Bulletin of Mathematical Biology

Gliomas, a type of brain tumor, have become increasingly important in oncology as they are difficult to treat due to their location deep in the brain. While some research has been done on spatiotemporal prediction of future glioma growth--something that can aid in surgical resections of gliomas once a patient has been diagnosed--modeling efforts for pre-diagnostic gliomas remain limited. The lack of retrospective, pre-diagnostic data makes this a challenging task; yet, pre-symptomatic serum glucose levels in patients have been shown to have a relationship with the emergence of gliomas, motivating this area of research. In 2015, Sturrock et al. presented an ordinary differential equation model of pre-diagnostic glioma growth that describes glioma-glucose-immune interactions. This report reproduces the major findings of Sturrock et al., revising their model to incorporate more biological phenomena--namely vascularization and mutational burden--testing additional medically relevant patient scenarios, and providing an extended discussion on the implications of the model. In-silico simulations performed in this report provide further insight into models describing glioma-glucose-immune interactions, and how they can be expanded to incorporate physiologically relevant features. Future work is necessary to refine model parameters and validate predictions with the limited, albeit steadily growing, amount of longitudinal patient data.

This paper focused on determining the asymptotical stability of the new model which establishes that a disease-free equilibrium state exists and is locally asymptotically stable when the basic reproduction number 0 [&le;] R0 < 1 and the following threshold conditions (0 [&le;] R1 < 1, 0 [&le;] R2 < 1, 0 [&le;] R3 < 1, 0 [&le;] R4 < 1 and 0 [&le;] R5 < 1, 0 [&le;] R6 < 1, 0 [&le;] R7 < 1, 0 [&le;] R8 < 1) are satisfied. Results from the model analysis shows that the proportion of infected juvenile and adult sub-population in the presence of High Active Anteritroviral Therapy (HAART) drastically reduce to a zero (R0 = 0) as compared to the infected age-structured population when treatment rate is low and the net transmission rate is near zero.

We present a multiscale model of HIV that couples within-host viral dynamics with population-level transmission to capture the interplay between individual infection and epidemic spread. The model is structured by treatment age, allowing viral load to influence both infectiousness and progression to AIDS. The model is fitted using both clinical data (viral load and target cell counts of treated individuals) and epidemiological data (HIV incidence, diagnoses, and AIDS classifications). We derive the basic reproduction number and establish threshold conditions for the existence and stability of disease-free and endemic equilibria. Structural identifiability is assessed via input-output equations, showing that the multiscale model is identifiable when the initial number of treated individuals is set to zero and the AIDS death rate is assumed known. Parameters are estimated sequentially: within-host parameters are obtained via nonlinear mixed-effects modeling of clinical data, followed by estimation of population-level parameters using CDC surveillance data. Numerical simulations are performed using a finite-difference scheme with Picard iteration, and practical identifiability is evaluated via Monte Carlo simulations across varying noise levels. Results indicate that current strategies are unlikely to meet the 2030 targets, while increasing diagnosis rates and reducing transmission from diagnosed individuals could significantly alter epidemic trajectories. These findings highlight the importance of multiscale modeling and identifiability in informing effective HIV intervention strategies.

The Human Immunodeficiency Virus (HIV) is one of the most common chronic infectious diseases of humans. Increasing the expected lifetime of the patients depends on the use of optimal antiretroviral therapies. The emergence of the drug-resistant strains may decrease the effects of treatments and lead to Acquired Immune Deficiency Syndrome (AIDS) even if the existence of antiretroviral therapy. Investigation of the genotype-phenotype relations is a crucial process to optimize the therapy protocols of the patients. Here we propose a mathematical modelling framework to address the effect of initial strains, initiation timing and adherence levels of nucleotide reverse transcriptase inhibitors (NRTI) on the emergence of a possible AIDS phase. For the first time, we have combined the existing Stanford HIV drug resistance data with a multi-strain within-host ordinary differential equation (ODE) model to track the dynamics of most common NRTI resistant strains. Regardless of the drug choice, the late initiation and poor adherence levels to the NRTI therapy increase the probability of the emergence of the AIDS phase. Overall, the 3TC, D4T-AZT and TDF-D4T drug combinations provide higher success rates. The results are in line with genotype-phenotype data and pharmacokinetic parameters of the NRTI inhibitors, but we show the heavy influence of neighbour viral strains of the initial ones has a considerable effect on the success/failure rates. Improving multiscale models can contribute to understanding the disease progression and treatment options.

