Inflammation is a physiological process aimed to protect the organism in various diseases and injuries. This work presents a generic inflammation model based on the reaction-diffusion equations for the concentrations of uninflamed cells, inflamed cells, immune cells and the inflammatory cytokines. The analysis of the model shows the existence of three different regimes of inflammation progression depending on the value of a parameter R called the inflammation number. If R>1, then inflammation propagates in cell culture or tissue as a reaction-diffusion wave due to diffusion of inflammatory cytokines produced by inflamed cells. If 0<R<1, then inflammation vanishes and the system converges to the stable inflammation-free equilibrium. Finally if R<0, inflammation also propagates as a reaction-diffusion wave, but the mechanism of propagation is different, it is determined by positive feedback between inflammation and immune response. From the biological point of view, these three regimes correspond to acute inflammation resolved due to the immune response, to the disappearance of inflammatory reaction, and to an auto-immune inflammatory reaction or cytokine storm. We focus on finding the wave speed and other characteristics of inflammation progression by analytical and numerical methods, in order to deduce a qualitative understanding of various inflammatory reactions.

We develop and analyze a mathematical model of oncolytic virotherapy in the treatment of melanoma. We begin with a special, local case of the model, in which we consider the dynamics of the tumour cells in the presence of an oncolytic virus at the primary tumour site. We then consider the more general regional model, in which we incorporate a linear network of lymph nodes through which the tumour cells and the oncolytic virus may spread. The modelling also considers the impact of hypoxia on the disease dynamics. The modelling takes into account both the effects of hypoxia on tumour growth and spreading, as well as the impact of hypoxia on oncolytic virotherapy as a treatment modality. We find that oxygen-rich environments are favourable for the use of adenoviruses as oncolytic agents, potentially suggesting the use of complementary external oxygenation as a key aspect of treatment. Furthermore, the delicate balance between a virus' infection capabilities and its oncolytic capabilities should be considered when engineering an oncolytic virus. If the virus is too potent at killing tumour cells while not being sufficiently effective at infecting them, the infected tumour cells are destroyed faster than they are able to infect additional tumour cells, leading less favourable clinical results. Numerical simulations are performed in order to support the analytic results and to further investigate the impact of various parameters on the outcomes of treatment. Our modelling provides further evidence indicating the importance of three key factors in treatment outcomes: tumour microenvironment oxygen concentration, viral infection rates, and viral oncolysis rates. The numerical results also provide some estimates on these key model parameters which may be useful in the engineering of oncolytic adenoviruses.

Atherosclerotic plaques form in artery walls due to a chronic inflammatory response driven by lipid accumulation. A key component of the inflammatory response is the interaction between monocyte-derived macrophages and extracellular lipid. Although concentrations of low-density lipoprotein (LDL) and high-density lipoprotein (HDL) particles in the blood are known to affect plaque progression, their impact on the lipid load of plaque macrophages remains unexplored. In this paper, we develop a lipid-structured mathematical model to investigate the impact of blood LDL/HDL levels on plaque composition, and lipid distribution in plaque macrophages. A reduced subsystem, derived by summing the equations of the full model, describes the dynamics of biophysical quantities relating to plaque composition (e.g. total number of macrophages, total amount of intracellular lipid). We also derive a continuum approximation of the model to facilitate analysis of the macrophage lipid distribution. The results, which include time-dependent numerical solutions and asymptotic analysis of the unique steady state solution, indicate that plaque lipid content is sensitive to the influx of LDL relative to HDL capacity. The macrophage lipid distribution evolves in a wave-like manner towards an equilibrium profile which may be monotone decreasing, quasi-uniform or unimodal, attaining its maximum value at a non-zero lipid level. Our model also reveals that macrophage uptake may be severely impaired by lipid accumulation. We conclude that lipid accumulation in plaque macrophages may serve as a partial explanation for the defective uptake of apoptotic cells (efferocytosis) often reported in atherosclerotic plaques.

