Mathematical Biosciences and Engineering

Cancer remains a global health challenge requiring sophisticated understanding of tumor-immune dynamics for effective treatment design. Mathematical oncology has emerged as a rapidly evolving interdisciplinary field that uses mathematical models to enhance our understanding of cancer dynamics, including tumor growth, metastasis, and treatment response. This paper presents a comprehensive multiscale framework integrating patient-specific data, machine learning, and optimal control for personalized immunotherapy design. We develop a hybrid model that combines deterministic dynamics with stochastic elements and time delays, capturing the inherent variability and temporal lags in biological processes. The model incorporates biologically realistic Holling Type-II functional responses and is validated against longitudinal clinical data from 100+ cancer patients and patient-derived organoid experiments. Using deep neural networks with Bayesian regularization, we learn patient-specific parameter distributions from clinical biomarkers and predict treatment responses with high accuracy. Our optimal control framework, incorporating clinical constraints and toxicity limits, generates personalized treatment protocols that stabilize otherwise unstable dynamics. The framework establishes a new paradigm for precision immuno-oncology, bridging mathematical theory, computational methods, and clinical practice. Author summaryCancer remains one of the leading causes of death worldwide, and the immune system plays a crucial role in controlling tumor growth. However, the complex interactions between tumor cells and immune cells make it difficult to predict how individual patients will respond to immunotherapy. In this work, we develop a mathematical framework that integrates patient-specific data, machine learning, and optimal control to design personalized immunotherapy strategies. Our model captures the realistic dynamics of tumor-immune interactions by incorporating biologically relevant features such as time delays (representing immune response lags) and stochastic effects (representing biological variability). Using deep learning, we estimate patient-specific parameters from clinical biomarkers, enabling personalized predictions of treatment outcomes. We validate our framework against data from over 100 cancer patients and patient-derived organoid experiments, demonstrating excellent agreement. Our optimal control approach generates personalized treatment protocols that stabilize otherwise unstable tumor dynamics, achieving 78% tumor reduction compared to 52% for standard-of-care protocols. These findings suggest that therapies targeting immunological thresholds may be as important as those directly killing tumor cells, providing a new perspective for immunotherapy design. This framework bridges mathematical theory, computational methods, and clinical practice, offering a pathway toward truly personalized cancer treatment.

Clinical decisions for determining optimal patient-specific interventions are complicated prediction tasks that rely on health care professionals' understanding of physiological mechanisms and their dynamics. These decisions are challenged by (a) observational data sparsity and (b) patient heterogeneity. Here, we focus on estimating and forecasting specific physiological properties--that are not explicitly present in clinical observations--to provide additional features using only data available bedside at the time of decision-making. Mechanistic models of physiological system(s), e.g., physiological ordinary differential equation (ODE) models, provide pathways to compensate for data sparsity by synchronizing the model with observations of an individual patient using data assimilation (DA). However, DA used in a standard computational workflow to estimate constant model parameters from presently-known data is less effective at optimizing state forecasts of the model governed by physiological processes that evolve before new observations are available. Stated simply, we cannot forecast the future evolution of the model because we cannot forecast model parameters. To support next-generation clinical decision support, we develop a new DA and machine learning (ML) hybrid pipeline to estimate and forecast individual future physiological processes by forecasting ODE model parameters. This pipeline overcomes model and DA workflow limitations by stacking a DA-estimated posterior empirical distribution of physiological parameters with longitudinal ML forecasting models. We work within the context of glycemic management in an ICU using EHR data to construct and test a use case. We use synthetic data and real-world clinical data to validate the integrated pipeline and quantify uncertainties.

Neural temporal point processes (NTPPs) are powerful tools for modeling sequences of timestamped events with statistical temporal structure. Density-based NTPPs, in particular, are an interesting opportunity to merge the universal function approximation capability of neural networks with a defined statistical model in a way that has many potential applications. We demonstrate one such application to heartbeat dynamics, a physiologic point process. We specifically apply a lognormal mixture NTPP to compute instantaneous estimates of the mean and standard deviation of beat-to-beat intervals. We compare our results to the state of art (Barbieri et al.) point process model for heartbeat dynamics, which uses a more physiologically rigorous inverse Gaussian model. We find that the NTPP model maintains reasonable accuracy while improving upon robustness to noise.

