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Abstract: Abstract We derive a discrete predator–prey model from first principles, assuming that the prey population grows to carrying capacity in the absence of predators and that the predator population requires prey in order to grow. The proposed derivation method exploits a technique known from economics that describes the relationship between continuous and discrete compounding of bonds. We extend standard phase plane analysis by introducing the next iterate root-curve associated with the nontrivial prey nullcline. Using this curve in combination with the nullclines and direction field, we show that the prey-only equilibrium is globally asymptotic stability if the prey consumption-energy rate of the predator is below a certain threshold that implies that the maximal rate of change of the predator is negative. We also use a Lyapunov function to provide an alternative proof. If the prey consumption-energy rate is above this threshold, and hence the maximal rate of change of the predator is positive, the discrete phase plane method introduced is used to show that the coexistence equilibrium exists and solutions oscillate around it. We provide the parameter values for which the coexistence equilibrium exists and determine when it is locally asymptotically stable and when it destabilizes by means of a supercritical Neimark–Sacker bifurcation. We bound the amplitude of the closed invariant curves born from the Neimark–Sacker bifurcation as a function of the model parameters. PubDate: 2022-05-21

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Abstract: Abstract Testing individuals for pathogens can affect the spread of epidemics. Understanding how individual-level processes of sampling and reporting test results can affect community- or population-level spread is a dynamical modeling question. The effect of testing processes on epidemic dynamics depends on factors underlying implementation, particularly testing intensity and on whom testing is focused. Here, we use a simple model to explore how the individual-level effects of testing might directly impact population-level spread. Our model development was motivated by the COVID-19 epidemic, but has generic epidemiological and testing structures. To the classic SIR framework we have added a per capita testing intensity, and compartment-specific testing weights, which can be adjusted to reflect different testing emphases—surveillance, diagnosis, or control. We derive an analytic expression for the relative reduction in the basic reproductive number due to testing, test-reporting and related isolation behaviours. Intensive testing and fast test reporting are expected to be beneficial at the community level because they can provide a rapid assessment of the situation, identify hot spots, and may enable rapid contact-tracing. Direct effects of fast testing at the individual level are less clear, and may depend on how individuals’ behaviour is affected by testing information. Our simple model shows that under some circumstances both increased testing intensity and faster test reporting can reduce the effectiveness of control, and allows us to explore the conditions under which this occurs. Conversely, we find that focusing testing on infected individuals always acts to increase effectiveness of control. PubDate: 2022-05-13

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Abstract: Abstract We consider a natural class of reaction networks which consist of reactions where either two species can inactivate each other (i.e., sequestration), or some species can be transformed into another (i.e., transmutation), in a way that gives rise to a feedback cycle. We completely characterize the capacity of multistationarity of these networks. This is especially interesting because such networks provide simple examples of “atoms of multistationarity”, i.e., minimal networks that can give rise to multiple positive steady states. PubDate: 2022-05-11

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Abstract: Abstract The present work studies models of oncolytic virotherapy without space variable in which virus replication occurs at a faster time scale than tumor growth. We address the questions of the modeling of virus injection in this slow–fast system and of the optimal timing for different treatment strategies. To this aim, we first derive the asymptotic of a three-species slow–fast model and obtain a two-species dynamical system, where the variables are tumor cells and infected tumor cells. We fully characterize the behavior of this system depending on the various biological parameters. In the second part, we address the modeling of virus injection and its expression in the two-species system, where the amount of virus does not appear explicitly. We prove that the injection can be described by an instantaneous jump in the phase plane, where a certain amount of tumors cells are transformed instantly into infected tumor cells. This description allows discussing qualitatively the timing of different injections in the frame of successive treatment strategies. This work is illustrated by numerical simulations. The timing and amount of injected virus may have counterintuitive optimal values; nevertheless, the understanding is clear from the phase space analysis. PubDate: 2022-05-10