The retinal vascular network supplies perfusion to vital visual structures, including retinal ganglion cells responsible for vision. Impairments in retinal blood flow and oxygenation are involved in the progression of many ocular diseases, including glaucoma. In this study, an established theoretical hybrid model of a retinal microvascular network is extended to include the effects of local blood flow regulation on oxygenation. A heterogeneous representation of the arterioles based on confocal microscopy images is combined with a compartmental description of the downstream capillaries and venules. A Green's function method is used to simulate oxygen transport in the arterioles, and a Krogh cylinder model is applied to the capillary and venular compartments. Acute blood flow regulation is simulated in response to changes in pressure, shear stress, and metabolism. Model results predict that both increased intraocular pressure and impairment of blood flow regulation can cause decreased tissue oxygenation, indicating that both mechanisms represent factors that could lead to impaired oxygenation characteristic of ocular disease. Results also indicate that the metabolic response mechanism reduces the fraction of poorly oxygenated tissue but that the pressure- and shear stress-dependent response mechanisms may hinder the vascular response to changes in oxygenation. Importantly, the heterogeneity of the vascular network demonstrates that traditionally reported average values of tissue oxygen levels hide significant localized defects in tissue oxygenation that may be involved in disease processes, including glaucoma. Ultimately, the model framework presented in this study will facilitate future comparisons to sectorial-specific clinical data to better assess the role of impaired blood flow regulation in ocular disease.

We consider a general mathematical model for protein subunit vaccine with a focus on the MF59-adjuvanted spike glycoprotein-clamp vaccine for SARS-CoV-2, and use the model to study immunological outcomes in the humoral and cell-mediated arms of the immune response from vaccination. The mathematical model is fit to vaccine clinical trial data. We elucidate the role of Interferon-γ and Interleukin-4 in stimulating the immune response of the host. Model results, and results from a sensitivity analysis, show that a balance between the TH1 and TH2 arms of the immune response is struck, with the TH1 response being dominant. The model predicts that two-doses of the vaccine at 28 days apart will result in approximately 85% humoral immunity loss relative to peak immunity approximately 6 months post dose 1.

Previous work showed that Gal-8 and Gal-1A, two proteins belonging to the galactoside-binding galectin family, are the earliest determinants of the patterning of the skeletal elements of embryonic chicken limbs, and further, that their experimentally determined interactions in the embryonic limb bud can be interpreted via a reaction-diffusion-adhesion (2GL: two galectin plus ligands) model. Here, we use an ordinary differential equation-based approach to analyze the intrinsic switching modality of the 2GL network and characterize the network behavior independent of the diffusive and adhesive arms of the patterning mechanism. We identify two states: where the concentrations of both the galectins are respectively, negligible, and very high. This bistable switch-like system arises via a saddle-node bifurcation from a monostable one. For the case of mass-action production terms, we provide an explicit Lyapunov function for the system, which shows that it has no periodic solutions. Our model therefore predicts that the galectin network may exist in low expression and high expression states separated in space or time, without any intermediate states. We test these predictions in experiments performed with high density cultures of chick limb mesenchymal cells and observe that cells inside precartilage protocondensations express Gal-1A at a much higher rate than those outside, for which it was negligible. The Gal-1 and -8-based patterning network is therefore sufficient to partition the mesenchymal cell population into two discrete cell states with different developmental (chondrogenic vs. non-chondrogenic) fates. When incorporated into an adhesion and diffusion-enabled framework this system can generate a spatially patterned limb skeleton.

Understanding pattern formation driven by cell-cell interactions has been a significant theme in cellular biology for many years. In particular, due to their implications within many biological contexts, lateral-inhibition mechanisms present in the Notch-Delta signalling pathway led to an extensive discussion between biologists and mathematicians. Deterministic and stochastic models have been developed as a consequence of this discussion, some of which address long-range signalling by considering cell protrusions reaching non-neighbouring cells. The dynamics of such signalling systems reveal intricate properties of the coupling terms involved in these models. In this work, we investigate the advantages and drawbacks of a single-parameter long-range signalling model across diverse scenarios. By employing linear and multi-scale analyses, we discover that pattern selection is not only partially explained but also depends on nonlinear effects that extend beyond the scope of these analytical techniques.