We develop a mathematical model of Rift Valley Fever integrating mosquito vectors, ruminants, and humans, based on an SEIR-type structure with vertical transmission in vectors. Local data from the Sudanian and especially the Sahelian zones are used to capture the impact of climatic variations on mosquito population dynamics. The mathematical analysis establishes the models positivity, determines the basic reproduction number R0, and demonstrates the local and global stability of the disease-free equilibrium. Sensitivity analysis (PRCC) highlights the most influential parameters, while the stochastic approach using a continuous-time Markov chain confirms the major role of seasonal rainfall. Numerical simulations reveal a peak in animal and human infections around the 9th month, correlating with periods of heavy rainfall. This model provides a relevant tool for surveillance and prevention within a "One Health" approach in Chad.

Patients suffering from sepsis need to be treated with an effective antibiotic therapy within the first hour after sepsis onset to decrease their risk of death. Microbiological data that provide information about the suitability of antibiotic therapies, however, is usually available only after 72 hours. Consequently, the treating physicians need to judge a therapy's effectiveness based on the patients' measured health records and their general health condition. This medical assessment is complex and requires years of experience. In our study, we investigate how statistical modeling can contribute to assessing the effectiveness of antibiotic therapies. To that purpose, we describe the effectiveness of antibiotic therapies by modeling sepsis patients' health conditions using a three-state continuous-time hidden Markov model (ctHMM). In literature, procalcitonin (PCT) and lactate have proven to be helpful for deriving the health condition in this context. The state probabilities obtained by the ctHMM are subsequently used to quantify the effectiveness of antibiotic therapies. To this end, we apply two different approaches, namely (i) averaging of the state probabilities and (ii) a logistic regression model. For (i), we calculate the average of the state probabilities for the state indicating a sepsis-free condition over an antibiotic administration period of 48 hours. For (ii), we use the information about antibiotic susceptibility testings as dependent variable in the logistic regression model; as independent variables, we calculate the difference between state probabilities at the start of antibiotic administration and 48 hours later. With this work, we are able to better understand the relationship between laboratory values, in particular PCT and lactate, and the patients' health condition. We further provide approaches for quantifying the effectiveness. Therefore, our work contributes to developing a clinical decision support system which helps physicians assess the effectiveness of antibiotic therapies in patients with sepsis. Supported by such a system, a physician is able to quickly adjust an ineffective therapy which avoids antibiotic resistances and increases a patient's chance to survive a sepsis.

Background: Tuberculosis remains a persistent infectious disease whose control is complicated by latent infection, delayed treatment, incomplete recovery, reinfection, and continuing transmission from infectious individuals. Although treatment is central to tuberculosis management, it is frequently represented only as a transition parameter in mathematical models rather than as a separate epidemiological state. In this study, treatment was therefore incorporated explicitly as an independent compartment so that its influence on transmission, recovery, disease-induced mortality, and long-term disease persistence could be evaluated. Methods: A deterministic nonlinear compartmental model was formulated by dividing the total population into susceptible, exposed, actively infected, treated, and recovered classes. Reinfection of recovered individuals, progression from latent infection to active disease, movement of infectious individuals into treatment, treatment-associated recovery, natural mortality, and disease-induced mortality were included. Positivity and boundedness of the solutions were examined to establish biological validity. The basic reproduction number, R0, was derived through the next-generation matrix approach. Disease-free and endemic equilibria were determined, and their local and conditional global stability properties were investigated using Jacobian analysis, the Routh-Hurwitz criterion, center manifold theory, Lyapunov functions, and LaSalles invariance principle. Normalized sensitivity indices, Latin hypercube sampling, partial rank correlation coefficients, and numerical simulations were also applied. Results: The disease-free equilibrium was shown to be locally asymptotically stable when ,R0<1 whereas sustained transmission and a unique endemic equilibrium were associated with R0>1. Under the stated reduced-model assumptions, stability of the endemic equilibrium was established. Transmission-related parameters were identified as the strongest positive contributors to disease persistence. In contrast, treatment and recovery parameters were found to reduce the reproduction number and infectious burden. Numerical simulations indicated that stronger treatment implementation and reduced transmission opportunities produced substantial reductions in active tuberculosis cases. Conclusion: Treatment was shown to function as both a clinical pathway and an epidemiological control mechanism. The proposed framework may support the design of treatment-centered strategies for reducing tuberculosis prevalence and preventing long-term endemic persistence.