Paclitaxel (taxol) is a commonly used chemotherapeutic that stabilizes microtubules to inhibit chromosome segregation during mitosis. Although it is used effectively to treat a wide variety of solid tumour types, it also impairs healthy cell proliferation leading to severe dose-limiting toxicities. Newer anti-mitotic drugs have been developed but these have so far failed to offer the same clinical benefit as paclitaxel, which begs the question of why this drug targets tumour cells so effectively? Here we develop a mathematical model of paclitaxel penetration and retention within 3D tumour environments following periodic drug-on/drug-off regimes typically used in the clinic. Our model suggests that during the drug-free periods, dense poorly-perfused tissue can retain paclitaxel for much longer than well-perfused tissue. This is due to paclitaxels ability to bind strongly to microtubules, which causes slower drug-release from densely packed tissue. Assuming that tumour cells are generally dense and less perfused than proliferative healthy tissues, this simple model suggests that tumour-selective killing could be achieved later in each chemotherapy cycle when the drug has otherwise cleared the healthy cell compartments. We use our model to optimize dosing regimens to allow paclitaxel to selectively kill tumour spheroids of different size, whilst sparing well-perfused healthy cells. Together, our model suggests paclitaxel could target key distinguishing features of many solid tumours: their size, 3D geometry and perfusion status. It is important to validate these predictions in cell models because, if correct, they could be harnessed to optimize paclitaxel use, to predict and enhance tumour responsiveness, and to develop newer drugs that are preferentially retained for longer within solid tumours.

We propose a model for in vitro glioblastoma multiforme brain tumor growth, which uses density-dependent diffusion to capture both the proliferative and migratory behavior of the cancer cells. The model is compared to well-known experimental data and is analyzed for the existence of traveling wave solutions.

Recently, glioblastoma tumors were shown to form tumor microtubes, which are thin long protrusions that help the tumor grow and spread. Follow-up experiments were conducted on mice in order to test what impact the tumor microtubes have on tumor regrowth after partial removal of a tumor region. The surgery was performed in isolation and along with growth-inhibiting treatments such as a tumor microtube inhibiting treatment and an anti-inflammatory treatment. Here, we propose a partial differential equation model applicable to describe the microtube driven regrowth of the cancer in the lesion. We find that the model is able to replicate the main trends seen in the experiments such as fast regrowth, larger cancer density in the lesion, and further spread into healthy tissue. The model indicates that the dominant mechanisms of re-growth are growth-inducing wound healing mechanisms. The tumor microtubes accelerate this process as the microtubes provide orientational guidance from the untreated tissue into the lesion.

This paper continues the study of a model which was introduced in earlier work by the authors to study spontaneous and induced evolution to drug resistance under chemotherapy. The model is fit to existing experimental data, and is then validated on additional data that had not been used when fitting. In addition, an optimal control problem is studied numerically.

We propose a numerical method to uniformly handle the random genetic drift model for pure drift with or without natural selection and mutation. For pure drift and natural selection case, the Dirac {delta} singularity will develop at two boundary ends and the mass lumped at the two ends stands for the fixation probability. For the one-way mutation case, known as Mullers ratchet, the accumulation of deleterious mutations leads to the loss of the fittest gene, the Dirac {delta} singularity will spike only at one boundary end, which stands for the fixation of the deleterious gene and loss of the fittest one. For two-way mutation case, the singularity with negative power law may emerge near boundary points. We first rewrite the original model on the probability density function (PDF) to one with respect to the cumulative distribution function (CDF). Dirac {delta} singularity of the PDF becomes the discontinuity of the CDF. Then we establish a revised finite difference method(rFDM), which keeps the total probability, is positivity-preserving and unconditionally stable. For pure drift, the scheme also keeps the conservation of expectation. It can catch the discontinuous jump of the CDF, then predicts accurately the fixation probability for pure drift with or without natural selection and one-way mutation. For two-way mutation case, it can catch the power law of the singularity. The numerical results show the effectiveness of the scheme. We also compare rFDM with a standard finite difference method(sFDM). We find that, for pure drift problem, sFDM fails to keeps the conservation of expectation and can not predict the fixation probability, because there is artificial mutation introduced near two boundary ends.

