Midbrain dopamine (DA) neurons exhibit spiking and bursting patterns under physiological conditions. Based on the data on electrophysiological recordings, Yu et al. developed a 13-dimensional mathematical model to capture the detailed characteristics of the DA neuronal firing activities. We use the fitting method to simplify the original model into a 4-dimensional model. Then, the spiking-to-bursting transition is detected from a simple and robust mathematical condition. Physiologically, this condition is a balance of the restorative and the regenerative ion channels at resting potential. Geometrically, this condition imposes a transcritical bifurcation. Moreover, we combine singularity theory and singular perturbation methods to capture the geometry of three-timescale firing attractors in a universal unfolding of a cusp singularity. In particular, the planar description of the corresponding firing patterns can generate the corresponding firing attractors. This analysis provides a new idea for understanding the firing activities of the DA neuron and the specific mechanisms for the switching and dynamic regulation among different patterns.

Elucidating the hydrodynamics of fish swimming is critical to identifying the processes underlying fish orientation and schooling. Due to their mathematical tractability, models based on potential flow are preferred in the study of bidirectional interactions of fish with their surroundings. Dipole-based models that assimilate fish to pairs of vortices are particularly enticing, but yet to be thoroughly validated. Here, we embark on a computational fluid dynamics (CFD) campaign informed by experimental data to validate the accuracy of dipole-based models. The locomotory patterns of a fish undergoing carangiform swimming are reconstructed from existing experimental data, which are used as inputs to CFD simulations of a fish swimming in a channel flow. We demonstrate that dipole-based models are accurate in capturing key features of the fluid flow, but cannot predict the elongated flow streamlines around the fish that are evident in CFD. To address this issue, we propose an alternative model that replaces each vortex in the pair with a sheet along the fish length. Using a pair of vortex sheets that span approximately 80% of the fish body length with a separation distance of approximately 50% of the body width, the model is successful in predicting the fluid flow around the swimming fish for a range of background flow speeds and channel widths. The proposed model shows improved accuracy at the cost of a mildly increased computational effort, thereby constituting an ideal basis for research on fish hydrodynamics.

One of the key features of any infectious disease is whether infection generates long-lasting immunity or whether repeated reinfection is common. In the former, the long-term dynamics are driven by the birth of susceptible individuals while in the latter the dynamics are governed by the speed of waning immunity. Between these two extremes a range of scenarios is possible. During the early waves of SARS-CoV-2, the underlying paradigm was for long-lasting immunity, but more recent data and in particular the 2022 Omicron waves have shown that reinfection can be relatively common. Here we investigate reported SARS-CoV-2 cases in England, partitioning the data into four main waves, and consider the temporal distribution of first and second reports of infection. We show that a simple low-dimensional statistical model of random (but scaled) reinfection captures much of the observed dynamics, with the value of this scaling, k, providing information of underlying epidemiological patterns. We conclude that there is considerable heterogeneity in risk of reporting reinfection by wave, age-group and location. The high levels of reinfection in the Omicron wave (we estimate that 18% of all Omicron cases had been previously infected, although not necessarily previously reported infection) point to reinfection events dominating future COVID-19 dynamics. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

Multi-omics clustering plays an important role in cancer subtyping. However, the data of different kinds of omics are often related, these correlations may reduce the clustering algorithm performance. It is crucial to eliminate the unexpected redundant information caused by these correlations between different omics. We proposed RSC-based differential model with correlation removal for improving multi-omics clustering (RSC-MCR). This method first introduced RSC to calculate the pairwise correlations of all features, and decomposed it to obtain the pairwise correlations of different omics features, thus built the connection between different omics based on the pairwise correlations of different omics features. Then, to remove the redundant correlation, we designed a differential model to calculate the degree of difference between the original feature matrix and the correlation matrix which contained the most relevant information between different omics. We compared the performance of RSC-MCR with decorrelation methods on different clustering methods (CC, FCM, SNF, NMF, LRAcluster). The experimental results on five cancer datasets show the efficiency of the RSC-MCR as well as improvements over other decorrelation methods.

Immune checkpoint inhibitors (ICIs) introduced in recent years have revolutionized the treatment of many metastatic cancers. However, data suggest that treatment has benefits only in a limited percentage of patients, and that this is due to immune suppression of the tumor microenvironment (TME). Anti-tumor inflammatory macrophages (M1), which are attracted to the TME, are converted by tumor secreted cytokines, such as CSF-1, to pro-tumor anti-inflammatory macrophages (M2), or tumor associated macrophages (TAMs), which block the anti-tumor T cells. In the present paper we develop a mathematical model that represents the interactions among the immune cells and cancer in terms of differential equations. The model can be used to assess treatments of combination therapy of anti-PD-1 with anti-CSF-1. Examples are given in comparing the efficacy among different strategies for anti-CSF-1 dosing in a setup of clinical trials.