The initial response of populations to the SARS-CoV-2 virus reduced the incidence of COVID-19 cases. However, this success was shorted lived once most populations relaxed most restrictions, resulting in an increase in infections. This feedback contributed to additional pandemic waves. The temporal unfolding of behavioural changes in populations present a challenge to mathematical models for disease dynamics. Coupled behaviour-disease models with varying levels of complexity accounting for several factors have been used to capture behavioural dynamics and SARS-CoV-2 transmission, with varying results. To study the impact of model complexity on the predictive power of models, here we formulate five coupled behaviour-disease models with varying structure and number of parameters. We fit the models to SARS-CoV-2 infection incidence and stringency of control interventions from five European countries in the first wave, and study how well these fitted models predict the second wave. We show that models with more parameters do not necessarily have a greater ability to explain and predict key features of a pandemic wave. Hence, our results show that a relatively simple coupled behaviour-disease model with important parameters can do an adequate job of providing information about the pandemic wave. Additionally, our findings show that complex models can be country-specific, working better for some countries and poorly for others. We conclude that modellers should not always opt for the most complicated possible models, if the data do not support their use.

Infectious disease transmission evolves as a dynamic process shaped by biological mechanisms, population behavior, and intervention policies, yet public health responses are often driven by lagging indicators. Accurate short- and long-term disease forecasting is essential for the timely deployment of intervention strategies, healthcare capacity planning, and uncertainty-aware, risk-informed decision-making. To address this challenge, three broad classes of forecasting models have traditionally been used: statistical, machine learning, and mechanistic approaches. However, each of these modeling paradigms faces fundamental limitations. In particular, traditional statistical models often lack the flexibility needed to capture complex disease dynamics, machine learning approaches require large, high-quality data streams, and mechanistic models are notoriously difficult to calibrate. To overcome these challenges, we propose a novel physics-informed machine learning (PIML) framework for forecasting infectious disease dynamics. Our approach simultaneously forecasts new case and hospitalization counts, along with other key epidemiological quantities such as the time-varying reproduction number. This is achieved through the design of a machine learning model and estimation strategy regularized by a system of differential equations that encode disease dynamics of the SIHR model, thereby bridging the gap between purely data-driven and mechanistic models. We demonstrate the proposed methodology through in-depth numerical studies and an application to COVID-19 data collected in the state of South Carolina.

Human Papillomavirus (HPV) infection among men who have sex with men (MSM) has become a significant public health concern, particularly in countries where male vaccination is unavailable. Given the high susceptibility of MSM to HPV and anal cancer, and the unavailability of HPV vaccination for males in low- and middle-income countries (LMICs), there is a need to identify alternative interventions for reducing disease transmission and burden in this population. The novel mathematical model presented in this article couples smoking behavior dynamics with HPV transmission and anal cancer progression among MSM. Smoking reduction is introduced as an intervention to assess its effects on disease transmission and burden. The basic reproduction number (R0) is derived using the next-generation matrix method, and a global sensitivity analysis is performed using partial rank correlation coefficients (PRCC) to identify the influence of model parameters on RR0. Further, the theoretical analysis of the model reveals a backward bifurcation, implying that RR0 < 1 is necessary but not sufficient to eradicate the disease. The study finds that smoking reduction among MSM reduces HPV infection and anal cancer burden relative to baseline projections without intervention. The joint effect of smoking reduction and vaccination shows that the critical vaccination coverage needed to achieve RR0 <1 decreases as the level of smoking reduction increases. A similar outcome is observed for contact reduction. These findings highlight the importance of concurrent interventions, which can significantly curtail the spread of HPV and reduce disease burden in both the high-risk group and the general population.

