Mathematical Medicine and Biology: A Journal of the IMA

Centrioles duplicate when a mother centriole gives birth to a daughter that grows from its side. Polo-like-kinase 4 (PLK4), the master regulator of centriole duplication, is recruited symmetrically around the mother centriole, but it then concentrates at a single focus that defines the daughter centriole assembly site. How PLK4 breaks symmetry is unclear. Here, we propose that phosphorylated and unphosphorylated species of PLK4 form the two components of a classical Turing reaction-diffusion system. These two components bind-to/unbind-from the surface of the mother centriole at different rates, allowing a slow-diffusing activator species of PLK4 to accumulate at a single site on the mother, while a fast-diffusing inhibitor species of PLK4 suppresses activator accumulation around the rest of the centriole. This "short-range activation/long-range inhibition", inherent to Turing-systems, can drive PLK4 symmetry breaking on a continuous centriole surface, with PLK4 overexpression producing multiple PLK4 foci and PLK4 kinase inhibition leading to uniform PLK4 accumulation--as observed experimentally.

A SEIR model with an added fomite term is used to constrain the contribution of fomites to the spread of COVID-19 under the Spring 2020 lockdown in the UK. Assuming uniform priors on the reproduction number in lockdown and the fomite transmission rate, an upper limit is found on the fomite transmission rate of less than 1 contaminated object in 7 per day per infectious person (95% CL). Basing the prior on the reproduction rate during lockdown instead on the CoMix study results for the reduction in social contacts under lockdown, and assuming the reproduction number scales with the number of social contacts, provides a much more restrictive upper limit on the transmission rate by contaminated objects of fewer than 1 in 30 per day per infectious person (95% CL). Applied to postal deliveries and groceries, the upper limit on the fomite transmission rate corresponds to a probability below 1 in 70 (95% CL) that a contaminated object transmits the infection. Fewer than about half (95% CL) of the total number of deaths during the lockdown are found to arise from fomites, and most likely fewer than a quarter. These findings apply only to fomites with a transmission rate that is unaffected by a lockdown.

AL amyloidosis is a rapidly progressive disorder characterized by clonal plasma cell expansion, excessive production of light chains (LCs), and their misfolding into aggregation-prone monomers. These monomers assemble into oligomers and ultimately deposit as amyloid fibrils, particularly within cardiac tissue, where they contribute to myocardial stiffening and direct cardiotoxicity. A mechanistic model was developed to quantify the interplay between these two pathogenic processes and to examine the kinetics of LC aggregation in different compartments. Simulations reveal that LC aggregation exhibits pronounced nonlinearity: oligomer concentrations remain low during early disease stages, followed by exponential growth driven by autocatalytic conversion. When aggregation is assumed to occur within cardiac tissue, fibril deposition is approximately 25 times greater, and oligomer-induced cardiotoxicity is about five times higher, compared with aggregation occurring in the blood plasma. These differences stem from the smaller cardiac volume, which accelerates autocatalytic oligomer formation. A combined cardiac damage criterion, integrating both oligomer-induced cardiotoxicity and fibril-associated myocardial stiffening, was introduced and found to reach values approximately tenfold higher when LC aggregation occurs within cardiac tissue compared with aggregation in the blood plasma. This parameter may serve as a quantitative measure of cardiac aging or disease severity. The model also predicts that therapeutic intervention markedly reduces, but does not eliminate cardiac injury, highlighting the importance of early treatment initiation in AL amyloidosis.

Individual variation in susceptibility and exposure is subject to selection by force of infection, accelerating the natural acquisition of immunity, and reducing herd immunity thresholds and epidemic final sizes. This is a manifestation of a wider population phenomenon known as "frailty variation" in demography. Despite this theoretical understanding, public health policies continue to be guided by mathematical models that leave out most of the relevant variation and as a result inflate projected infection burdens. Here we focus on the trajectories of the coronavirus disease (COVID-19) pandemic in England and Scotland. We fit models to series of daily deaths and estimate relevant epidemiological parameters, including coefficients of variation which we find in agreement with direct measurements based on published contact surveys. Our estimates are robust to whether the data series encompass one or two pandemic waves. We conclude that herd immunity thresholds are being reached with a larger contribution of vaccination in Scotland than in England, where naturally acquired immunity is higher. These results are relevant to global vaccination policies.