We provide a critique of mathematical biology in light of rapid developments in modern machine learning. We argue that out of the three modelling activities - (1) formulating models; (2) analysing models; and (3) fitting or comparing models to data - inherent to mathematical biology, researchers currently focus too much on activity (2) at the cost of (1). This trend, we propose, can be reversed by realising that any given biological phenomenon can be modelled in an infinite number of different ways, through the adoption of a pluralistic approach, where we view a system from multiple, different points of view. We explain this pluralistic approach using fish locomotion as a case study and illustrate some of the pitfalls - universalism, creating models of models, etc. - that hinder mathematical biology. We then ask how we might rediscover a lost art: that of creative mathematical modelling.

We present a detailed physiological model of the (human) retina that includes the biochemistry and electrophysiology of phototransduction, neuronal electrical coupling, and the spherical geometry of the eye. The model is a parabolic-elliptic system of partial differential equations based on the mathematical framework of the bi-domain equations, which we have generalized to account for multiple cell-types. We discretize in space with non-uniform finite differences and step through time with a custom adaptive time-stepper that employs a backward differentiation formula and an inexact Newton method. A refinement study confirms the accuracy and efficiency of our numerical method. Numerical simulations using the model compare favorably with experimental findings, such as desensitization to light stimuli and calcium buffering in photoreceptors. Other numerical simulations suggest an interplay between photoreceptor gap junctions and inner segment, but not outer segment, calcium concentration. Applications of this model and simulation include analysis of retinal calcium imaging experiments, the design of electroretinograms, the design of visual prosthetics, and studies of ephaptic coupling within the retina.

One mode by which infection-derived immunity fails is when recovery leads to a reduced but nonzero risk of reinfection. This type of partial protection is called leaky immunity with the degree of leakiness quantified by the relative probability a previously infected individual will get infected upon exposure compared to a naively susceptible individual. Previous authors have defined the reinfection threshold, which occurs when the basic reproduction number equals the inverse of the leakiness, however, there has been some debate about whether or not this is a real threshold. Here we show how the reinfection threshold relates to two important occurrences: (1) the point at which the endemic equilibrium changes from being a stable spiral to a stable node, and (2) the point at which the rate of change of the prevalence increases the most relative to leakiness. When the recovery period is short relative to the average lifetime then both occurrences are close to the reinfection threshold. We show how these results are related to the reinfection threshold found in other models of imperfect immunity. To further demonstrate the significance of this threshold in modeling, we conducted a simulation study to evaluate some of the consequences the reinfection threshold might have in parameter estimation and modeling. Using specific parameter values chosen to reflect an acute infection, we found that the basic reproduction number values larger than that of the reinfection threshold value were less identifiable than those below the threshold.

Defending against novel, repeated, or unpredictable attacks, while avoiding attacks on the 'self', are the central problems of both mammalian immune systems and computer systems. Both systems have been studied in great detail, but with little exchange of information across the different disciplines. Here, we present a conceptual framework for structured comparisons across the fields of biological immunity and cybersecurity, by framing the context of defense, considering different (combinations of) defensive strategies, and evaluating defensive performance. Throughout this paper, we pose open questions for further exploration. We hope to spark the interdisciplinary discovery of general principles of optimal defense, which can be understood and applied in biological immunity, cybersecurity, and other defensive realms.

Though overfishing and climate change are the primary reasons for a regime shift in the fishery, we demonstrate here a different reason for the regime shift, not reported earlier to the best of our knowledge. We show that high demand for fish may cause a regime shift in a fishery in a shorter time. For this, a four-dimensional bioeconomic fishery model is considered and analyzed to explore the system's dynamic behavior. The objective is to demonstrate how increasing demand may cause a catastrophic change in the fish and fishery. We provide the local and global stabilities of different equilibrium points, guaranteeing the stable coexistence of ecological and economic states. Our bifurcation analysis revealed that the demand parameter might play positive and negative roles in the system dynamics. Demand can make an unstable fishery stable. It can also help remove the infection from the system. On the flip side, high demand may cause a regime shift from a harvested state to a non-harvested state, making the price unbounded. Using Pontryagin's maximum principle, we further discussed optimal revenue generation.

