Exponential dynamics of DNA methylation with age.

The association of DNA methylation with age has been extensively studied. Previous work has investigated the trajectories of methylation with age, and developed predictive biomarkers of age. However, we still have a limited understanding of the functional form of methylation-age dynamics. To address this we present a theoretical framework to model the dynamics of DNA methylation at single sites. We show that this model leads to convergence to a steady-state methylation level at an exponential rate. By fitting the model to a dataset that measures changes in DNA methylation in the brain from birth to old age, we show that the timescales of this exponential convergence are heterogeneous across sites. To model this heterogeneity we generated a simulation of CpG Methylation changes with time and investigated the functional form of the dynamics of methylation with age under the empirical distribution of timescales estimated from the dataset. The resulting dynamics of the average methylation of the system were characterized and were found to closely follow an exponential trajectory. We conclude that DNA methylation can be modeled as a system that starts out of equilibrium at birth and approaches equilibrium with age in an exponential fashion. These insights illustrate the importance of accounting for nonlinear dynamics when utilizing age associated DNA methylation changes for constructing biomarkers of aging. Thus DNA methylation, along with the exponentially increasing risk of mortality with age, further establishes the exponential nature of aging.

Optimizing Antimicrobial Treatment Schedules: Some Fundamental Analytical Results.

This work studies fundamental questions regarding the optimal design of antimicrobial treatment protocols, using pharmacodynamic and pharmacokinetic mathematical models. We consider the problem of designing an antimicrobial treatment schedule to achieve eradication of a microbial infection, while minimizing the area under the time-concentration curve (AUC), which is equivalent to minimizing the cumulative dosage. We first solve this problem under the assumption that an arbitrary antimicrobial concentration profile may be chosen, and prove that the ideal concentration profile consists of a constant concentration over a finite time duration, where explicit expressions for the optimal concentration and the time duration are given in terms of the pharmacodynamic parameters. Since antimicrobial concentration profiles are induced by a dosing schedule and the antimicrobial pharmacokinetics, the 'ideal' concentration profile is not strictly feasible. We therefore also investigate the possibility of achieving outcomes which are close to those provided by the 'ideal' concentration profile, using a bolus+continuous dosing schedule, which consists of a loading dose followed by infusion of the antimicrobial at a constant rate. We explicitly find the optimal bolus+continuous dosing schedule, and show that, for realistic parameter ranges, this schedule achieves results which are nearly as efficient as those attained by the 'ideal' concentration profile. The optimality results obtained here provide a baseline and reference point for comparison and evaluation of antimicrobial treatment plans.

Gene Transfer-Based Phylogenetics: Analytical Expressions and Additivity via Birth-Death Theory.

The genomic era has opened up vast opportunities in molecular systematics, one of which is deciphering the evolutionary history in fine detail. Under this mass of data, analyzing the point mutations of standard markers is often too crude and slow for fine-scale phylogenetics. Nevertheless, genome dynamics (GD) events provide alternative, often richer information. The synteny index (SI) between a pair of genomes combines gene order and gene content information, allowing the comparison of genomes of unequal gene content, together with order considerations of their common genes. Recently, genome dynamics has been modeled as a continuous-time Markov process, and gene distance in the genome as a birth-death-immigration process. Nevertheless, due to complexities arising in this setting, no precise and provably consistent estimators could be derived, resulting in heuristic solutions. Here, we extend this modeling approach by using techniques from birth-death theory to derive explicit expressions of the system's probabilistic dynamics in the form of rational functions of the model parameters. This, in turn, allows us to infer analytically accurate distances between organisms based on their SI. Subsequently, we establish additivity of this estimated evolutionary distance (a desirable property yielding phylogenetic consistency). Applying the new measure in simulation studies shows that it provides accurate results in realistic settings and even under model extensions such as gene gain/loss or over a tree structure. In the real-data realm, we applied the new formulation to unique data structure that we constructed-the ordered orthology DB-based on a new version of the EggNOG database, to construct a tree with more than 4.5K taxa. To the best of our knowledge, this is the largest gene-order-based tree constructed and it overcomes shortcomings found in previous approaches. Constructing a GD-based tree allows to confirm and contrast findings based on other phylogenetic approaches, as we show.

Optimal vaccination at high reproductive numbers: sharp transitions and counterintuitive allocations.