{blacksquare} We augment the well-known susceptible - infected - recovered - deceased (SIRD) epidemiological model to include vaccination dynamics, implemented as a piecewise continuous simulation. We calibrate this model to reported case data in the UK at a national level, {blacksquare}Our modelling approach decouples the inherent characteristics of the infection from the degree of human interaction (as defined by the effective reproduction number, Re). This allows us to detect and infer a change in the characteristic of the infection, for example the emergence of the Kent variant, {blacksquare}We find that that the infection rate constant (k) increases by around 89% as a result of the B.1.1.7 (Kent) COVID-19 variant in England, {blacksquare}Through retrospective analysis and modelling of early epidemic case data (between March 2020 and May 2020) we estimate that [~]1.2M COVID-19 infections were unreported in the early phase of the epidemic in the UK. We also obtain an estimate of the basic reproduction number as, R0 = 3.23, {blacksquare}We use our model to assess the UK Governments roadmap for easing the third national lockdown as a result of the current vaccination programme. To do this we use our estimated model parameters and a future forecast of the daily vaccination rates of the next few months, {blacksquare}Our modelling predicts an increased number of daily cases as NPIs are lifted in May and June 2021, {blacksquare}We quantify this increase in terms of the vaccine rollout rate and in particular the percentage vaccine uptake rate of eligible individuals, and show that a reduced take up of vaccination by eligible adults may lead to a significant increase in new infections.

Despite the United States Center for Disease Control (CDC)s May 2023 expiration of the declared public health emergency pertaining to the COVID-19 pandemic (Silk 2023), approximately 3 years after the first cases of SARS-CoV-2 appeared in the United Sates, thousands of new cases persist daily. Many questions persist about the future dynamics of SARS-CoV-2s in the United States, including: will COVID continue to circulate as a seasonal disease like influenza, and will annual vaccinations be required to prevent outbreaks? In response, we present an Agent Based Networked Simulation of COVID-19 transmission to evaluate recurrent future outbreaks of the disease, accounting for contact heterogeneity and waning vaccine-derived and natural immunity. Our model is parameterized with data collected as part of the Berkeley Interpersonal Contact Survey (BICS; Feehan and Mahmud 2021) and is used to simulate time series of confirmed cases of and deaths due to SARS-CoV-2, paying special attention to seasonal forces and waning immunity (Kronfeld-Schor et al. 2021; X. Liu et al. 2021; Nichols et al. 2021). From the BICS ABM model we simulate SARS-CoV-2 dynamics over the 10-year period beginning in 2021 with waning immunity and inclusion of annual booster doses under a variety of transmission scenarios. We find that SARS-CoV-2 outbreaks are likely to occur frequently, and that distribution of booster doses during certain times of the year--notably in the late winter/early spring--may reduce the severity of a wintertime outbreak depending on the seasonal epidemiology of the pathogen.

A second dose deferred strategy has been proposed to increase initial population immunity as an alternative to the default two dose vaccine regimen with spacing of 21 or 28 days between vaccine doses for the mRNA vaccines from Pfizer and Moderna. This increased initial population immunity is only of value if one dose immunity does not decay so fast as to nullify the benefit. Because decay rates of one dose and two dose efficacy are currently unknown, a model to project population immunity between the two strategies was created. By evaluating the decay rate of one dose efficacy, two dose efficacy, and time until the second dose is given, the model shows that if there is an increased decay rate of one dose efficacy relative to the two dose decay rate, it is highly unlikely to nullify the benefit of increased population immunity seen in a second dose deferral strategy. Rather, all reasonable scenarios strongly favour a second dose deferral strategy with much higher projected population immunity in comparison to the default regimen.

