are less sensitive, particularly for recent infection, than gold-standard laboratory tests. It is important to understand the impact if some portion of standard testing is replaced by self-tests. We introduced a novel compartmental model to evaluate the effects of self-testing among gay, bisexual and other men who have sex with men (MSM) in the United States for the period 2020 to 2030. We varied the model for different screening rates, self-test proportions, and delays to diagnosis for those identified through self-tests to determine the potential impact on HIV incidence and awareness of status. When HIV self-tests are strictly supplemental, self-testing can decrease HIV incidence among MSM in the US by up to 10% and increase awareness of status among MSM from 85% to 91% over a 10-year period, provided linkage to care and formal diagnosis occur promptly following a positive self-test (90 days or less). As self-tests replace a higher percentage laboratory-based testing algorithms, increases in overall testing rates were necessary to ensure reductions in HIV incidence. However, such increases were small (under 10% for prompt engagement in care and moderate levels of replacement). Improvements in self-test sensitivity and/or decreases in the detection period may further reduce any necessary increases in overall testing. Our study suggests that, if properly utilized, self-testing can provide significant long-term reductions to HIV incidence and improve awareness of HIV status.

As the outbreak of COVID-19 enters its third year, we have now enough data to analyse the behavior of the pandemic with mathematical models over a long period of time. The pandemic alternates periods of high and low infections, in a way that sheds a light on the nature of mathematical model that can be used for reliable predictions. The main hypothesis of the model presented here is that the oscillatory behavior is a structural feature of the outbreak, even without postulating a time-dependence of the coefficients. As such, it should be reflected by the presence of limit cycles as asymptotic solutions. This stems from the introduction of (i) a non-linear waning immunity based on the concept of immunity booster (already used for other pathologies); (ii) a fine description of the compartments with a discrimination between individuals infected/vaccinated for the first time, and individuals already infected/vaccinated, undergoing to new infections/doses. We provide a proof-of-concept that our novel model is capable of reproducing long-term oscillatory behavior of many infectious diseases, and, in particular, the periodic nature of the waves of infection. Periodic solutions are inherent to the model, and achieved without changing parameter values in time. This may represent an important step in the long-term modeling of COVID-19 and similar diseases, as the natural, unforced behavior of the solution shows the qualitative characteristics observed during the COVID-19 pandemic.

COVID-19 data released by public health authorities features the presence of notable time-delays, corresponding to the difference between actual time of infection and identification of infection. These delays have several causes, including the natural incubation period of the virus, availability and speed of testing facilities, population demographics, and testing center capacity, among others. Such delays have important ramifications for both the mathematical modeling of COVID-19 contagion and the design and evaluation of intervention strategies. In the present work, we introduce a novel optimization technique for the identification of time delays in COVID-19 data, making use of a delay-differential equation model. The proposed method is general in nature and may be applied not only to COVID-19, but for generic dynamical systems in which time delays may be present.

Dynamic Mode Decomposition (DMD) is an unsupervised machine learning method that has attracted considerable attention in recent years owing to its equation-free structure, ability to easily identify coherent spatio-temporal structures in data, and effectiveness in providing reasonably accurate predictions for certain problems. Despite these successes, the application of DMD to certain problems featuring highly nonlinear transient dynamics remains challenging. In such cases, DMD may not only fail to provide acceptable predictions but may indeed fail to recreate the data in which it was trained, restricting its application to diagnostic purposes. For many problems in the biological and physical sciences, the structure of the system obeys a compartmental framework, in which the transfer of mass within the system moves within states. In these cases, the behavior of the system may not be accurately recreated by applying DMD to a single quantity within the system, as proper knowledge of the system dynamics, even for a single compartment, requires that the behavior of other compartments is taken into account in the DMD process. In this work, we demonstrate, theoretically and numerically, that, when performing DMD on a fully coupled PDE system with compartmental structure, one may recover useful predictive behavior, even when DMD performs poorly when acting compartment-wise. We also establish that important physical quantities, as mass conservation, are maintained in the coupled-DMD extrapolation. The mathematical and numerical analysis suggests that DMD may be a powerful tool when applied to this common class of problems. In particular, we show interesting numerical applications to a continuous delayed-SIRD model for Covid-19, and to a problem from additive manufacturing considering a nonlinear temperature field and the resulting change of material phase from powder, liquid, and solid states.