Optimization of vaccine allocations among different segments of a heterogeneous population is important for enhancing the effectiveness of vaccination campaigns in reducing the burden of epidemics. Intuitively, it would seem that allocations designed to minimize infections should prioritize those with the highest risk of being infected and infecting others. This prescription is well supported by vaccination theory, e.g. when the vaccination campaign aims to reach herd immunity. In this work, we show, however, that for vaccines providing partial protection (leaky vaccines) and for sufficiently high values of the basic reproduction number, intuition is overturned: the optimal allocation minimizing the number of infections prioritizes the vaccination of those who are least likely to be infected. The work combines numerical investigations, asymptotic analysis for a general model, and complete mathematical analysis in a two-group model. The results point to important considerations in managing vaccination campaigns for infections with high transmissibility.

Dispersal-induced growth in a time-periodic environment.

Dispersal-induced growth (DIG) occurs when several populations with time-varying growth rates, each of which, when isolated, would become extinct, are able to persist and grow exponentially when dispersal among the populations is present. This work provides a mathematical exploration of this surprising phenomenon, in the context of a deterministic model with periodic variation of growth rates, and characterizes the factors which are important in generating the DIG effect, and the corresponding conditions on the parameters involved.

Population-level implications of the Israeli booster campaign to curtail COVID-19 resurgence.

Israel was one of the first countries to administer mass vaccination against severe acute respiratory syndrome coronavirus 2. Consequently, it was among the first countries to experience substantial breakthrough infections due to the waning of vaccine-induced immunity, which led to a resurgence of the epidemic. In response, Israel launched a booster campaign to mitigate the outbreak and was the first country to do so. Israel's success in curtailing the Delta resurgence while imposing only mild nonpharmaceutical interventions influenced the decision of many countries to initiate a booster campaign. By constructing a detailed mathematical model and calibrating it to the Israeli data, we extend the understanding of the impact of the booster campaign from the individual to the population level. We used the calibrated model to explore counterfactual scenarios in which the booster vaccination campaign is altered by changing the eligibility criteria or the start time of the campaign and to assess the direct and indirect effects in the different scenarios. The results point to the vast benefits of vaccinating younger age groups that are not at a high risk of developing severe disease but play an important role in transmission. We further show that, when the epidemic is exponentially growing, the success of the booster campaign is highly sensitive to the timing of its initiation. Hence, a rapid response is an important factor in reducing disease burden using booster vaccination.

The role of childrens' vaccination for COVID-19-Pareto-optimal allocations of vaccines.

COVID-19 vaccines have been approved for children of age five and older in many countries. However, there is an ongoing debate as to whether children should be vaccinated and at what priority. In this work, we use mathematical modeling and optimization to study how vaccine allocations to different age groups effect epidemic outcomes. In particular, we consider the effect of extending vaccination campaigns to include the vaccination of children. When vaccine availability is limited, we consider Pareto-optimal allocations with respect to competing measures of the number of infections and mortality and systematically study the trade-offs among them. In the scenarios considered, when some weight is given to the number of infections, we find that it is optimal to allocate vaccines to adolescents in the age group 10-19, even when they are assumed to be less susceptible than adults. We further find that age group 0-9 is included in the optimal allocation for sufficiently high values of the basic reproduction number.

The role of children in the spread of COVID-19: Using household data from Bnei Brak, Israel, to estimate the relative susceptibility and infectivity of children.

One of the significant unanswered questions about COVID-19 epidemiology relates to the role of children in transmission. This study uses data on infections within households in order to estimate the susceptibility and infectivity of children compared to those of adults. The data were collected from households in the city of Bnei Brak, Israel, in which all household members were tested for COVID-19 using PCR (637 households, average household size of 5.3). In addition, serological tests were performed on a subset of the individuals in the study. Inspection of the PCR data shows that children are less likely to be tested positive compared to adults (25% of children positive over all households, 44% of adults positive over all households, excluding index cases), and the chance of being positive increases with age. Analysis of joint PCR/serological data shows that there is under-detection of infections in the PCR testing, which is more substantial in children. However, the differences in detection rates are not sufficient to account for the differences in PCR positive rates in the two age groups. To estimate relative transmission parameters, we employ a discrete stochastic model of the spread of infection within a household, allowing for susceptibility and infectivity parameters to differ among children and adults. The model is fitted to the household data using a simulated maximum likelihood approach. To adjust parameter estimates for under-detection of infections in the PCR results, we employ a multiple imputation procedure using estimates of under-detection in children and adults, based on the available serological data. We estimate that the susceptibility of children (under 20 years old) is 43% (95% CI: [31%, 55%]) of the susceptibility of adults. The infectivity of children was estimated to be 63% (95% CI: [37%, 88%]) relative to that of adults.