Understanding the incubation period of Legionnaires disease is vital for accurate source-term identification. Traditionally, researchers estimate the dose-dependent incubation period from human outbreak data, but this method suffers from the inability to estimate the exposure dose retrospectively for each case. This challenge limit the precision of incubation-period analysis using human case data. Existing within-host models, such as ordinary differential equation (ODE)-based and discrete-event stochastic approaches, estimate the dose-dependent incubation period of Legionnaires disease. However, discrete-event models, while useful, are so computationally costly that the within-host dynamics must be simplified to solely the Legionella and macrophage interactions. This simplification makes the computation feasible, but precludes cytokine interactions and adaptive immune response modelling. In this paper, we develop a new approach to model the within-host dynamics of Legion-naires disease that focuses on reducing computational cost while maintaining accuracy. Specifically, we propose a hybrid framework that integrates and improves upon existing ODE and discrete event within-host models of Legionnaires disease. By integrating the previously developed ODE and discrete-event stochastic models with stochastic differential equation (SDE) models, we create a unified system that adapts dynamically throughout the infection process. We quantify the points at which each model becomes the optimal tool for describing the infection, resulting in a flexible simulation of disease dynamics. Our hybrid model aligns with observed human incubation-period data and is the first framework of its kind in this context. This advancement offers a more robust platform for testing additional biological assumptions and improving our understanding of Legionnaires disease.

In this paper, for an infectious disease such as Covid-19, we present a SIR model which examines the impact of waning immunity, vaccination rates, vaccine efficacy, and the proportion of the susceptible population who aspire to be vaccinated. Under an assumed constant control reproduction number, we provide simple conditions for the disease to be eliminated, and conversely for it to exhibit the more likely endemic behaviour. With regard to Covid-19, it is shown that if the control reproduction number is set to the basic reproduction number (say 6) of the dominant delta (B1.617.2) variant, vaccination alone, even under the most optimistic of assumptions about vaccine efficacy and high vaccine coverage, is very unlikely to lead to elimination of the disease. The model is not intended to be predictive but more an aid to understanding the relative importance of various biological and control parameters. For example, from a long-term perspective, it may be found that in the UK, through changes in societal behaviour (such as mask use, ventilation, and level of homeworking), without formal government interventions such as on-off lockdowns, the control reproduction number can still be maintained at a level significantly below the basic reproduction number. Even so, our simulations show that endemic behaviour ensues. The model obtains equilibrium values of the state variables such as the infection prevalence and mortality rate under various scenarios.

Recent clinical research finds that rapid transmission of SARS-CoV-2 is facilitated by substantial undocumented asymptomatic infections. Asymptomatic infections have implications for behavioral response to voluntary testing. The paper argues that a substantial proportion of SARS-CoV-2 infections are hidden due to rational test avoidance behavior, especially among those without perceptible disease symptoms. However, if perception of disease threat is prevalence dependent, testing compliance increases in response to reported infection prevalence rate in the population. This behavior, in turn, affects infection and mortality dynamics. This paper proposes an analytical framework that explicitly incorporates prevalence-dependent testing behavior in a standard epidemiological model, generating distinctive equilibrium epidemiological outcomes with significant policy implications. Numerical simulations show that failure to consider endogenous testing behavior among asymptomatic individuals leads to over- and underestimation of infection rates at the peaks and troughs, respectively, thereby distorting the disease containment policies. The results underscore the importance of augmenting testing capacity as an effective mitigation policy for COVID-19 and similar infectious diseases. JEL CodesI12, I18

This article proposes (and demonstrates the effectiveness of) a new strategy for assessing the results of epidemic models which we designate reproduction. The strategy is to build an independent model that uses (as far as possible) only the published information about the model to be assessed. In the example presented here, the independent model also follows a different modelling approach (agent-based modelling) to the model being assessed (the London School of Hygiene and Tropical Medicine compartmental model which has been influential in COVID lockdown policy). The argument runs that if the policy prescriptions of the two models match then this independently supports them (and reduces the chance that they are artefacts of assumptions, modelling approach or programming bugs). If, on the other hand, they do not match then either the model being assessed is not provided with sufficient information to be relied on or (perhaps) there is something wrong with it. In addition to justifying the approach, describing the two models and demonstrating the success of the approach, the article also discusses additional benefits of the reproduction strategy independent of whether match between policy prescriptions is actually achieved.