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Abstract: Abstract We extended a class of coupled PDE–ODE models for studying the spatial spread of airborne diseases by incorporating human mobility. Human populations are modeled with patches, and a Lagrangian perspective is used to keep track of individuals’ places of residence. The movement of pathogens in the air is modeled with linear diffusion and coupled to the SIR dynamics of each human population through an integral of the density of pathogens around the population patches. In the limit of fast diffusion pathogens, the method of matched asymptotic analysis is used to reduce the coupled PDE–ODE model to a nonlinear system of ODEs for the average density of pathogens in the air. The reduced system of ODEs is used to derive the basic reproduction number and the final size relation for the model. Numerical simulations of the full PDE–ODE model and the reduced system of ODEs are used to assess the impact of human mobility, together with the diffusion of pathogens on the dynamics of the disease. Results from the two models are consistent and show that human mobility significantly affects disease dynamics. In addition, we show that an increase in the diffusion rate of pathogen leads to a lower epidemic. PubDate: 2022-05-04

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Abstract: Abstract Polio can circulate unobserved in regions that are challenging to monitor. To assess the probability of silent circulation, simulation models can be used to understand transmission dynamics when detection is unreliable. Model assumptions, however, impact the estimated probability of silent circulation. Here, we examine the impact of having distinct populations, rather than a single well-mixed population, with a discrete-individual model including environmental surveillance. We show that partitioning a well-mixed population into networks of distinct communities may result in a higher probability of silent circulation as a result of the time it takes for the detection of a circulation event. Population structure should be considered when assessing polio control in a region with many loosely interacting communities. PubDate: 2022-05-04

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Abstract: Abstract Although great progress has been made in the prevention and mitigation of TB in the past 20 years, China is still the third largest contributor to the global burden of new TB cases, accounting for 833,000 new cases in 2019. Improved mitigation strategies, such as vaccines, diagnostics, and treatment, are needed to meet goals of WHO. Given the huge variability in the prevalence of TB across age-groups in China, the vaccination, diagnostic techniques, and treatment for different age-groups may have different effects. Moreover, the statistics data of TB cases show significant seasonal fluctuations in China. In view of the above facts, we propose a non-autonomous differential equation model with age structure and seasonal transmission rate. We derive the basic reproduction number, \({\mathcal {R}}_{0}\) , and prove that the unique disease-free periodic solution, \({\mathcal {P}}_{0}\) is globally asymptotically stable when \({\mathcal {R}}_{0}<1\) , while the disease is uniformly persistent and at least one positive periodic solution exists when \({\mathcal {R}}_{0}>1\) . We estimate that the basic reproduction number \({\mathcal {R}}_{0}=1.3935\) ( \(95\%\text{ CI }:(1.3729, 1.4087)\) ), which means that TB is uniformly persistent. Our results demonstrate that vaccinating susceptible individuals whose ages are over 65 and between 20 and 24 is much more effective in reducing the prevalence of TB, and each of the improved vaccination strategy, diagnostic strategy, and treatment strategy leads to substantial reductions in the prevalence of TB per 100,000 individuals compared with current approaches, and the combination of the three strategies is more effective. Scenario A (i.e., coverage rate \(85\%\) , diagnosis rate \(5\theta _{k}\) , relapse rate \(0.1\chi _{k}\) ) is the best and can reduce the prevalence of TB per 100,000 individuals by \(98.91\%\) and \(99.07\%\) in 2035 and 2050, respectively. Although the improved strategies will significantly reduce the incidence rate of TB, it is challenging to achieve the goal of WHO in 2050. Our findings can provide guidance for public health authorities in projecting effective mitigation strategies of TB. PubDate: 2022-04-29

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Abstract: Abstract We show that the combination of Allee effects and noise can produce a stochastic process with alternating sudden decline to a low population phase, followed, after a random time, by abrupt increase in population density. We introduce a new, flexible, deterministic model of attenuated Allee effects, which interpolates between the logistic and a usual Allee model. Into this model, we incorporate environmental and demographic noise. The solution of the resulting Kolmogorov forward equation shows a dichotomous distribution of residence times with heavy occupation of high, near saturation, and low population states. Investigation of simulated sample paths reveals that indeed attenuated Allee effects and noise, acting together, produce alternating, sustained, low and high population levels. We find that the transition times between the two types of states are approximately exponentially distributed, with different parameters, rendering the embedded hi-low process approximately Markov. PubDate: 2022-04-24