Modeling the spread of polio in an IPV-vaccinated population: lessons learned from the 2013 silent outbreak in southern Israel.

BACKGROUND: Polio eradication is an extraordinary globally coordinated health program in terms of its magnitude and reach, leading to the elimination of wild poliovirus (WPV) in most parts of the world. In 2013, a silent outbreak of WPV was detected in Israel, a country using an inactivated polio vaccine (IPV) exclusively since 2005. The outbreak was detected using environmental surveillance (ES) of sewage reservoirs. Stool surveys indicated the outbreak to be restricted mainly to children under the age of 10 in the Bedouin population of southern Israel. In order to curtail the outbreak, a nationwide vaccination campaign using oral polio vaccine (OPV) was conducted, targeting all children under 10. METHODS: A transmission model, fitted to the results of the stool surveys, with additional conditions set by the ES measurements, was used to evaluate the prevalence of WPV in Bedouin children and the effectiveness of the vaccination campaign. Employing the parameter estimates of the model fitting, the model was used to investigate the effect of alternative timings, coverages and dosages of the OPV campaign on the outcome of the outbreak. RESULTS: The mean estimate for the mean reproductive number was 1.77 (95 % credible interval, 1.46-2.30). With seasonal variation, the reproductive number maximum range was between zero and six. The mean estimate for the mean infectious periods was 16.8 (8.6-24.9) days. The modeling indicates the OPV campaign was effective in curtailing the outbreak. The mean estimate for the attack rate in Bedouin children under 10 at the end of 2014 was 42 % (22-65 %), whereas without the campaign the mean projected attack rate was 57 % (35-74 %). The campaign also likely shortened the duration of the outbreak by a mean estimate of 309 (2-846) days. A faster initiation of the OPV campaign could have reduced the incidence of WPV even if a lower coverage was reached, at the risk of prolonging the outbreak. CONCLUSIONS: OPV campaigns are essential for interrupting WPV transmission, even in a developed country setting with a high coverage of IPV. In this setting, establishing ES of WPV circulation is particularly crucial for early detection and containment of an outbreak.

Modeling the Impact of White-Plague Coral Disease in Climate Change Scenarios.

Coral reefs are in global decline, with coral diseases increasing both in prevalence and in space, a situation that is expected only to worsen as future thermal stressors increase. Through intense surveillance, we have collected a unique and highly resolved dataset from the coral reef of Eilat (Israel, Red Sea), that documents the spatiotemporal dynamics of a White Plague Disease (WPD) outbreak over the course of a full season. Based on modern statistical methodologies, we develop a novel spatial epidemiological model that uses a maximum-likelihood procedure to fit the data and assess the transmission pattern of WPD. We link the model to sea surface temperature (SST) and test the possible effect of increasing temperatures on disease dynamics. Our results reveal that the likelihood of a susceptible coral to become infected is governed both by SST and by its spatial location relative to nearby infected corals. The model shows that the magnitude of WPD epidemics strongly depends on demographic circumstances; under one extreme, when recruitment is free-space regulated and coral density remains relatively constant, even an increase of only 0.5°C in SST can cause epidemics to double in magnitude. In reality, however, the spatial nature of transmission can effectively protect the community, restricting the magnitude of annual epidemics. This is because the probability of susceptible corals to become infected is negatively associated with coral density. Based on our findings, we expect that infectious diseases having a significant spatial component, such as Red-Sea WPD, will never lead to a complete destruction of the coral community under increased thermal stress. However, this also implies that signs of recovery of local coral communities may be misleading; indicative more of spatial dynamics than true rehabilitation of these communities. In contrast to earlier generic models, our approach captures dynamics of WPD both in space and time, accounting for the highly seasonal nature of annual WPD outbreaks.

Spatio-temporal synchrony of influenza in cities across Israel: the "Israel is one city" hypothesis.