The formation of amyloid beta (A{beta}) deposits (senile plaques) is one of the hallmarks of Alzheimers disease (AD). This study investigates what processes are primarily responsible for their formation. A model is developed to simulate the diffusion of amyloid beta (A{beta}) monomers, the production of free A{beta} aggregates through nucleation and autocatalytic processes, and the deposition of these aggregates into senile plaques. The model suggests that efficient degradation of A{beta} monomers alone may suffice to prevent the growth of senile plaques, even without degrading A{beta} aggregates and existing plaques. This is because the degradation of A{beta} monomers interrupts the supply of reactants needed for plaque formation. The impact of A{beta} monomer diffusivity is demonstrated to be small, enabling the application of the lumped capacitance approximation and the derivation of approximate analytical solutions for limiting cases with both small and large rates of A{beta} aggregate deposition into plaques. It is found that the rate of plaque growth is governed by two competing processes. One is the deposition rate of free A{beta} aggregates into senile plaques. If this rate is small, the plaque grows slowly. However, if the rate of deposition of A{beta} aggregates into senile plaques is very large, the free A{beta} aggregates are removed from the intracellular fluid by deposition into the plaques, leaving insufficient free A{beta} aggregates to catalyze the production of new aggregates. This suggests that under certain conditions, A{beta} plaques may offer neuroprotection and impede their own growth. Additionally, it indicates that there exists an optimal rate of deposition of free A{beta} aggregates into the plaques, at which the plaques attain their maximum size.

This study proposes using accumulated neurotoxicity, defined as the time integral of A{beta} oligomer concentration, as a biomarker for neuronal aging. A relationship between biological age and accumulated neurotoxicity is proposed. Numerical analysis guided the development of a new analytical solution linking the biological and calendar ages of neurons. The effects of A{beta} monomer and oligomer half-lives--key indicators of proteolytic efficiency--on biological age are examined. Both constant and age-dependent (exponentially increasing) half-life scenarios are considered. The findings indicate that increasing the half-life of A{beta} monomers and oligomers with age accelerates biological aging. Reducing A{beta} monomer production is shown to slow biological aging, with a linear relationship established between these two quantities. Additionally, biological age is found to depend linearly on the half-deposition time of A{beta} oligomers into senile plaques. The model demonstrates that biological age is irreversible, providing a theoretical explanation for why plaque-clearing therapies cannot reverse established cognitive impairment. The model also demonstrates that biological age is path-dependent rather than state-dependent.

When effective vaccines are available, vaccination programs are typically one of the best defences against the spread of an infectious disease. Unfortunately, vaccination rates may be suboptimal for a prolonged duration as a result of slow uptake of vaccines by the public. Key factors driving slow vaccination uptake can be a complex interaction of vaccine roll-out policies and logistics, and vaccine hesitancy behaviours potentially caused by an inflated sense of risk in adverse reactions in some populations or community complacency in communities that have not yet experienced a large outbreak. In the recent COVID-19 pandemic, public health responses around the world began to include vaccination programs from late 2020 to early 2021 with an aim of relaxing non-pharmaceutical interventions such as lockdowns and travel restrictions. For many jurisdictions there have been challenges in getting vaccination rates high enough to enable the relaxation of restrictions based on non-pharmaceutical interventions. A key concern during this time was vaccine hestitancy behaviours potentially caused by vaccine safety concerns fuelled by misinformation and community complacency in jurisdictions that had seen very low COVID-19 case numbers throughout 2020, such as Australia and New Zealand. We develop a novel stochastic epidemiological model of COVID-19 transmission that incorporates changes in population behaviour relating to responses based on non-pharmaceutical interventions and community vaccine uptake as functions of the reported COVID-19 cases, deaths, and vaccination rates. Through a simulation study, we develop a Bayesian analysis approach to demonstrate that different factors inhibiting the uptake of vaccines by the population can be isolated despite key model parameters being subject to substantial uncertainty. In particular, we are able to identify the presence of vaccine hesitancy in a population using reported case, death and vaccination count data alone. Furthermore, our approach provides insight as to whether the dominant concerns driving hesitancy are related to vaccine safety or complacency. While our simulation study is inspired by the COVID-19 pandemic, our tools and techniques are general and could be enable vaccination programs of various infectious diseases to be adapted rapidly in response to community behaviours moving forward into the future.