Compelling evidence continues to build to support the idea that SARS-CoV-2 Neutralizing Antibody (NAb) levels in an individual can serve as an important indicator of the strength of protective immunity against infection. It is not well understood why NAb levels in some individuals remain high over time, while in others levels decline rapidly. In this work, we present a two-state mathematical model of within-host NAb dynamics in response to vaccination. By fitting only four host-specific parameters, the model is able to capture individual-specific NAb levels over time as measured by the AditxtScore™ for NAbs. The model can serve as a foundation for predicting NAb levels in the long-term, understanding connections between NAb levels, protective immunity, and breakthrough infections, and potentially guiding decisions about whether and when a booster vaccination may be warranted.

We present a mathematical model that describes how tumour heterogeneity evolves in a tissue slice that is oxygenated by a single blood vessel. Phenotype is identified with the stemness level of a cell and determines its proliferative capacity, apoptosis propensity and response to treatment. Our study is based on numerical bifurcation analysis and dynamical simulations of a system of coupled, non-local (in phenotypic "space") partial differential equations that link the phenotypic evolution of the tumour cells to local tissue oxygen levels. In our formulation, we consider a 1D geometry where oxygen is supplied by a blood vessel located on the domain boundary and consumed by the tumour cells as it diffuses through the tissue. For biologically relevant parameter values, the system exhibits multiple steady states; in particular, depending on the initial conditions, the tumour is either eliminated ("tumour-extinction") or it persists ("tumour-invasion"). We conclude by using the model to investigate tumour responses to radiotherapy, and focus on identifying radiotherapy strategies which can eliminate the tumour. Numerical simulations reveal how phenotypic heterogeneity evolves during treatment and highlight the critical role of tissue oxygen levels on the efficacy of radiation protocols that are commonly used in the clinic.

The nervous system is today recognized to play an important role in the development of cancer. Indeed, neurons extend long processes (axons) that grow and infiltrate tumors in order to regulate the progression of the disease in a positive or negative way, depending on the type of neuron considered. Mathematical modeling of this biological process allows to formalize the nerve-tumor interactions and to test hypotheses in silico to better understand this phenomenon. In this work, we introduce a system of differential equations modeling the progression of pancreatic ductal adenocarcinoma (PDAC) coupled with associated changes in axonal innervation. The study of the asymptotic behavior of the model confirms the experimental observations that PDAC development is correlated with the type and densities of axons in the tissue. We study then the identifiability and the sensitivity of the model parameters. The identifiability analysis informs on the adequacy between the parameters of the model and the experimental data and the sensitivity analysis on the most contributing factors on the development of cancer. It leads to significant insights on the main neural checkpoints and mechanisms controlling the progression of pancreatic cancer. Finally, we give an example of a simulation of the effects of partial or complete denervation that sheds lights on complex correlation between the healthy, pre-cancerous and cancerous cell densities and axons with opposite functions.

Modeling of the biological neurons is a way to understand the architecture of neural networks of the brain. A complex brain network includes the synchronization between some groups of neurons. The dynamic behavior of interactions between groups of slave-master neurons in the neocortical network is unpredictable and challenging. The purpose of synchronizing a neural interaction is to reduce the synchronization error between the chaotic slave-master neurons. This paper uses a proportional-integral-derivative (PID) controller to synchronize master-slave neurons in the fractional-order of the neocortical network model based on dendritic spike frequency adaptation (DSFA) uncertainties and unknown disturbance effects. The purpose of this article is in two parts: First, we implemented the effect of previous states of the neuron conditions by fractional-order of the differential equations in the neocortical network model. Second, by synchronizing the FO neocortical master-slave model by PID controller, we investigated the connection strength of the complex network in chaotic point of view. The optimized PID coefficients and fractional-order were calculated using root mean square error (RMSE) criteria to control the membrane voltage synchronization. The chaotic behavior of the system was evaluated by numerical techniques such as attractor analysis and time series diagrams. The optimal RMSE value for master-slave neurons occurred at fractional-orders 0.89. It is shown that the synchronization of master-slave neurons improves over time, and eventually they are fully synchronized while the controller error is reduced.

Cancer is one of the leading causes of mortality and morbidity among people worldwide. Cancer appears as solid tumors in many cases. Angiogenesis is the growth of blood vessels from the existing vasculature and is one of the imperative processes in tumor growth. Another vital phenomenon for formation and functionality of this vasculature network is lumen formation. The results of recent studies indicate the importance of blood pressure in this mechanism. Computational modeling can study these processes in different scales. Hence, wide varieties of these models have been proposed during recent years. In this research, a multi-scale model is developed for the angiogenesis process. In the extracellular scale, the growth factor concentration is calculated via the reaction diffusion equation. At the cellular scale, growth, migration, and the adhesion of endothelial cells are modeled by the Potts cellular model. At the intra-cellular scale by considering biochemical signals, a Boolean network model describes migration, division, or apoptosis of endothelial cells. A stochastic model developed for lumen formation via inverse membrane blebbing mechanism. A CFD simulation was also used to investigate the role of pulsated blood pressure in the inverse membrane blebbing mechanism. The lumen formation model shows stochastic behavior in blebs expansion and lumen expansion. Comparing the stochastic model's results with the CFD simulation also shows the vital role of pressure pulse and the topology of the blebs in bleb retraction.