We analysed an 11-year dataset (1998-2009) of Influenza-Like Illness (ILI) that was based on surveillance of ∽23% of Israel's population. We examined whether the level of synchrony of ILI epidemics in Israel's 12 largest cities is high enough to view Israel as a single epidemiological unit. Two methods were developed to assess the synchrony: (1) City-specific attack rates were fitted to a simple model in order to estimate the temporal differences in attack rates and spatial differences in reporting rates of ILI. The model showed good fit to the data (R2  =  0.76) and revealed considerable differences in reporting rates of ILI in different cities (up to a factor of 2.2). (2) A statistical test was developed to examine the null hypothesis (H0) that ILI incidence curves in two cities are essentially identical, and was tested using ILI data. Upon examining all possible pairs of incidence curves, 77.4% of pairs were found not to be different (H0 was not rejected). It was concluded that all cities generally have the same attack rate and follow the same epidemic curve each season, although the attack rate changes from season to season, providing strong support for the "Israel is one city" hypothesis. The cities which were the most out of synchronization were Bnei Brak, Beersheba and Haifa, the latter two being geographically remote from all other cities in the dataset and the former geographically very close to several other cities but socially separate due to being populated almost exclusively by ultra-orthodox Jews. Further evidence of assortative mixing of the ultra-orthodox population can be found in the 2001-2002 season, when ultra-orthodox cities and neighborhoods showed distinctly different incidence curves compared to the general population.

Modeling and statistical analysis of the spatio-temporal patterns of seasonal influenza in Israel.

BACKGROUND: Seasonal influenza outbreaks are a serious burden for public health worldwide and cause morbidity to millions of people each year. In the temperate zone influenza is predominantly seasonal, with epidemics occurring every winter, but the severity of the outbreaks vary substantially between years. In this study we used a highly detailed database, which gave us both temporal and spatial information of influenza dynamics in Israel in the years 1998-2009. We use a discrete-time stochastic epidemic SIR model to find estimates and credible confidence intervals of key epidemiological parameters. FINDINGS: Despite the biological complexity of the disease we found that a simple SIR-type model can be fitted successfully to the seasonal influenza data. This was true at both the national levels and at the scale of single cities.The effective reproductive number R(e) varies between the different years both nationally and among Israeli cities. However, we did not find differences in R(e) between different Israeli cities within a year. R(e) was positively correlated to the strength of the spatial synchronization in Israel. For those years in which the disease was more "infectious", then outbreaks in different cities tended to occur with smaller time lags. Our spatial analysis demonstrates that both the timing and the strength of the outbreak within a year are highly synchronized between the Israeli cities. We extend the spatial analysis to demonstrate the existence of high synchrony between Israeli and French influenza outbreaks. CONCLUSIONS: The data analysis combined with mathematical modeling provided a better understanding of the spatio-temporal and synchronization dynamics of influenza in Israel and between Israel and France. Altogether, we show that despite major differences in demography and weather conditions intra-annual influenza epidemics are tightly synchronized in both their timing and magnitude, while they may vary greatly between years. The predominance of a similar main strain of influenza, combined with population mixing serve to enhance local and global influenza synchronization within an influenza season.

The size of epidemics in populations with heterogeneous susceptibility.

We formulate and study a general epidemic model allowing for an arbitrary distribution of susceptibility in the population. We derive the final-size equation which determines the attack rate of the epidemic, somewhat generalizing previous work. Our main aim is to use this equation to investigate how properties of the susceptibility distribution affect the attack rate. Defining an ordering among susceptibility distributions in terms of their Laplace transforms, we show that a susceptibility distribution dominates another in this ordering if and only if the corresponding attack rates are ordered for every value of the reproductive number R0. This result is used to prove a sharp universal upper bound for the attack rate valid for any susceptibility distribution, in terms of R0 alone, and a sharp lower bound in terms of R0 and the coefficient of variation of the susceptibility distribution. We apply some of these results to study two issues of epidemiological interest in a population with heterogeneous susceptibility: (1) the effect of vaccination of a fraction of the population with a partially effective vaccine, (2) the effect of an epidemic of a pathogen inducing partial immunity on the possibility and size of a future epidemic. In the latter case, we prove a surprising '50% law': if infection by a pathogen induces a partial immunity reducing susceptibility by less than 50%, then, whatever the value of R0>1 before the first epidemic, a second epidemic will occur, while if susceptibility is reduced by more than 50%, then a second epidemic will only occur if R0 is larger than a certain critical value greater than 1.

Attack rates of seasonal epidemics.

We study the attack rate, that is the total fraction of the population infected each year, for a disease with seasonally varying transmission rate. The attack rate is shown to be governed by both the reproductive number, reflecting the transmissibility of the disease, and the birth rate, which provides a source of new susceptibles. For the case of epidemics which have an annual period (like the seasonality), we prove inequalities which show that the attack rate is close to that of the non-seasonal model, so that it is nearly independent of the strength of the forcing, despite the fact that the shape of the epidemic curve depends strongly on the degree of seasonality of the forcing. Numerical simulations show that this holds to an even stronger extent than is implied by our rigorous results. When the system has subharmonic or chaotic solutions, we show that similar results hold when the attack rate is replaced by the average attack rate over several years. Consequences of these findings for analyzing the effect of vaccination in seasonally-forced models are noted.