Determining optimal antibiotic dosing strategies is complex. Clinically, some antibiotics work best in continuous low doses, while others require high repeated pulses. However, a rational understanding of the best approach depending on the specific pairing of antibiotic and bacterial species remains unclear. Using mathematical models, we analyze bacterial populations under two strategies - constant concentration and repeated dosing - given fixed pharmacodynamic and pharmacokinetic properties. Our results reveal that the shape of the dose-response curve, which measures bacterial net growth rate against antibiotic concentration, is crucial. Specifically, its concavity determines the best strategy. In cases where the curve exhibits multiple concavities, additional factors such as tolerable dosing range influence the regimen. These findings challenge the universal application of "hit hard and hit early," as some recommended schedules include lower, constant doses. This work contributes to the literature on rational antibiotic prescription, aiming to minimize antibiotic use and combat antimicrobial resistance.

HIV can be transmitted from a HIV infected mother to her child during pregnancy, delivery, or breastfeeding. According to NSDCC 2023, Kenya has estimated PMTCT coverage of 89.56% and PMTCT transmission rate of 8.6%. Even though there has been strides to address PMTCT, there is need to gear up approaches in addressing MTCT in order to significantly advance elimination. This research formulates a mathematical model to represent the dynamics of MTCT. Equilibrium points of the model are computed and the stability of HIV-free point is investigated. The numerical results show that a 50% decrease in maternal HIV transmission lowers infant infection rates by about 17.7%, whereas the same reduction in infant transmission decreases infections by nearly 39%, highlighting the greater sensitivity of infant transmission rates to direct interventions. While combination of strategies achieves the highest HIV minimization rates of up to 99.89% on infants, ART adherence alone significantly reduces transmission, particularly on infants (91.42%) while use of post-exposure prophylaxis (PEP) shows limited effectiveness when used alone(39.65%), suggesting that it should be complemented with other strategies for optimal impact. These findings emphasize the critical need for integrated interventions, where combining multiple prevention methods yields the best outcomes in reducing HIV infections on infants and moving closer to the elimination of pediatric HIV. These findings align with global recommendations from World Health Organization (WHO). This research can be used by the ministry of health to inform policy as well as recreated for other maternal infections. Author summaryHIV can be transmitted from a mother to her child during pregnancy, delivery, or breastfeeding. In Kenya, despite efforts to prevent mother-to-child transmission (PMTCT), HIV transmission rates remain a concern. In this study, we developed a mathematical model to understand how HIV spreads from mothers to infants and to evaluate the effectiveness of different prevention strategies. Our findings highlight that reducing HIV transmission in mothers lowers infant infection rates, but direct interventions for infants, such as early antiretroviral therapy (ART) and post-exposure prophylaxis (PEP), have an even greater impact. A combination of strategies--ensuring mothers adhere to ART, providing PEP for infants, and promoting safe breastfeeding practices--was found to reduce HIV infections in infants by up to 99.89%. These results support the need for integrated approaches to HIV prevention. Policymakers and healthcare providers can use this research to refine HIV prevention programs, ensuring better maternal and infant health outcomes. Our model can also be adapted for other maternal infections, contributing to broader public health efforts in disease prevention.

The authors have withdrawn their manuscript due to the discovery of fundamental issues with the implementation of the ODE/PDE model, whose output was shown in Figure 5 of the original version of this paper, rendering the results and conclusion unreliable. Therefore, the authors do not wish this work to be cited as reference for the project. If you have any questions, please contact the corresponding author.

Oncolytic virotherapy (OVT) represents an innovative and promising therapeutic method for cancer treatment. This approach involves the introduction of oncolytic viruses into the patient, which are engineered to selectively target and lyse tumor cells. Based on previous mathematical modelling of oncolytic viruses, we consider a mathematical model that describes the intricate interactions between the oncolytic virus, cancer cell populations, and the immune system. Our study includes a detailed qualitative and quantitative analysis of the model to explain why, despite their promise, oncolytic viruses alone rarely lead to complete and lasting regression of established tumors in vivo. We use parameter sensitivity analysis to support our findings. Furthermore, we consider the spatial version of the model in the form of a system of reaction-diffusion equations. A travelling wave analysis shows an unexpected phenomenon. The solution components do not necessarily evolve into a single traveling front; rather, they develop into stacked fronts, where each front propagates at a different speed. We give explicit formulas for these different invasion speeds, confirm those through numerical simulations, and discuss their significance for OVT.