Models of populations in habitat networks are vital for understanding and linking processes and patterns across individuals, environments, ecological interactions, and population structures. River ecosystem models combine the physical structure of the networks with the biological processes of the organisms using structural and functional models, respectively. Previous studies on dendritic river networks have employed different functional (population) models and either directly claimed or implied that the results illustrate general properties of actual river systems. However, these studies have used different approaches and assumptions when modeling population characteristics and behavior, and it is possible that inferences regarding a system may vary based on the combination of functional model and the spatial structure of a network. This study aims to understand if different functional models in river systems produce substantially different model results and, therefore, whether conclusions are model-dependent. We compare variation in extinction time and occupancy proportion of river networks with linear, trellis, dendritic and ring-lattice topologies, using three population models (uniform, age-class and individual based) and one metapopulation-based (patch-occupancy) model. Dendritic, linear, and trellis structures did not show notable differences among extinction times for any of the four models. The difference between topologies was higher for the patch-occupancy model compared to the three population models. There were significant differences in the variations of patch-occupancy between the metapopulation and the population models, but the three population models of differing complexity produced broadly similar results. Therefore, if the occupancy data is obtained based on local subpopulations, spatial arrangement and connectivity does not appear to be the sole predictor of single-species metapopulation responses. We conclude that the outputs from functional models are robust to assumptions and varying levels of detail as long as they contain at least some detail at the level of individuals within habitat nodes. Also, if we are modeling network-scale populations, models that include at least some detailed information on individuals are a far better choice than considering populations implicitly.

Seroprevalence studies can estimate proportions of the population that have been infected or vaccinated, including infections that were not reported because of the lack of symptoms or testing. Based on information from studies in the United States from mid-summer 2020 through the end of 2021, we describe proportions of the population with antibodies to SARS-CoV-2 as functions of age and time. Slices through these surfaces at arbitrary times provide initial and target conditions for simulation modeling. They also provide the information needed to calculate age-specific forces of infection, attack rates, and - together with contact rates - age-specific probabilities of infection on contact between susceptible and infectious people. We modified the familiar Susceptible-Exposed-Infectious-Removed (SEIR) model to include features of the biology of COVID-19 that might affect transmission of SARS-CoV-2 and stratified by age and location. We consulted the primary literature or subject matter experts for contact rates and other parameter values. Using time-varying Oxford COVID-19 Government Response Tracker assessments of US state and DC efforts to mitigate the pandemic and compliance with non-pharmaceutical interventions (NPIs) from a YouGov survey fielded in the US during 2020, we estimate that the efficacy of social-distancing when possible and mask-wearing otherwise at reducing susceptibility or infectiousness was 31% during the fall of 2020. Initialized from seroprevalence among people having commercial laboratory tests for purposes other than SARS-CoV-2 infection assessments on 7 September 2020, our age- and location-stratified SEIR population model reproduces seroprevalence among members of the same population on 25 December 2020 quite well. Introducing vaccination mid-December 2020, first of healthcare and other essential workers, followed by older adults, people who were otherwise immunocompromised, and then progressively younger people, our metapopulation model reproduces seroprevalence among blood donors on 4 April 2021 less well, but we believe that the discrepancy is due to vaccinations being under-reported or blood donors being disproportionately vaccinated, if not both. As experimenting with reliable transmission models is the best way to assess the indirect effects of mitigation measures, we determined the impact of vaccination, conditional on NPIs. Results indicate that, during this period, vaccination substantially reduced infections, hospitalizations and deaths. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics.

The synaptic tagging and capture (STC) hypothesis not only explain the integration and association of synaptic activities, but also the formation of learning and memory. The synaptic pathways involved in the synaptic tagging and capture phenomenon are called STC pathways. The STC hypothesis provides a potential explanation of the neuronal and synaptic processes underlying the synaptic consolidation of memories. Several mechanisms and molecules have been proposed to explain the process of memory allocation and synaptic tags, respectively. However, a clear link between the STC hypothesis and memory allocation is still missing because the encoding of memories in neural circuits is mainly associated with strongly recurrently connected groups of neurons. To explore the mechanisms of potential synaptic tagging candidates and their involvement in the process of memory allocation, we develop a mathematical model for a single dendritic spine based on five essential criteria of a synaptic tag. By developing a mathematical model, we attempt to understand the roles of the potentially critical molecular networks underlying the STC and the essential attributes of a synaptic tag. We include essential memory molecules in the STC model that have been identified in earlier studies as crucial for STC pathways. CaMKII activation is critical for the setting of the initial tag; however, coordinated activities with other kinases and the biochemical pathways are necessary for the tag to be stable. PKA modulates NMDAR-mediated Ca2+ signalling. Similarly, PKA and ERK crosstalk is essential for Ca2+ - mediated protein synthesis during l-LTP. Our theoretical model explains the quantitative contribution of Tags and protein synthesis during l-LTP in synaptic strength.