Modelling seasonal influenza in Israel.

Mathematical modeling approaches are used to study the epidemic dynamics of seasonal influenza in Israel. The recent availability of highly resolved ten year timeseries of influenza cases provides an opportunity for modeling and estimating important epidemiological parameters in the Israeli population. A simple but well known SIR discrete-time deterministic model was fitted to consecutive epidemics allowing estimation of the initial number of susceptibles in the population S0, as well as the reproductive number R0 each year. The results were corroborated by implementing a stochastic model and using a maximum likelihood approach. The paper discusses the difficulties in estimating these important parameters especially when the reporting rate of influenza cases might only be known with limited accuracy, as is generally the case. In such situations invariant parameters such as the percentage of susceptibles infected, and the effective reproductive rate might be preferred, as they do not depend on reporting rate. Results are given based on the Israeli timeseries.

Onset of a pandemic: characterizing the initial phase of the swine flu (H1N1) epidemic in Israel.

BACKGROUND: The swine influenza H1N1 first identified in Mexico, spread rapidly across the globe and is considered the fastest moving pandemic in history. The early phase of an outbreak, in which data is relatively scarce, presents scientific challenges on key issues such as: scale, severity and immunity which are fundamental for establishing sound and rapid policy schemes. Our analysis of an Israeli dataset aims at understanding the spatio-temporal dynamics of H1N1 in its initial phase. METHODS: We constructed and analyzed a unique dataset from Israel on all confirmed cases (between April 26 to July 7, 2009), representing most swine flu cases in this period. We estimated and characterized fundamental epidemiological features of the pandemic in Israel (e.g. effective reproductive number, age-class distribution, at-risk social groups, infections between sexes, and spatial dynamics). Contact data collected during this stage was used to estimate the generation time distribution of the pandemic. RESULTS: We found a low effective reproductive number (Re=1.06), an age-class distribution of infected individuals (skewed towards ages 18-25), at-risk social groups (soldiers and ultra Orthodox Jews), and significant differences in infections between sexes (skewed towards males). In terms of spatial dynamics, the pandemic spread from the central coastal plain of Israel to other regions, with higher infection rates in more densely populated sub-districts with higher income households. CONCLUSIONS: Analysis of high quality data holds much promise in reducing uncertainty regarding fundamental aspects of the initial phase of an outbreak (e.g. the effective reproductive number Re, age-class distribution, at-risk social groups). The formulation for determining the effective reproductive number Re used here has many advantages for studying the initial phase of the outbreak since it neither assumes exponential growth of infectives and is independent of the reporting rate. The finding of a low Re (close to unity threshold), combined with identification of social groups with high transmission rates would have enabled the containment of swine flu during the summer in Israel. Our unique use of contact data provided new insights into the differential dynamics of influenza in different ages and sexes, and should be promoted in future epidemiological studies. Thus our work highlights the importance of conducting a comprehensive study of the initial stage of a pandemic in real time.

Epidemics with partial immunity to reinfection.

We obtain analytical results about epidemics generated by the partial immunity model of Gomes et al. [3], in which infection confers partial immunity to reinfection. When the demographic process is excluded, the behavior switches from epidemic to endemic as the basic reproduction number R0 crosses the reinfection threshold R0=1σ. We derive formulas for two quantities characterizing the size of the epidemic below the reinfection threshold: the attack rate A, which is the fraction of the population infected at least once, and the final size Z, which is the average number of infections per individual. We also derive a system of differential equations which can be used to obtain more detailed information, such as the fraction of the population infected n times throughout the epidemic, for every n.

Pandemic dynamics and the breakdown of herd immunity.

In this note we discuss the issues involved in attempting to model pandemic dynamics. More specifically, we show how it may be possible to make projections for the ongoing H1N1 pandemic as extrapolated from knowledge of seasonal influenza. We derive first-approximation parameter estimates for the SIR model to describe seasonal influenza, and then explore the implications of the existing classical epidemiological theory for the case of a pandemic virus. In particular, we note the dramatic nonlinear increase in attack rate as a function of the percentage of susceptibles initially present in the population. This has severe consequences for the pandemic, given the general lack of immunity in the global population.