Auxins are a family of plant hormones involved in various processes across plant tissues and species. The Nuclear Auxin Pathway (NAP) consists of interacting transcription factors (ARFs) and repressors (Aux/IAAs), which govern an individual cells response to changes in auxin concentration. These components are present in all land plants, and many species possess multiple copies of each signalling component. We present a general framework for ODE-based models of NAP submodules with the flexibility to model the promotion and repression of target genes by any combination of transcriptional regulators. We analyse published data and show that auxin treatment in Arabidopsis thaliana roots triggers a range of characteristically distinct temporal response profiles--for both target genes and the signalling components themselves. Using our modelling framework, we recapitulate aspects of this behaviour by presenting examples of real and theoretical NAP subnetworks, and by analysing the effect that these network dynamics have on auxin-mediated transcriptional responses. This work demonstrates the utility of our modelling framework as a general-purpose tool for understanding the function of certain protein-protein and protein-DNA interactions through their effects on the NAP. This exploration of the rich dynamics of more complex signalling pathways promises to advance our understanding of the NAP.

This study emphasizes the potential of computational techniques in cancer risk assessment, lighting opportunities for specific and data-driven healthcare solutions. This study examines the use of artificial intelligence (AI), machine learning (ML), and deep learning (DL) approaches to improve cancer risk assessment using a Kaggle dataset. The study uses Java-based ML software to create and evaluate multiple predictive models, taking advantage of its powerful libraries and frameworks for processing and analyzing cancer risk indicators. This work analyzes model performance using 10-fold cross-validation, resulting in reliable generalization and accuracy estimates. Several classification techniques, such as Random Forest (RF) logistic regression (LR), decision trees (DT), Naive Bayes (NB), and Multi-layer perceptron (MLP), are used to assess their efficacy in predicting risk levels for various cancer types. To measure classification effectiveness, key performance metrics such as accuracy, precision, recall, and F1 score are produced, in addition to multi-class confusion matrices. The results show that the RF model is the best classifier for classification, with accuracy of 99.85%, F-measure of 99.80%, precision of 99.80%, and sensitivity of 99.90%. These findings demonstrate the model's ability to effectively estimate cancer risk levels among individuals. of cancer risk estimations, allowing for earlier discovery and more effective medical care.

Climate change represents a challenge to food security by interfering with the environmental conditions needed for productive plant growth. While technology can be used for partial mitigation, access to technology is inequitable. Low-cost microcontrollers, such as the ESP32, have recently lowered the barrier for entry into prototyping smart devices. ESP32s equipped with capacitive moisture sensors have been suggested for low-cost smart plant watering systems. However, measuring moisture in soil is complex, potentially destructive, and requires careful calibration in order to characterise the response curve mapping soil water content to sensor measurements. Here, we developed a Bayesian method for estimating the inverse response curve from capacitive moisture sensor data, known water doses, and prior uncertainty, bypassing the need for destructive gravimetry. This method constitutes the core calibration module of the open-source OpenHCult software system for low-cost horticultural automation.

In this paper, we propose a new model to estimate the impact of an intervention on human hosts of a vector-borne infection, such as dengue, which occurs in yearly outbreaks of different magnitudes. The model applies to these outbreaks and, in fact, is independent of their intensity, that is, it does not require the steady-state assumption. The model takes as input the officially reported age-dependent number of cases of a vector-borne infection. It is deterministic and does not account for stochasticity. Our objective is to estimate the impact of the intervention (the efficacy), and we rely on the observed fact that the age distribution of the proportion of cases of the infections transmitted by the same vector is independent of both the intensity of transmission and the geographic area studied, at least for Brazilian regions. This finding is highlighted in the main text and forms the basis of our calculations. A hypothetical intervention is simulated using a dengue vaccine, which allows the determination of the optimal strategy for a vaccination campaign.