Atopic dermatitis (AD) is an immune-driven inflammatory skin disease that is known to have a significantly high life-time prevalence in the human population. T-helper (Th) immune cells play a key role in the pathogenesis of AD which is marked by defects in the skin barrier function along with a significant increase in the population of either Th1 or Th2 sub-types of Th cells. The progression of AD from the acute to chronic phase is still poorly understood, and here we explore the mechanism of this transition through the study of a mathematical model for indirect cell-cell interactions among Th and skin cells via the secreted cytokines IFNγ and IL-4, both known to have therapeutic potential. An increase in the level of cytokine IFN γ can catalyse the transition of AD from an acute to a chronic stage, while an increase in the level of cytokine IL-4 has the reverse effect. In our model, the transition of AD from the acute to chronic stage and vice versa can be abrupt (switch-like) with hysteresis: this bistable behaviour can potentially be used to keep AD in the acute phase since therapy based on suppression of IFNγ can retard the transition to the chronic phase.

Phylogenetic trees visually represent evolution and diversification. While many studies have focused on the number and length of edges (branches), topological properties, such as edge connection patterns, are also important. In this study, the topological properties of phylogenetic trees were quantified, focusing on edge connection patterns. Horton's first law was applied to quantify the overall, rather than local, topological properties of phylogenetic trees. The topological properties of vertebrate phylogenetic trees for spiny-rayed fishes, Amphibians, turtles, Squamata, Aves, and placental mammals were analyzed. The topological features discussed herein include the number of first-order edges, maximum order, and bifurcation ratio. The average bifurcation ratio of all trees was approximately 3, suggesting that phylogenetic trees for different taxa have a common mechanism of evolution. Vertebrate phylogenetic trees were compared with artificial branching objects created from neutral stochastic branching model simulations. The topological properties of the actual vertebrate phylogenetic trees agreed with those of the artificial branching objects. Our study suggests that evolutionary events do not change the overall topological properties of actual phylogenetic trees, even if the number and length of the edges change. Specifically, non-neutral events (e.g., environmental changes and mass extinction) are not main factors associated with topological properties. The results instead demonstrate a relationship between the bifurcation ratio and symmetricity in the context of temporal changes of topological properties. When the number of first-order edges increased and the maximum order remained constant, the bifurcation ratio increased and symmetricity decreased. When the number of first-order edges increased and the maximum order increased by one, the bifurcation ratio decreased and symmetricity increased.

The fact that people often have preference rankings for their partners is a distinctive aspect of human behavior. Little is known, however, about how this talent as a powerful force shapes human behavioral traits, including those which should not have been favored by selection, such as cooperation in social dilemma situations. Here we propose a dynamic model in which network-structured individuals can switch their interaction partners within neighborhoods based on their preferences. For the partner switching, we propose two interruption regimes: dictatorial regime and negotiating regime. In the dictatorial regime, focal individuals are able to suspend interactions out of preferences unilaterally. In the negotiating regime, either focal individuals or the associated partners agree to suspend, then these interactions can be successfully suspended. We investigate the evolution of cooperation under both preference-driven partner switching regimes in the context of both the weakened variant of the donation game and the standard one. Specifically, we theoretically approximate the critical conditions for cooperation to be favored by weak selection in the weakened donation game where cooperators bear a unit cost to provide a benefit for each active neighbor and simulate the evolutionary dynamics of cooperation in the standard donation game to test the robustness of the analytical results. Under dictatorial regime, selection of cooperation becomes harder when individuals have preferences for either cooperator or defector partners, implying that the expulsion of defectors by cooperators is overwhelmed by the chasing of defectors towards cooperators. Under negotiating regime, both preferences for cooperator and defector partners can significantly favor the evolution of cooperation, yet underlying mechanisms differ greatly. For preferences over cooperator partners, cooperator-cooperator interaction relationships are reinforced and the associated mutual reciprocity can resist and assimilate defectors. For preferences over defector partners, defector-defector interaction relationships are anchored, weakening defectors' exploitation over cooperators. Cooperators are thus offered much time space to interact among cospecies and spread. Our work may help better understand the critical role of preference-based adaptive partner switching in promoting the evolution of cooperation.

Several creatures can climb on smooth surfaces with the help of hairy adhesive pads on their legs. A rapid change from strong attachment to effortless detachment of the leg is enabled by the asymmetric geometry of the tarsal hairs. In this study, we propose mechanisms by which the hairy pad can be easily detached, even when the hairs possess no asymmetry. Here, we examine the possible function of the tibia-tarsus leg joint and the claws. Based on a spring-based model, we consider three modes of detachment: vertically pulling the pad while maintaining either a (1) fixed or a (2) free joint, or by (3) flexing the pad about the claw. We show that in all cases, the adhesion force can be significantly reduced due to elastic forces when the hairs deform non-uniformly across the array. Our proposed model illustrates the design advantage of such fibrillar adhesive systems, that not only provide strong adhesion, but also allow easy detachment, making them suitable as organs for fast locomotion and reliable hold. The presented approaches can be potentially used to switch the adhesion state in bio-inspired fibrillar adhesives, by incorporating artificial joints and claws into their design, without the need of asymmetric or stimuli-responsive fibrillar structures.