Two different epidemiological models of vaccination are commonly used in dynamical modeling studies. The leaky vaccination model assumes that all vaccinated individuals experience a reduced force of infection by the same amount. The polarized vaccination model assumes that some fraction of vaccinated individuals are completely protected, while the remaining fraction remains completely susceptible; this seemingly extreme assumption causes the polarized model to always predict lower final epidemic size than the leaky model under the same vaccine efficacy. However, the leaky model also makes an implicit, unrealistic assumption: vaccinated individuals who are exposed to infection but not infected remain just as susceptible as they were prior to exposures (i.e., independent of previous exposures). To resolve the independence assumption, we introduce an immune boosting mechanism, through which vaccinated, yet susceptible, individuals can gain protection without developing a transmissible infection. The boosting model further predicts identical epidemic dynamics as the polarized vaccination model, thereby bridging the differences between two models. We further develop a generalized vaccination model to explore how the assumptions of immunity affect epidemic dynamics and estimates of vaccine effectiveness. Significance statementDifferent assumptions about the long- and medium-term effects of protective vaccination can predict sharply different epidemiological dynamics. However, there has been limited discussion about which assumptions are more realistic and therefore more appropriate for making public health decisions. Here, we show that the differences between the two most common assumptions (the "leaky" and "polarized" vaccination models) are bridged by immune boosting, a mechanism by which individuals who resist infectious challenge due to partial immunity have their immunity increased. We demonstrate that this mechanism has important implications for measuring vaccine effectiveness. Our study challenges fundamental assumptions about commonly used vaccination models and provides a novel framework for understanding the epidemiological impact of vaccination.

This work proposes that epidemiological features of both endemic coronaviruses and the recent highly pathogenic outbreak coronaviruses can be combined within an integrated framework. In this framework, mortality amongst those infected for the first time is mostly amongst the old but survivors acquire fatal infection immunity (FII). Subjects with FII can subsequently be infected and infect others without suffering significant mortality. Under these conditions, coronaviruses induce endemic infections that elicit FII in individuals during childhood when the risk of mortality is low and maintain it throughout their lifetime, thereby protecting the population against the worst effects of infection. A multi-compartment ODE model was constructed to explore the implications of this proposal on the evolution of a zoonosis sharing properties of both SARS-CoV-2 and endemic coronaviruses. The results show that mortality has two components, the first incurred during transition to endemicity and the other is exacted on a continuing basis. The relative contribution of each depends on the longevity of the FII state. In particular, a one-time vaccination of the older subpopulation is sufficient to reduce total mortality if FII is long-lived. The effect of a regular vaccination was also examined when FII was shorter lived. Herd immunity was not achieved. The validity of this proposal with regard to Covid-19 depends on whether reinfection with SARS-CoV-2 behaves in the manner expected of FII. If it does, then certain considerations apply to how Covid-19 is to be managed and how vaccine choice could influence that.