The COVID-19 Vaccines Global Access (COVAX) is an initiative led by the World Health Organization (WHO) and other partners that aims for an equitable access of COVID-19 vaccines. Despite a potential heterogeneous disease burden across space, countries receiving allotments of vaccines via COVAX may want to follow WHOs allocation rule and distribute vaccines to their jurisdictions based on the jurisdictions relative population size. Utilizing economic-epidemiological modeling, we benchmark the performance of this ad hoc allocation rule by comparing it to the rule that minimizes the economic damages and expenditures over time, including a penalty cost representing the social costs of deviating from the ad hoc allocation. Under different levels of vaccine scarcity and different demographic characteristics, we consider scenarios where length of immunity and compliance to travel restrictions vary, and consider the robustness of the rules when assumptions regarding these factors are incorrect. The benefits from deviating are especially high when immunity is permanent, when there is compliance to travel restrictions, when the supply of vaccine is low, and when there is heterogeneity in demographic characteristics. Interestingly, a lack of compliance to travel restrictions pushes the optimal allocations of vaccine towards the ad hoc and improves the relative robustness of the ad hoc rule, as the mixing of the populations reduces the spatial heterogeneity in disease burden. JEL ClassificationC61, H12, H84, I18, Q54

We model interactions between cancer cells and free virus during oncolytic viral therapy. One of our main goals is to identify parameter regions that yield treatment failure or success. We show that the tumor size under therapy at a particular time is less than the size without therapy. Our analysis shows there are two thresholds for the horizontal transmission rate: a "Control threshold", the threshold above which treatment is efficient, and an "optimum threshold", the threshold beyond which infection prevalence reaches 100% and the tumor shrinks to its smallest size. Moreover, we explain how changes in the virulence level of the free virus alter the optimum threshold and the minimum tumor size. We identify a threshold for the virulence level of the virus and show how this threshold depends on the timescale of virus dynamics. Our results suggest that when the timescale of virus dynamics is fast, the administration of a more virulent virus leads to more tumor reduction. Conversely, when the viral timescale is slow, a higher virulence will have drawbacks on the results, such as high amplitude oscillations. Furthermore, our numerical observation depicts fast and slow dynamics. Our numerical simulations indicate there exists a two-dimensional globally attracting surface that includes the unstable manifold of the interior equilibrium. All solutions with positive initial conditions rapidly approach this two-dimensional attracting surface. In contrast, the trajectories on the attracting surface slowly tend to the periodic solution. HighlightsO_LIThe assumption that the viral load is in a quasi-steady state is relaxed, and infected cells are assumed to be mitotic. C_LIO_LIOur model strongly suggests that the tumor size is always reduced by therapy. C_LIO_LIWe identify minimum tumor size, control threshold, and optimum threshold of the therapy. C_LIO_LIOur analysis shows optimal virulence level of oncolytic virus depends on the time scale of virus dynamics. C_LIO_LIWhen virus dynamic is slow, highly virulent virus causes long remission before relapse. C_LI

Tumors consist of heterogeneous phenotypic cells, such as normoxic cells, which are highly proliferative, and hypoxic cells, which are less proliferative. Their phenotypic switching depends on tumor microenvironmental factors, such as oxygen and nutrient concentrations supplied by local blood vessels. However, during ongoing angiogenesis, the process of sprouting new blood vessels at the tumor site from pre-existing blood vessels, and how this phenotypic switching affects and impacts tumor growth, remains poorly understood. In this article, we formulate a mathematical model to elucidate the crosstalk between vasculature and tumor cellular heterogeneity during tumor progression. The model results show a strong agreement with the experimental data. Our simulation results demonstrate that ongoing angiogenesis increases tumor growth rate. In addition, we observe that the influence of hypoxic cells on phenotypic switching from normoxic to hypoxic is more pronounced than their influence on the transition from hypoxic to normoxic. Furthermore, we perform a global sensitivity analysis using the Sobol's method to assess the importance of the model's parameters. It highlights that the volume at which blood vessels attain half-maximal rate has the maximum effect on the model.