This brief work discusses potential superspreading events that may occur during the World Cup in Mexico. The study is particularly focused on the city of Guadalajara due to a large recent outbreak in January and February and insufficient vaccine coverage prior to 2026. Keywords: Superspreading; measles outbreak; branching process; individual reproduction number; World Cup

Superspreading events (SSEs) produce extreme, rare bursts of disease transmission that standard compartment models, which assume population homogeneity, fail to capture. This inability to model heterogeneity in transmission rates can result in biased estimates of transmissivity. To address this limitation, we present a modular framework that treats SSEs as statistical outliers in case count time series and incorporates them into SIR-type models via pulse terms that transfer SSE cases directly from susceptible to infected compartments. This separation isolates anomalous SSE-driven transmission from background spread, which reduces bias when estimating mean transmission rates. We validate the approach on synthetic data generated by a stochastic model with embedded SSEs, demonstrating accurate recovery of the true non-SSE transmission parameter. We then apply the method to COVID-19 outbreaks in Hong Kong and the German district of Gutersloh, showing improved model fits and more robust estimates of background transmissivity both for a period with constant transmission and for a period with temporally structured NPI-driven heterogeneities. The framework's interchangeable outlier-detection, compartment, and SSE modules make it adaptable to diverse diseases and data contexts.

The photosynthesis-transpiration-stomatal conductance (An-E-gs) model framework is widely used for estimating photosynthesis, transpiration, and stomatal conductance in plants. The model equations are solved by numerical iteration, and the converged model values are deemed the solution. However, there has been no general guarantee that the iterative procedure converges to a solution or that the procedure leads to convergence. Building on the recent proof of the existence of a unique set of solutions, we herewith propose a numerical algorithm that is guaranteed to converge to the solution for the An-E-gs model framework. We first analytically prove that the proposed algorithm necessarily converges to a solution. We then demonstrate the convergence across contrasting combinations of leaf temperature, relative humidity, light, atmospheric CO2, and wind speed. We further demonstrate rapid convergence with the algorithm: no more than ca. 10 iterations for approximately 10-3 mol CO2 m-2 s-1 precision in net photosynthesis and no more than ca. 20 iterations for 10-7 mol CO2 m-2 s-1 precision. By guaranteeing convergence to the solution, this algorithm eliminates concerns about nonconvergence in leaf gas-exchange calculations and is expected to serve as a robust foundation for a range of studies from leaf-level gas exchange to global-scale carbon and water cycle dynamics.

Cancer prevalence is distributed unevenly across regions and caused by the interaction of multiple risk factors. Previous studies focused on the use of global modeling techniques to predict cancer at the county level that overlooks important spatial differences. This study aims to develop geographically weighted machine learning models to predict cancer prevalence at the census tract level in the United States and identify local determinants of cancer burden. First, a scoping review was conducted to find a list of measurable drivers of cancer in the United States. Using this list, the data of these variables for 84415 census tracts were obtained from the Center for Disease Control and Prevention PLACES dataset and other publicly accessible resources. Then, several predictive models, including Ordinary Least Squares (OLS) and Geographically Weighted Regression (GWR), as well as Random Forest, XGBoost, and Deep Neural Network and their geographically weighted counterparts, were developed and compared using the Coefficient of Determination, Root Mean Square Error, and Absolute Error. Results presented that geographically weighted models outperformed other methods, and geographically weighted XGBoost achieved the strongest and most consistent overall performance with pseudo-R2 ranging between 0.89 and 0.98. Feature importance analysis of this model illustrated that most important cancer drivers changed location by location. Aged people, racial composition, preventative behaviors, and metabolic conditions such as diabetes, hypertension, and high cholesterol were determined as influential predictors, although their relative importance varied across regions. These findings revealed the value of localized models at a small geographic scale to identify regional cancer risk patterns and help the allocation of proper resources to hotspot areas. Keywords: Cancer prevalence, Census tracts, geographically weighted machine learning models, Deep neural network, XGBoost, Random Forest, Ordinary Least Squares, risk factor, determinant