Differentiated cancer cells may regain stem cell characteristics; however, the effects of such a cellular dedifferentiation on tumoural growth and treatment are currently understudied. Thus, we here extend a mathematical model of cancer stem cell (CSC) driven tumour growth to also include dedifferentiation. We show that dedifferentiation increases the likelihood of tumorigenesis and the speed of tumoural growth, both modulated by the proliferative potential of the non-stem cancer cells (NSCCs). We demonstrate that dedifferentiation also may lead to treatment evasion, especially when a treatment solely targets CSCs. Conversely, targeting both CSCs and NSCCs in parallel is shown to be more robust to dedifferentiation. Despite dedifferentiation, perturbing CSC-related parameters continues to exert the largest relative effect on tumoural growth; however, we show the existence of synergies between specific CSC- and NSCC-directed treatments which cause superadditive reductions of tumoural growth. Overall, our study demonstrates various effects of dedifferentiation on growth and treatment of tumoural lesions, and we anticipate our results to be helpful in guiding future molecular and clinical research on limiting tumoural growth in vivo.

Land conversion and the resulting contact between domesticated and wild species has arguably been the single largest contributor to the emergence of novel epizootic and zoonotic diseases in the past century. An unintended consequence of these interactions is zoonotic or epizootic disease spillovers from wild species to humans and their domesticates. Disease spillovers are edge effects of land conversion and are sensitive to the size and shape of converted areas. We combine spatial metrics from landscape ecology with theoretical epidemiological models to understand how the size and shape of land conversion affect epizootic and zoonotic disease transmission of single and two species populations. We show that the less compact the converted area, and the greater the depth of the contact zone, the more rapidly will an introduced disease spread through the domesticated population.

The COVID-19 pandemic has underscored the need to understand the dynamics of SARS-CoV-2 respiratory infection and protection provided by the immune response. SARS-CoV-2 infections are characterized by a particularly high viral load, and further by the small number of inhaled virions sufficient to generate a high viral titer in the nasal passage a few days after exposure. SARS-CoV-2 specific antibodies (Ab), induced from vaccines, previous infection, or inhaled monoclonal Ab, have proven effective against SARS-CoV-2 infection. Our goal in this work is to model the protective mechanisms that Ab can provide and to assess the degree of protection from individual and combined mechanisms at different locations in the respiratory tract. Neutralization, in which Ab bind to virion spikes and inhibit them from binding to and infecting target cells, is one widely reported protective mechanism. A second mechanism of Ab protection is muco-trapping, in which Ab crosslink virions to domains on mucin polymers, effectively immobilizing them in the mucus layer. When muco-trapped, the continuous clearance of the mucus barrier by coordinated ciliary propulsion entrains the trapped viral load toward the esophagus to be swallowed. We model and simulate the protection provided by either and both mechanisms at different locations in the respiratory tract, parametrized by the Ab titer and binding-unbinding rates of Ab to viral spikes and mucin domains. Our results illustrate limits in the degree of protection by neutralizing Ab alone, the powerful protection afforded by muco-trapping Ab, and the potential for dual protection by muco-trapping and neutralizing Ab to arrest a SARS-CoV-2 infection. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

Glioblastoma multiforme (GBM) is one of the most deadly forms of cancer. Methods of characterizing these tumours are valuable for improving predictions of their progression and response to treatment. A mathematical model called the proliferation-invasion (PI) model has been used extensively in the literature to model the growth of these tumours, though it relies on known values of two key parameters: the tumour cell diffusivity and proliferation rate. Unfortunately, these parameters are difficult to estimate in a patient-specific manner, making personalized tumour forecasting challenging. In this paper, we develop and apply a deep learning model capable of making accurate estimates of these key GBM-characterizing parameters while simultaneously producing a full prediction of the tumour progression curve. Our method uses two sets of multi sequence MRI in order to produce estimations and relies on a preprocessing pipeline which includes brain tumour segmentation and conversion to tumour cellularity. We first apply our deep learning model to synthetic tumours to showcase the model's capabilities and identify situations where prediction errors are likely to occur. We then apply our model to a clinical dataset consisting of five patients diagnosed with GBM. For all patients, we derive evidence-based estimates for each of the PI model parameters and predictions for the future progression of the tumour, along with estimates of the parameter uncertainties. Our work provides a new, easily generalizable method for the estimation of patient-specific tumour parameters, which can be built upon to aid physicians in designing personalized treatments.