We introduce a general phenomenological framework for understanding how phenotypic plasticity gives rise to drug persisters. These persisters, often quiescent but sometimes which again return to cycling, survive in the presence of treatment and eventually can lead to mutants with true resistance. Our framework builds on recent experimental observations regarding variations between and among single-cell clones and the possible role of the drug itself in enhancing the survival strategy. Predictions of our approach include the existence of an optimum drug concentration as well as an optimum drug holiday schedule to minimize the persistence-based threat.

BackgroundTo control leishmaniasis, chemotherapy drugs are currently under development. However, these drugs often exhibit poor efficacy and are associated with toxicity, adverse effects, and drug resistance. At present, no specific drug is available for the treatment of leishmaniasis. Meanwhile, vaccine research is ongoing. Recent studies have analysed some experimental vaccines using mathematical models. AimIn previous work, drug targeting was focused on the entire human body rather than specifically addressing infected macrophages and parasites. In our current approach, we aim to eliminate infected macrophages and parasites through nano-drug design. Specifically, we utilise two types of nanoparticles: iron oxide and citric acid-coated iron oxide. Moving forward, we plan to advance this strategy using mathematical modelling of macrophage-parasite interactions. MethodsWe design PDE-based models of macrophages and parasites, incorporating cytokine dynamics, to support nano-drug development. Drug efficacy is estimated using posterior distributions to analyse phenotypic fluctuations of macrophages and parasites during the design phase. We investigate implicit and semi-implicit treatment schemes, focusing on energy decay properties. To model drug flow during treatment, we introduce a three-phase moving boundary problem. Comparative analyses are conducted to evaluate macrophage and parasite behaviour with and without treatment. Finally, the entire framework is implemented within a virtual lab environment. ResultsThe results show that the nano-drug exhibits better efficacy compared to combined drug doses. We analysed and compared two types of nano-drug particles: iron oxide and citric acid-coated iron oxide. We discuss how the drug effectively targets and eliminates infected macrophages and parasites. ConclusionOur models results and simulations will support researchers conducting further studies in nano-drug design for leishmaniasis. These simulations are performed within a virtual lab environment.

One of the most difficult challenges in cancer therapy is the emergence of drug resistance within tumors. Sometimes drug resistance can emerge as the result of mutations and Darwinian selection. However, recently another phenomenon has been discovered, in which tumor cells switch back and forth between drug-sensitive and pre-resistant states. Upon exposure to the drug, sensitive cells die off, and pre-resistant cells become locked in to a state of permanent drug resistance. In this paper, we explore the implications of this transient state switching for therapy scheduling. We propose a model to describe the phenomenon and estimate parameters from experimental melanoma data. We then compare the performance of continuous and alternating drug schedules, and use sensitivity analysis to explore how different conditions affect the efficacy of each schedule. We find that for our estimated parameters, a continuous therapy schedule is optimal. However we also find that an alternating schedule can be optimal for other, hypothetical parameter sets, depending on the difference in growth rate between pre-drug and post-drug cells, the delay between exposure to the drug and emergence of resistance, and the rate at which pre-resistant cells become resistant relative to the rate at which they switch back to the sensitive state.

Foot-and-mouth disease (FMD) is a highly contagious disease that spreads among cloven-hoofed animals. Although not deadly, FMD can cause major delays in meat and dairy production. One major concern is that the foot-and-mouth disease virus (FMDV) can persist in African buffalo hosts as natural reservoirs for long periods of time, causing the pathogens to reemerge in susceptible populations. In this paper, we present a novel immuno-epidemiological model of FMD in the African buffalo host populations. Upon infection, the hosts can undergo two phases, namely the acute and the carrier stages. In our model, we divide the infectious population based upon these two stages so that we can dynamically capture the immunological characteristics of both phases of the disease to better understand the carriers role in disease transmission. We first define the within-host viral-immune kinetics dependent epidemiological basic reproduction number[R] 0 and show that it is a threshold condition for the local stability of the disease-free equilibrium and existence of the endemic equilibrium. We also analytically show that the system always displays forward bifurcation with respect to between-host epidemic parameters. Later, by using a sensitivity analysis (SA) approach developed for multi-scale models, we assess the impact of the acute infection and carrier phase immunological parameters on[R] 0. Interestingly, our numerical results show that the within-carrier infected host immune kinetics parameters and the susceptible individual recruitment rates play significant roles in disease persistence, which are consistent with experimental and field studies.