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Abstract: Abstract The rapid rise of antibiotic resistance is a serious threat to global public health. The situation is exacerbated by the “antibiotics dilemma”: Developing narrow-spectrum antibiotics against resistant bacteria is most beneficial for society, but least attractive for companies, since their usage and sales volumes are more limited than for broad-spectrum drugs. After developing a general mathematical framework for the study of antibiotic resistance dynamics with an arbitrary number of antibiotics, we identify efficient treatment protocols. Then, we introduce a market-based refunding scheme that incentivizes pharmaceutical companies to develop new antibiotics against resistant bacteria and, in particular, narrow-spectrum antibiotics that target specific bacterial strains. We illustrate how such a refunding scheme can solve the antibiotics dilemma and cope with various sources of uncertainty that impede antibiotic R &D. Finally, connecting our refunding approach to the recently established Antimicrobial Resistance (AMR) Action Fund, we discuss how our proposed incentivization scheme could be financed. PubDate: 2022-04-22

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Abstract: Abstract The stem cell hypothesis suggests that there is a small group of malignant cells, the cancer stem cells, that initiate the development of tumors, encourage its growth, and may even be the cause of metastases. Traditional treatments, such as chemotherapy and radiation, primarily target the tumor cells leaving the stem cells to potentially cause a recurrence. Chimeric antigen receptor (CAR) T-cell therapy is a form of immunotherapy where the immune cells are genetically modified to fight the tumor cells. Traditionally, the CAR T-cell therapy has been used to treat blood cancers and only recently has shown promising results against solid tumors. We create an ordinary differential equations model which allows for the infusion of trained CAR-T cells to specifically attack the cancer stem cells that are present in the solid tumor microenvironment. Additionally, we incorporate the influence of TGF- \(\beta \) which inhibits the CAR-T cells and thus promotes the growth of the tumor. We verify the model by comparing it to available data and then examine combinations of CAR-T cell treatment targeting both non-stem and stem cancer cells and a treatment that reduces the effectiveness of TGF- \(\beta \) to determine the scenarios that eliminate the tumor. PubDate: 2022-04-16

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Abstract: Abstract Accurate prediction of the number of daily or weekly confirmed cases of COVID-19 is critical to the control of the pandemic. Existing mechanistic models nicely capture the disease dynamics. However, to forecast the future, they require the transmission rate to be known, limiting their prediction power. Typically, a hypothesis is made on the form of the transmission rate with respect to time. Yet the real form is too complex to be mechanistically modeled due to the unknown dynamics of many influential factors. We tackle this problem by using a hypothesis-free machine-learning algorithm to estimate the transmission rate from data on non-pharmaceutical policies, and in turn forecast the confirmed cases using a mechanistic disease model. More specifically, we build a hybrid model consisting of a mechanistic ordinary differential equation (ODE) model and a gradient boosting model (GBM). To calibrate the parameters, we develop an “inverse method” that obtains the transmission rate inversely from the other variables in the ODE model and then feed it into the GBM to connect with the policy data. The resulting model forecasted the number of daily confirmed cases up to 35 days in the future in the USA with an averaged mean absolute percentage error of 27%. It can identify the most informative predictive variables, which can be helpful in designing improved forecasters as well as informing policymakers. PubDate: 2022-04-08

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Abstract: Abstract Bacteria have developed resistance to antibiotics by various mechanisms, notable amongst these is the use of permeation barriers and the expulsion of antibiotics via efflux pumps. The resistance-nodulation-division (RND) family of efflux pumps is found in Gram-negative bacteria and a major contributor to multidrug resistance (MDR). In particular, Salmonella encodes five RND efflux pump systems: AcrAB, AcrAD, AcrEF, MdsAB and MdtAB which have different substrate ranges including many antibiotics. We produce a spatial partial differential equation (PDE) model governing the diffusion and efflux of antibiotic in Salmonella, via these RND efflux pumps. Using parameter fitting techniques on experimental data, we are able to establish the behaviour of multiple wild-type and efflux mutant Salmonella strains, which enables us to produce efflux profiles for each individual efflux pump system. By combining the model with a gene regulatory network (GRN) model of efflux regulation, we simulate how the bacteria respond to their environment. Finally, performing a parameter sensitivity analysis, we look into various different targets to inhibit the efflux pumps. The model provides an in silico framework with which to test these potential adjuvants to counter MDR. PubDate: 2022-04-05