A primary driver of species extinctions and declining biodiversity is loss and fragmentation of habitats owing to human activities. Many studies spanning a wide diversity of taxa have described the relationship between population density and habitat patch area, i.e., the density-area relationship (DAR), as positive, neutral, negative or some combination of the three. However, the mechanisms responsible for these relationships remain elusive. We employ a theoretical spatially explicit population model built upon the reaction-diffusion framework with absorbing boundary conditions to model a habitat specialist dwelling in islands of habitat surrounded by a hostile matrix. We consider patches with a convex or non-convex geometry. Our results show that a single species following logistic-type population growth exhibits a strictly positive and continuous DAR. However, when multiple asymptotically stable steady states are preset in the system, a discontinuous DAR arises. In the case of two species governed by diffusive Lotka-Volterra growth and competitive interactions, we observe that overall DAR structure can be either (1) positive, (2) positive for small areas and neutral for large, or (3) hump-shaped, i.e., positive for area below a threshold and negative for area above. Patch complexity such as non-convex geometry can cause discontinuities in DAR slope for a single species and create qualitatively different patterns in a competitive system as compared to a convex patch. We also compared our theoretical results with two empirical studies (Anolis lizards on islands and crossbills and pine squirrels in forest fragments) where the pragmatic view of DAR fails to give a mechanistic understanding of what was observed. Close qualitative agreement between theoretical and observed DAR indicates that our model gives a reasonable explanation of the mechanisms underpinning DAR found in those studies. From a conservation perspective, the DAR is crucial to the identification of valuable habitat fragments that favor high abundance and the design of a reserve for a target species. When it comes to protecting a single species, these results suggest that there is unlikely to be a simple solution and that conservation decisions should always be made on a case-by-case basis.

The moral rule "Risk your life to save your family members" is, at the same time, a biological phenomenon. The prominent population geneticist, J.B.S. Haldane told his friends that he would risk his life to save two drowning brothers, but not one - so the story goes. In biological terms, Haldane's arithmetic claims that sib altruism is evolutionarily rational, whenever by "self-sacrifice" an altruistic gene "rescues", on average, more than one copy of itself in its lineage. Here, we derive conditions for evolutionary stability of sib altruism, using population genetic models for three mating systems (monogamy, promiscuity and polygyny) with linear and non-linear group effect on the siblings' survival rate. We show that for all considered selection situations, the condition of evolutionary stability is equivalent to Haldane's arithmetic. The condition for evolutionary stability is formulated in terms of genetic relatedness and the group effect on the survival probability, similarly to the classical Hamilton's rule. We can set up a "scale of mating systems", since in pairwise interactions the chance of evolutionary stability of sib altruism decreases in this order: monogamy, polygyny and promiscuity. Practice of marrying and siblings' solidarity are moral rules in a secular world and in various religious traditions. These moral rules are not evolutionarily independent, in the sense that the subsistence of sib altruism is more likely in a monogamous population.

Repeat asymptomatic testing in order to identify and quarantine infectious individuals has become a widely-used intervention to control SARS-CoV-2 transmission. In some workplaces, and in particular health and social care settings with vulnerable patients, regular asymptomatic testing has been deployed to staff to reduce the likelihood of workplace outbreaks. We have developed a model based on data available in the literature to predict the potential impact of repeat asymptomatic testing on SARS-CoV-2 transmission. The results highlight features that are important to consider when modelling testing interventions, including population heterogeneity of infectiousness and correlation with test-positive probability, as well as adherence behaviours in response to policy. Furthermore, the model based on the reduction in transmission potential presented here can be used to parameterise existing epidemiological models without them having to explicitly simulate the testing process. Overall, we find that even with different model paramterisations, in theory, regular asymptomatic testing is likely to be a highly effective measure to reduce transmission in workplaces, subject to adherence. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

Susceptible-exposed-infectious-recovered (SEIR) models were applied to assess the effectiveness of non-pharmaceutical interventions (NPIs) and to study the dynamic behavior of the COVID-19 pandemic. Recently, SEIR models have evolved to address the change of human mobility by some NPIs for predicting the new confirmed cases. However, the models have serious limitations when applied to Seoul. Seoul has two representative quarantine policies, i.e. social distancing and the ban on gatherings. Effects of the two policies need to be reflected in different functional forms in the model because changes in human mobility do not fully reflect the ban on gatherings. Thus we propose a modified SEIR model to assess the effectiveness of social distancing, ban on gatherings and vaccination strategies. The application of the modified SEIR model was illustrated by comparing the model output with real data.

Multiple sclerosis (MS) is a disease of the central nervous system (CNS) that affects the brain and spinal cord. It is estimated that the average prevalence of MS is 35.9 cases per 100,000 and a total of 2.8 million people worldwide have MS. Brain atrophy is usually seen in the early stages of MS, and its progress is faster than healthy people. The present study was a numerical study that uses the Fluid-structure interaction (FSI) model to investigate the effect of brain atrophy on brain injury in MS. Firstly, a healthy model was constructed from MRI images and validated by experimental data. Then three models with different degrees of brain atrophy, which showed the rate of brain atrophy in different years in MS patients, were developed to model the brain atrophy in MS. The models were subjected to two different types of impact conditions. Type I, which only produced a translational motion and the HIC value of 744, was applied to each model. Type II produced both translational and rotational motion. In this type of impact, the experimental kinematics, with peaks of 450 g for the translational acceleration and 26.2 krad/s2 for the rotational acceleration, were applied to the nodes that located in the center of gravity of the head models and the results were extracted from each one. According to the results of impact type I, the pressure of the frontal lobe of the brain is 149,647 Pa in the health model and 137,690 Pa in the model with severe atrophy.