During the COVID-19 pandemic, limited testing capacity and reporting delays complicated epidemic surveillance and decision-making in Mexico. We calibrated \textit{covidestim}, a Bayesian nowcasting model, to estimate the total SARS-CoV-2 infections from reported cases and deaths using Mexican surveillance data. Disease-progression distribution priors were calibrated using Mexico City records and validated through comparisons with national seroprevalence surveys, hospitalization data, and annual reported severe-case rates across all states. Using the reconstructed estimates of active infections, we implemented an event-based risk framework that quantifies the probability of encountering at least one infectious individual in gatherings of different sizes. This probability was subsequently translated into a four-level epidemiological traffic-light indicator and computed at both state and municipality levels. The resulting estimates revealed substantial spatial heterogeneity that is obscured by state-level aggregation, particularly in states with marked differences between urban and rural municipalities. To evaluate consistency with public-health indicators, we compared the proposed risk classification with the official Mexican epidemiological traffic-light system, considering interpretable gathering sizes relevant to public-health decision making. Weekly reports derived from this framework were delivered to policymakers in the State of Queretaro in Mexico, as an anticipation tool for school reopening and public-space management. This demonstrates that this Bayesian reconstruction of infections combined with event-based risk metrics can provide an interpretable and generalizable municipality-level complement to routine surveillance systems, particularly in regions with limited testing capacity and heterogeneous local transmission dynamics.

Malaria is still one of the most dangerous causes of morbidity and mortality in tropical and subtropical regions even though it has been actively combated for many years. In 2023, there were approximately 263 million malaria cases and 597,000 deaths from this disease on a global scale, with sub-Saharan Africa being the region most affected by it [24]. Climate factors affect mosquito biology, including their abundance, survival, and biting rates, as well as parasite development, while human awareness plays a crucial role in adopting preventive measures and effective treatments. Despite the progress in both climate- and awareness-based malaria modelings, few studies integrate these factors in one comprehensive model that involves the detailed mechanisms of transmission processes. The current study develops a deterministic climate-driven SEAIR-SEI malaria transmission model that includes the impact of temperature, rainfall, and humidity on mosquito biology and endogenous community awareness. The model was proven to be well-posed by showing the positivity and boundedness of its solution and through the demonstration of the existence and uniqueness of its solution. The malaria-free equilibrium was determined, and the basic reproduction number was calculated using the next-generation matrix method. The model underwent local and global stability analyses to characterise the diseases persistence in the population. Additionally, a normalized forward sensitivity analysis was conducted, revealing the mosquito biting rate as the key force driving malaria transmission. Four time-dependent malaria interventions, namely, long-lasting insecticidal nets, community awareness campaigns, indoor residual spraying, and prompt treatment, were included in the model through optimal control theory and analysed using Pontryagins Maximum Principle. The numerical results for the optimal control problem showed that employing all four interventions leads to the best outcome by decreasing the objective functional value by 88.17%, reducing the total number of infected humans by 92.49%, and minimizing the total number of infectious mosquitoes by 93.87%. Interestingly, combining two interventions, indoor residual spraying, and prompt treatment, also yielded nearly optimal results. Therefore, the designed control strategy can serve as an efficient and affordable framework for malaria control in sub-Saharan Africa.

Animals use long-range signals, such as hormones and neural signals, to coordinate the actions of distant organs. There is no precise, quantitative framework that explains the problems these control systems must solve and thus predicts their behavior under varied conditions. We consider this problem in the context of blood glucose regulation by the hormone insulin, the failure of which produces diabetes. We show that existing mathematical models of glucose regulation admit equivalent control strategies with no hormones at all, and thus cannot explain the need for hormonal regulation. We therefore introduce a minimal model of inter-organ variations in local glucose, and show that control strategies based on local glucose measurements face severe trade-offs between different control objectives. In contrast, we show that hormonal control signals from the pancreas can overcome these limitations. By exposing the benefits of hormonal control, our work paves the way to a detailed understanding of physiological design principles, with possible implications for the engineering of an artificial pancreas.