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Abstract: Abstract The sudden outbreak of SARS-CoV-2 has caused the shortage of medical resources around the world, especially in developing countries and underdeveloped regions. With the continuous increase in the duration of this disease, the control of migration of humans between regions or countries has to be relaxed. Based on this, we propose a two-patches mathematical model to simulate the transmission of SARS-CoV-2 among two-patches, asymptomatic infected humans and symptomatic infected humans, where a half-saturated detection rate function is also introduced to describe the effect of medical resources. By applying the methods of linearization and constructing a suitable Lyapunov function, the local and global stability of the disease-free equilibrium of this model without migration is obtained. Further, the existence of forward/backward bifurcation is analyzed, which is caused by the limited medical resources. This means that the elimination or prevalence of the disease no longer depends on the basic reproduction number but is closely related to the initial state of asymptomatic and symptomatic infected humans and the supply of medical resources. Finally, the global dynamics of the full model are discussed, and some numerical simulations are carried to explain the main results and the effects of migration and supply of medical resources on the transmission of disease. PubDate: 2022-04-04

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Abstract: Abstract As antibiotic resistance grows more frequent for common bacterial infections, alternative treatment strategies such as phage therapy have become more widely studied in the medical field. While many studies have explored the efficacy of antibiotics, phage therapy, or synergistic combinations of phages and antibiotics, the impact of virus competition on the efficacy of antibiotic treatment has not yet been considered. Here, we model the synergy between antibiotics and two viral types, temperate and chronic, in controlling bacterial infections. We demonstrate that while combinations of antibiotic and temperate viruses exhibit synergy, competition between temperate and chronic viruses inhibits bacterial control with antibiotics. In fact, our model reveals that antibiotic treatment may counterintuitively increase the bacterial load when a large fraction of the bacteria are antibiotic resistant, and both chronic and temperate phages are present. PubDate: 2022-03-22

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Abstract: Abstract We show that under the assumption of weak frequency-dependent selection a wide class of population dynamical models can be analysed using perturbation theory. The inner solution corresponds to the ecological dynamics, where to zeroth order, the genotype frequencies remain constant. The outer solution provides the evolutionary dynamics and corresponds, to zeroth order, to a generalisation of the replicator equation. We apply this method to a model of public goods dynamics and construct, using matched asymptotic expansions, a composite solution valid for all times. We also analyse a Lotka–Volterra model of predator competition and show that to zeroth order the fraction of wild-type predators follows a replicator equation with a constant selection coefficient given by the predator death rate. For both models, we investigate how the error between approximate solutions and the solution to the full model depend on the order of the approximation and show using numerical comparison, for \(k=1\) and 2, that the error scales according to \(\varepsilon ^{k+1}\) , where \(\varepsilon \) is the strength of selection and k is the order of the approximation. PubDate: 2022-03-19

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Abstract: Abstract We introduce a class of linear compartmental models called identifiable path/cycle models which have the property that all of the monomial functions of parameters associated to the directed cycles and paths from input compartments to output compartments are identifiable and give sufficient conditions to obtain an identifiable path/cycle model. Removing leaks, we then show how one can obtain a locally identifiable model from an identifiable path/cycle model. These identifiable path/cycle models yield the only identifiable models with certain conditions on their graph structure and thus we provide necessary and sufficient conditions for identifiable models with certain graph properties. A sufficient condition based on the graph structure of the model is also provided so that one can test if a model is an identifiable path/cycle model by examining the graph itself. We also provide some necessary conditions for identifiability based on graph structure. Our proofs use algebraic and combinatorial techniques. PubDate: 2022-03-19