The emergence of SARS-CoV-2 saw severe detriments to public health being inflicted by COVID-19 disease throughout 2020. In the lead up to Christmas 2020, the UK Government sought an easement of social restrictions that would permit spending time with others over the Christmas period, whilst limiting the risk of spreading SARS-CoV-2. In November 2020, plans were published to allow individuals to socialise within 'Christmas bubbles' with friends and family. This policy involved a planned easing of restrictions in England between 23-27 December 2020, with Christmas bubbles allowing people from up to three households to meet throughout the holiday period. We estimated the epidemiological impact of both this and alternative bubble strategies that allowed extending contacts beyond the immediate household. We used a stochastic individual-based model for a synthetic population of 100,000 households, with demographic and SARS-CoV-2 epidemiological characteristics comparable to England as of November 2020. We evaluated five Christmas bubble scenarios for the period 23-27 December 2020, assuming our populations of households did not have symptomatic infection present and were not in isolation as the eased social restrictions began. Assessment comprised incidence and cumulative infection metrics. We tested the sensitivity of the results to a situation where it was possible for households to be in isolation at the beginning of the Christmas bubble period and also when there was lower adherence to testing, contact tracing and isolation interventions. We found that visiting family and friends over the holiday period for a shorter duration and in smaller groups was less risky than spending the entire five days together. The increases in infection from greater amounts of social mixing disproportionately impacted the eldest. We provide this account as an illustration of a real-time contribution of modelling insights to a scientific advisory group, the Scientific Pandemic Influenza Group on Modelling, Operational sub-group (SPI-M-O) for the Scientific Advisory Group for Emergencies (SAGE) in the UK, during the COVID-19 pandemic. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

Dopamine D1 receptor (D1DR) is proved to be a promising target to prevent tumor metastasis, and our previous studies showed that QAP14, a potent anti-cancer agent, exerted inhibitory effect on lung metastasis via D1DR activation. Therefore, the purpose of the study was to establish count data models to quantitatively characterize the disease progression of lung metastasis and assess the anti-metastatic effect of QAP14. Data of metastatic progression were collected in 4T1 tumor-bearing mice. Generalized Poisson distribution best described the variability of metastasis counts among the individuals. An empirical PK/PD model was developed to establish mathematical relationships between steady plasma concentrations of QAP14 and metastasis growth dynamics. The latency period of metastasis was estimated to be 12 days after tumor implantation. Our model structure also fitted well to other D1DR agonists (fenoldopam and l-stepholidine) which had inhibitory impact on breast cancer lung metastasis likewise. QAP14 40 mg/kg showed the best inhibitory efficacy, for it provided the longest prolongation of metastasis-free periods compared with fenoldopam or l-stepholidine. This study provides a quantitative method to describe the lung metastasis progression of 4T1 allografts, as well as an alternative PD model structure to evaluate anti-metastatic efficacy.

The COVID-19 epidemic has lasted for more than two years since the outbreak in late 2019. An urgent and challenging question is how to systematically evaluate epidemic developments in different countries, during different periods, and to determine which measures that could be implemented are key for successful epidemic prevention. In this study, SBD distance-based K-shape clustering and hierarchical clustering methods were used to analyse epidemics in Asian countries. For the hierarchical clustering, epidemic time series were divided into three periods (epidemics induced by the Original/Alpha, Delta and Omicron variants separately). Standard deviations, the Hurst index, mortality rates, peak value of confirmed cases per capita, average growth rates, and the control efficiency of each period were used to characterize the epidemics. In addition, the total numbers of cases in the different countries were analysed by correlation and regression in relation to 15 variables that could have impacts on COVID-19. Finally, some suggestions on prevention and control measures for each category of country are given. We found that the total numbers of cases per million of a population, total deaths per million and mortality rates were highly correlated with the proportion of people aged over 65 years, the prevalence of multiple diseases, and the national GDP. We also found significant associations between case numbers and vaccination rates, health expenditures, and stringency of control measures. Vaccinations have played a positive role in COVID-19, with a gradual decline in mortality rates in later periods, and are still playing protective roles against the Delta and Omicron strains. The stringency of control measures taken by a government is not an indicator of the appropriateness of a country's response to the outbreak, and a higher index does not necessarily mean more effective measures; a combination of factors such as national vaccination rates, the country's economic foundation and the availability of medical equipment is also needed. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