Humans are a hyper social species, which greatly impacts the spread of infectious diseases. How do social dynamics impact epidemiology? How does public health policy best take into account these impacts? Here we develop a model of disease transmission that incorporates human behaviour and social dynamics. We use a "tipping-point" dynamic, previously used in the sociological literature, where individuals adopt a behaviour given a sufficient frequency of the behaviour in the population. The thresholds at which individuals adopt behaviours is modulated by the perceived risks of infection, i.e. the disease prevalence and transmission rate, and the behaviour of others. Social conformity creates a type of "stickiness" whereby individuals are resistant to changing their behaviour due to the populations inertia. In this model, the epidemic attack rate is sensitive to the timing of the behavioural response. Near the optimal response, small errors can result in large increases in the total number infected during the epidemic. And, more surprisingly, we observe a non-monotinicity in the attack rate as a function of various biological and social parameters such as the transmission rate, efficacy of social distancing, the costs to social distancing, the weight of social consequences of shirking the norm, and the degree of heterogeneity in the population.

Prion diseases are lethal neurodegenerative disorders such as mad cow disease in bovines, chronic wasting disease in cervids, and Creutzfeldt-Jakob disease in humans. They are caused when the prion protein PrPC misfolds into PrPSc, which is capable of inducing further misfolding in healthy PrPC proteins. Recent in vivo experiments show that pharmacological chaperones can temporarily prevent this conversion by binding to PrPC molecules, and thus constitute a possible treatment. A second strategic approach uses interferons to decrease the concentration of PrPSc. In order to study the quantitative effects of these treatments on prion proliferation, we develop a model using a non-linear system of ordinary differential equations. By evaluating their efficacy and potency, we find that interferons act at lower doses and achieve greater prion decay rates. However, there are benefits in combining them with pharmacological chaperones in a two-fold therapy. This research is crucial to guide future prion experiments and inform potential treatment protocols.Competing Interest StatementThe authors have declared no competing interest.View Full Text

This report considers three mechanisms that might underlie the course of the secondary peak of coronavirus infections in the United Kingdom. It considers: (i) fluctuations in transmission strength; (ii) seasonal fluctuations in contact rates and (iii) fluctuations in testing. Using dynamic causal modelling, we evaluated the contribution of all combinations of these three mechanisms using Bayesian model comparison. We found overwhelming evidence for the combination of all mechanisms, when explaining 16 types of data. Quantitatively, there was clear evidence for an increase in transmission strength of 57% over the past months (e.g., due to viral mutation), in the context of increased contact rates (e.g., rebound from national lockdowns) and increased test rates (e.g., due to the inclusion of lateral flow tests). Models with fluctuating transmission strength outperformed models with fluctuating contact rates. However, the best model included all three mechanisms suggesting that the resurgence during the second peak can be explained by an increase in effective contact rate that is the product of a rebound of contact rates following a national lockdown and increased transmission risk due to viral mutation.

Alzheimers disease (AD) is characterized by the accumulation of amyloid-{beta} (A{beta}), yet the specific link between plaque burden and cognitive decline remains a subject of intense investigation. This paper presents a mathematical model that simulates the coupled dynamics of A{beta} monomers, soluble oligomers, and fibrillar species in the brain tissue. By modifying existing moment equations to include a dedicated conservation equation for A{beta} monomers, the model explores how various microscopic processes, such as primary nucleation, surface-catalyzed secondary nucleation, fibril elongation, and fragmentation, contribute to macroscopic disease progression. Central to this study is the concept of "accumulated neurotoxicity" as a surrogate marker of biological age, defined as the time-integrated concentration of soluble A{beta} oligomers. Unlike plaque burden, accumulated neurotoxicity cannot be reversed, and the harm it causes depends critically on the sequence of events that produced it. Numerical results demonstrate that while plaque burden and neurotoxicity both increase over time, their relationship is non-linear and highly sensitive to the efficiency of protein degradation machinery. Specifically, impaired degradation leads to a rapid advancement of biological age relative to calendar age. The model further identifies oligomer dissociation and fibril fragmentation as potential protective mechanisms that can counterintuitively reduce neurotoxic burden by diverting monomers away from the soluble oligomer pool. These findings provide a quantitative framework for understanding why individuals with similar plaque burdens may experience vastly different cognitive outcomes, underscoring the importance of targeting soluble oligomers early in therapeutic interventions.