Dengue disease transmission is a complex vector-borne disease, mainly due to the co-circulation of four serotypes of the virus. Mathematical models have proved to be a useful tool to understand the complexity of this disease. In this work, we extend the model studied by Esteva et al., 2003, originally proposed for two serotypes, to four circulating serotypes. Using epidemic data of dengue fever in Iquitos (Peru) and San Juan (Puerto Rico), we estimate numerically the co-circulation parameter values for selected outbreaks using a bootstrap method, and we also obtained the Basic Reproduction Number, R0, for each serotype, using both analytical calculations and numerical simulations. Our results indicate that the impact of co-circulation of serotypes in population dynamics of dengue infection is such that there is a reduced effect from DENV-3 to DENV-4 in comparison to no-cross effect for epidemics in Iquitos. Concerning San Juan epidemics, also comparing to no-cross effect, we also observed a reduced effect from the predominant serotype DENV-3 to both DENV-2 and DENV-1 epidemics neglecting the very small number of cases of DENV-4.

At active zones of chemical synapses, an arriving electric signal induces the fusion of vesicles with the presynaptic membrane, thereby releasing neurotransmitters into the synaptic cleft. After a fusion event, both the release site and the vesicle undergo a recovery process before becoming available for reuse again. Of central interest is the question which of the two restoration steps acts as the limiting factor during neurotransmission under high-frequency sustained stimulation. In order to investigate this problem, we introduce a non-linear reaction network which involves explicit recovery steps for both the vesicles and the release sites, and includes the induced time-dependent output current. The associated reaction dynamics are formulated by means of ordinary differential equations (ODEs), as well as via the associated stochastic jump process. While the stochastic jump model describes the dynamics at a single active zone, the average over many active zones is close to the ODE solution and shares its periodic structure. The reason for this can be traced back to the insight that recovery dynamics of vesicles and release sites are statistically almost independent. A sensitivity analysis on the recovery rates based on the ODE formulation reveals that neither the vesicle nor the release site recovery step can be identified as the essential rate-limiting step but that the rate-limiting feature changes over the course of stimulation. Under sustained stimulation, the dynamics given by the ODEs exhibit transient changes leading from an initial depression of the postsynaptic response to an asymptotic periodic orbit, while the individual trajectories of the stochastic jump model lack the oscillatory behavior and asymptotic periodicity of the ODE-solution.

A normally functioning menstrual cycle requires significant crosstalk between hormones originating in ovarian and brain tissues. Reproductive hormone dysregulation may cause abnormal function and sometimes infertility. The inherent complexity in this endocrine system is a challenge to identifying mechanisms of cycle disruption, particularly given the large number of unknown parameters in existing mathematical models. We develop a new endocrine model to limit model complexity and use simulated distributions of unknown parameters for model analysis. By employing a comprehensive model evaluation, we identify a collection of mechanisms that differentiate normal and abnormal phenotypes. We also discover an intermediate phenotype-displaying relatively normal hormone levels and cycle dynamics-that is grouped statistically with the irregular phenotype. Results provide insight into how clinical symptoms associated with ovulatory disruption may not be detected through hormone measurements alone.

We analyse and mutually compare time series of covid-19-related data and mobility data across Belgium's 43 arrondissements (NUTS 3). In this way, we reach three conclusions. First, we could detect a decrease in mobility during high-incidence stages of the pandemic. This is expressed as a sizeable change in the average amount of time spent outside one's home arrondissement, investigated over five distinct periods, and in more detail using an inter-arrondissement "connectivity index" (CI). Second, we analyse spatio-temporal covid-19-related hospitalisation time series, after smoothing them using a generalise additive mixed model (GAMM). We confirm that some arrondissements are ahead of others and morphologically dissimilar to others, in terms of epidemiological progression. The tools used to quantify this are time-lagged cross-correlation (TLCC) and dynamic time warping (DTW), respectively. Third, we demonstrate that an arrondissement's CI with one of the three identified first-outbreak arrondissements is correlated to a substantial local excess mortality some five to six weeks after the first outbreak. More generally, we couple results leading to the first and second conclusion, in order to demonstrate an overall correlation between CI values on the one hand, and TLCC and DTW values on the other. We conclude that there is a strong correlation between physical movement of people and viral spread in the early stage of the sars-cov-2 epidemic in Belgium, though its strength weakens as the virus spreads.