In March 2020 mathematics became a key part of the scientific advice to the UK government on the pandemic response to COVID-19. Mathematical and statistical modelling provided critical information on the spread of the virus and the potential impact of different interventions. The unprecedented scale of the challenge led the epidemiological modelling community in the UK to be pushed to its limits. At the same time, mathematical modellers across the country were keen to use their knowledge and skills to support the COVID-19 modelling effort. However, this sudden great interest in epidemiological modelling needed to be coordinated to provide much-needed support, and to limit the burden on epidemiological modellers already very stretched for time. In this paper we describe three initiatives set up in the UK in spring 2020 to coordinate the mathematical sciences research community in supporting mathematical modelling of COVID-19. Each initiative had different primary aims and worked to maximise synergies between the various projects. We reflect on the lessons learnt, highlighting the key roles of pre-existing research collaborations and focal centres of coordination in contributing to the success of these initiatives. We conclude with recommendations about important ways in which the scientific research community could be better prepared for future pandemics. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

A reliable theory of biological tissues growth and organization, a fundamental tool for a comprehensive interpretation of experimental observations and a guide to progress in life sciences, is definitively missing. This would support the advancement of knowledge for both normal and pathological expansion and regulation of tissues and organisms. In this work is presented a computational model of cell culture that describes its growth and organization using cell proliferation as its default state, constrained by contact inhibition, closely connected to the cell bioelectric state. The model results describe in a correct way the reported experimental results, involving contact inhibition due to the presence of other cells, and gap junctions for signaling, molecules exchange and extracellular environment sensing. Starting from depolarized cells (in this model considered tantamount to proliferative), the cell culture grows until it fills the available domain and, due to the contact inhibition constraint, it turns into quiescence (a consequence of cell polarization), except on the periphery. Using drugs or via protein expression manipulation, it is possible to change the final tissue state, to fully polarized or depolarized. Other experimental tests are proposed and the expected results simulated. This model can be extended to pathological events, such as carcinogenesis, with cells homeostasis perturbed by a cell depolarizing (carcinogenic) event and express its default proliferative state without adequate control. This simplified model of tissue organization, regulated by the cell's bioelectric state and their interaction with vicinity, is an alternative to the description of the experimental results by mechanical stress, and can be further tested and extended in dedicated experiments.

Mathematical models are increasingly used throughout infectious disease outbreaks to guide control measures. In this review article, we focus on the initial stages of an outbreak, when a pathogen has just been observed in a new location (e.g., a town, region or country). We provide a beginner's guide to two methods for estimating the risk that introduced cases lead to sustained local transmission (i.e., the probability of a major outbreak), as opposed to the outbreak fading out with only a small number of cases. We discuss how these simple methods can be extended for epidemiological models with any level of complexity, facilitating their wider use, and describe how estimates of the probability of a major outbreak can be used to guide pathogen surveillance and control strategies. We also give an overview of previous applications of these approaches. This guide is intended to help quantitative researchers develop their own epidemiological models and use them to estimate the risks associated with pathogens arriving in new host populations. The development of these models is crucial for future outbreak preparedness. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

Developing a functional description of the neural control circuits and visual feedback paths underlying insect flight behaviors is an active research area. Feedback controllers incorporating engineering models of the insect visual system outputs have described some flight behaviors, yet they do not explain how insects are able to stabilize their body position relative to nearby targets such as neighbors or forage sources, especially in challenging environments in which optic flow is poor. The insect experimental community is simultaneously recording a growing library of in-flight head and eye motions that may be linked to increased perception. This study develops a quantitative model of the optic flow experienced by a flying insect or robot during head yawing rotations (distinct from lateral peering motions in previous work) with a single other target in view. This study then applies a model of insect visuomotor feedback to show via analysis and simulation of five species that these head motions sufficiently enrich the optic flow and that the output feedback can provide relative position regulation relative to the single target (asymptotic stability). In the simplifying case of pure rotation relative to the body, theoretical analysis provides a stronger stability guarantee. The results are shown to be robust to anatomical neck angle limits and body vibrations, persist with more detailed Drosophila lateral-directional flight dynamics simulations, and generalize to recent retinal motion studies. Together, these results suggest that the optic flow enrichment provided by head or pseudopupil rotation could be used in an insect's neural processing circuit to enable position regulation.

Several key plant hormones are synthesised in the shoot and are advected within the phloem to the root tip. In the root tip, these hormones regulate growth and developmental processes, and responses to environmental cues. However, we lack understanding of how environmental factors and biological parameters affect the delivery of hormones to the root tip. In this study, we build on existing models of phloem flow to develop a mathematical model of sugar transport alongside the transport of a generic hormone. We derive the equations for osmotically driven flow in a long, thin pipe with spatially varying membrane properties to capture the phloem loading and unloading zones. Motivated by experimental findings, we formulate solute membrane transport in terms of passive and active components, and incorporate solute unloading via bulk flow (i.e. advection with the water efflux) by including the Staverman reflection coefficient. We use the model to investigate the coupling between the sugar and hormone dynamics. The model predicts that environmental cues that lead to an increase in active sugar loading, an increase in bulk flow sugar unloading or a decrease in the relative root sugar concentration result in an increase in phloem transport velocity. Furthermore, the model reveals that such increases in phloem transport velocity result in an increase in hormone delivery to the root tip for passively loaded hormones.