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Abstract: Abstract We develop a novel eco-evolutionary modelling framework and demonstrate its efficacy by simulating the evolution of trait distributions in predator and prey populations. The eco-evolutionary modelling framework assumes that population traits have beta distributions and defines canonical equations for the dynamics of each total population size, the population’s average trait value, and a measure of the population’s trait differentiation. The trait differentiation is included in the modelling framework as a phenotype analogue, Q, of Wright’s fixation index \(F_\mathrm{ST}\) , which is inversely related to the sum of the beta distribution shape parameters. The canonical equations may be used as templates to describe the evolution of population trait distributions in many ecosystems that are subject to stabilising selection. The solutions of the “population model” are compared with those of a “phenotype model” that simulates the growth of each phenotype as it interacts with every other phenotype under the same trade-offs. The models assume no sources of new phenotypic variance, such as mutation or gene flow. We examine a predator–prey system in which each population trades off growth against mortality: the prey optimises devoting resources to growth or defence against predation; and the predator trades off increasing its attack rate against increased mortality. Computer solutions with stabilising selection reveal very close agreement between the phenotype and population model results, which both predict that evolution operates to stabilise an initially oscillatory system. The population model reduces the number of equations required to simulate the eco-evolutionary system by several orders of magnitude, without losing verisimilitude for the overarching population properties. The population model also allows insights into the properties of the system that are not available from the equivalent phenotype model. PubDate: 2022-03-07 DOI: 10.1007/s11538-022-01004-8

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Abstract: Abstract We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model, which we call the substrate model, involves a partial differential equation describing the density of tissue, \({\hat{u}}(\hat{{\mathbf {x}}},{\hat{t}})\) that is coupled to the concentration of an immobile extracellular substrate, \({\hat{s}}(\hat{{\mathbf {x}}},{\hat{t}})\) . Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to \({\hat{s}}\) , while a logistic growth term models cell proliferation. The extracellular substrate \({\hat{s}}\) is produced by cells and undergoes linear decay. Preliminary numerical simulations show that this mathematical model is able to recapitulate key features of recent tissue growth experiments, including the formation of sharp fronts. To provide a deeper understanding of the model we analyse travelling wave solutions of the substrate model, showing that the model supports both sharp-fronted travelling wave solutions that move with a minimum wave speed, \(c = c_{\mathrm{min}}\) , as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, \(c > c_{\mathrm{min}}\) . We provide a geometric interpretation that explains the difference between smooth and sharp-fronted travelling wave solutions that is based on a slow manifold reduction of the desingularised three-dimensional phase space. In addition, we also develop and test a series of useful approximations that describe the shape of the travelling wave solutions in various limits. These approximations apply to both the sharp-fronted and smooth-fronted travelling wave solutions. Software to implement all calculations is available at GitHub. PubDate: 2022-03-02 DOI: 10.1007/s11538-022-01005-7

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Abstract: Abstract Opioid use disorder (OUD) has become a serious leading health issue in the USA leading to addiction, disability, or death by overdose. Research has shown that OUD can lead to a chronic lifelong disorder with greater risk for relapse and accidental overdose deaths. While the prescription opioid epidemic is a relatively new phenomenon, illicit opioid use via heroin has been around for decades. Recently, additional illicit opioids such as fentanyl have become increasingly available and problematic. We propose a mathematical model that focuses on illicit OUD and includes a class for recovered users but allows for individuals to either remain in or relapse back to the illicit OUD class. Therefore, in our model, individuals may cycle in and out of three different classes: illicit OUD, treatment, and recovered. We additionally include a treatment function with saturation, as it has been shown there is limited accessibility to specialty treatment facilities. We used 2002–2019 SAMHSA and CDC data for the US population, scaled to a medium-sized city, to obtain parameter estimates for the specific case of heroin. We found that the overdose death rate has been increasing linearly since around 2011, likely due to the increased presence of fentanyl in the heroin supply. Extrapolation of this overdose death rate, together with the obtained parameter estimates, predict that by 2038 no endemic equilibrium will exist and the only stable equilibrium will correspond to the absence of heroin use disorder in the population. There is a range of parameter values that will give rise to a backward bifurcation above a critical saturation of treatment availability. We show this for a range of overdose death rate values, thus illustrating the critical role played by the availability of specialty treatment facilities. Sensitivity analysis consistently shows the significant role of people entering treatment on their own accord, which suggests the importance of removing two of the most prevalent SAMHSA-determined reasons that individuals do not enter treatment: financial constraints and the stigma of seeking treatment for heroin use disorder. PubDate: 2022-03-02 DOI: 10.1007/s11538-022-01002-w