From Leonhard Euler to Alfred Lotka and in recent years understanding the stationary process of the human population has been of central interest to scientists. Population reproductive measure NRR (net reproductive rate) has been widely associated with measuring the status of population stationarity and it is also included as one of the measures in the millennium development goals. This article argues how the partition theorem-based approach provides more up-to-date and timely measures to find the status of the population stationarity of a country better than the NRR-based approach. We question the timeliness of the value of NRR in deciding the stationary process of the country. We prove associated theorems on discrete and continuous age distributions and derive measurable functional properties. The partitioning metric captures the underlying age structure dynamic of populations at or near stationarity. As the population growth rates for an ever-increasing number of countries trend towards replacement levels and below, new demographic concepts and metrics are needed to better characterize this emerging global demography.

The COVID-19 pandemic continues to have a devastating impact on health systems and economies across the globe. Implementing public health measures in tandem with effective vaccination strategies have been instrumental in curtailing the burden of the pandemic. With the three vaccines authorized for use in the U.S. having varying efficacies and waning effects against major COVID-19 strains, understanding the impact of these vaccines on COVID-19 incidence and fatalities is critical. Here, we formulate and use mathematical models to assess the impact of vaccine type, vaccination and booster uptake, and waning of natural and vaccine-induced immunity on the incidence and fatalities of COVID-19 and to predict future trends of the disease in the U.S. when existing control measures are reinforced or relaxed. The results show a 5-fold reduction in the control reproduction number during the initial vaccination period and a 1.8-fold (2-fold) reduction in the control reproduction number during the initial first booster (second booster) uptake period, compared to the respective previous periods. Due to waning of vaccine-induced immunity, vaccinating up to 96% of the U.S. population might be required to attain herd immunity, if booster uptake is low. Additionally, vaccinating and boosting more people from the onset of vaccination and booster uptake, especially with the Pfizer-BioNTech and Moderna vaccines (which confer superior protection than the Johnson & Johnson vaccine) would have led to a significant reduction in COVID-19 cases and deaths in the U.S. Furthermore, adopting natural immunity-boosting measures is important in fighting COVID-19 and transmission rate reduction measures such as mask-use are critical in combating COVID-19. The emergence of a more transmissible COVID-19 variant, or early relaxation of existing control measures can lead to a more devastating wave, especially if transmission rate reduction measures and vaccination are relaxed simultaneously, while chances of containing the pandemic are enhanced if both vaccination and transmission rate reduction measures are reinforced simultaneously. We conclude that maintaining or improving existing control measures, and boosting with mRNA vaccines are critical in curtailing the burden of the pandemic in the U.S.

During several epidemics, transmission from deceased people significantly contributed to disease spread, but mathematical analysis of this transmission has not been seen in the literature numerously. Transmission of Ebola during traditional burials was the most well-known example; however, there are several other diseases, such as hepatitis, plague or Nipah virus, that can potentially be transmitted from disease victims. This is especially true in the case of serious epidemics when healthcare is overwhelmed and the operative capacity of the health sector is diminished, such as seen during the COVID-19 pandemic. We present a compartmental model for the spread of a disease with an imperfect vaccine available, also considering transmission from deceased infected in general. The global dynamics of the system are completely described by constructing appropriate Lyapunov functions. To support our analytical results, we perform numerical simulations to assess the importance of transmission from the deceased, considering the data collected from three infectious diseases, Ebola virus disease, COVID-19, and Nipah fever.

The COVID-19 pandemic is a significant public health threat with unanswered questions regarding the immune system's role in the disease's severity level. Here, based on antibody kinetic data of severe and non-severe COVID-19 patients, topological data analysis (TDA) highlights that severity is not binary. However, there are differences in the shape of antibody responses that further classify COVID-19 patients into non-severe, severe, and intermediate cases of severity. Based on the results of TDA, different mathematical models were developed to represent the dynamics between the different severity groups. The best model was the one with the lowest average value of the Akaike Information Criterion for all groups of patients. Our results suggest that different immune mechanisms drive differences between the severity groups. Further inclusion of different components of the immune system will be central for a holistic way of tackling COVID-19.