Introducing the Outbreak Threshold in Epidemiology

When a pathogen is rare in a host population, there is a chance that it will die out because of stochastic effects instead of causing a major epidemic. Yet no criteria exist to determine when the pathogen increases to a risky level, from which it has a large chance of dying out, to when a major outbreak is almost certain. We introduce such an outbreak threshold (T₀), and find that for large and homogeneous host populations, in which the pathogen has a reproductive ratio R₀, on the order of 1/Log(R₀) infected individuals are needed to prevent stochastic fade-out during the early stages of an epidemic. We also show how this threshold scales with higher heterogeneity and R₀ in the host population. These results have implications for controlling emerging and re-emerging pathogens.With the constant risk of pathogens emerging [1], such as Severe Acute Respiratory Syndrome (SARS) or avian influenza virus in humans, foot-and-mouth disease virus in cattle in the United Kingdom [2], or various plant pathogens [3], it is imperative to understand how novel strains gain their initial foothold at the onset of an epidemic. Despite this importance, it has seldom been addressed how many infected individuals are needed to declare that an outbreak is occurring: that is, when the pathogen can go extinct due to stochastic effects, to when it infects a high enough number of hosts such that the outbreak size increases in a deterministic manner (Figure 1A). Generally, the presence of a single infected individual is not sufficient to be classified as an outbreak, so how many infected individuals need to be present to cause this deterministic increase? Understanding at what point this change arises is key in preventing and controlling nascent outbreaks as they are detected, as well as determining the best course of action for prevention or treatment.Open in a separate windowFigure 1The outbreak threshold in homogeneous and heterogeneous populations.(A) A schematic of pathogen emergence. This graph shows the early stages of several strains of an epidemic, where R₀ = 1.25. The black line denotes the outbreak threshold (T ₀ = 1/Log(R₀) = 4.48). Blue thin lines show cases in which the pathogen goes extinct and does not exceed the threshold; the red thick line shows an epidemic that exceeds the threshold and persists for a long period of time. Simulations were based on the Gillespie algorithm [22]. (B) Outbreak threshold in a homogeneous (black thick line) or in a heterogeneous population, for increasing R₀. The threshold was calculated following the method described by Lloyd-Smith et al. [11] and is shown for different values of k, the dispersion parameter of the offspring distribution, as obtained from data on previous epidemics [11]. If the threshold lies below one, this means that around only one infected individual is needed to give a high outbreak probability.The classic prediction for pathogen outbreak is that the pathogen''s reproductive ratio (R₀), the number of secondary infections caused by an infected host in a susceptible population, has to exceed one [4], [5]. This criterion only strictly holds in deterministic (infinite population) models; in finite populations, there is still a chance that the infection will go extinct by chance rather than sustain itself [4]–[6]. Existing studies usually consider random drift affecting outbreaks in the context of estimating how large a host population needs to be to sustain an epidemic (the “Critical Community Size” [4], [7], [8]), calculating the outbreak probability in general [9]–[12], or ascertaining whether a sustained increase in cases over an area has occurred [13]. Here we discuss the fundamental question of how many infected individuals are needed to almost guarantee that a pathogen will cause an outbreak, as opposed to the population size needed to maintain an epidemic once it has appeared (Critical Community Size; see also Box 1). We find that only a small number of infected individuals are often needed to ensure that an epidemic will spread.

Box 1. Glossary of Key Terms

• The Basic Reproductive Ratio (R₀) is the number of secondary infections caused by a single infected individual, in a susceptible population. It is classically used to measure the rate of pathogen spread. In infinite-population models, a pathogen can emerge if R₀>1. In a finite population, the pathogen can emerge from a single infection with probability 1-1/R₀ if R₀>1, otherwise extinction is certain.
• The Critical Community Size (CCS) is defined as the total population size (of susceptible and infected individuals, or others) needed to sustain an outbreak once it has appeared. This idea was classically applied to determining what towns were most likely to maintain measles epidemics [7], so that there would always be some infected individuals present, unless intervention measures were taken.
• The Outbreak Threshold (T₀) has a similar definition to the CSS, but is instead for use at the onset of an outbreak, rather than once it has appeared. It measures how many infected individuals (not the total population size) are needed to ensure that an outbreak is very unlikely to go extinct by drift. Note that the outbreak can still go extinct in the long term, even if T₀ is exceeded, if there are not enough susceptible individuals present to carry the infection afterwards.

We introduce the concept of the outbreak threshold (denoted T₀), which we define as the number of infected individuals needed for the disease to spread in an approximately deterministic manner. T₀ can be given by simple equations. Indeed, if the host population is homogeneous (that is, where there is no individual variability in reproductive rates) and large enough so that depletion of the pool of susceptible hosts is negligible, then the probability of pathogen extinction if I infected hosts are present is (1/R₀)^I ([6], details in Material S.1 in Text S1). By solving this equation in the limit of extinction probability going to zero, we find that on the order of 1/Log(R₀) infected hosts are needed for an outbreak to be likely (black thick curve in Figure 1B), a result that reflects similar theory from population genetics [14]–[16]. Note that this result only holds in a finite population, as an outbreak in a fully susceptible infinite population is certain if R₀>1 ([4], see also Material S.1 in Text S1).This basic result can be modified to consider more realistic or precise cases, and T₀ can be scaled up if an exact outbreak risk is desired. For example, for the pathogen extinction probability to be less than 1%, there needs to be at least 5/Log(R₀) infected individuals. More generally, the pathogen extinction probability is lower than a given threshold c if there are at least −Log(c)/Log(R₀) infected individuals. Furthermore, if only a proportion p<1 of all infected individuals are detected, then the outbreak threshold order is p/Log(R₀). Also, if there exists a time-lag τ between an infection occurring and its report, then the order of T₀ is e^−τ(β-δ)/Log(R₀), where β is the infection transmission rate and 1/δ the mean duration of the infectious period (Material S.1 in Text S1). Finally, we can estimate how long it would take, on average, for the threshold to be reached and find that, if the depletion in susceptible hosts is negligible, this duration is on the order of 1/(β-δ) (Material S.1 in Text S1).So far we have only considered homogenous outbreaks, where on average each individual has the same pathogen transmission rate. In reality, there will be a large variance among individual transmission rates, especially if “super-spreaders” are present [17]. This population heterogeneity can either be deterministic, due to differences in immune history among hosts or differences in host behavior, or stochastic, due to sudden environmental or social changes. Spatial structure can also act as a form of heterogeneity, if each region or infected individual is subject to different transmission rates, or degree of contact with other individuals [18]. In such heterogeneous host populations, the number of secondary cases an infected individual engenders is jointly captured by R₀ and a dispersion parameter k (see Box 2). This dispersion parameter controls the degree of variation in individual transmission rates, while fixing the average R₀. The consequence of this model is that the majority of infected hosts tend to cause few secondary infections, while the minority behave as super-spreaders, causing many secondary infections. Host population heterogeneity (obtained with lower values of k) increases the probability that an outbreak will go extinct, as the pathogen can only really spread via one of the dwindling super-spreading individuals. In this heterogeneous case, we can find accurate values of T₀ numerically. As shown in Figure 1B, if R₀ is close to 1, host heterogeneity (k) does not really matter (T₀ tends to be high). However, if the pathogen has a high R₀ and thus spreads well, then host heterogeneity strongly affects T₀. Additionally, we find that the heterogeneous threshold simply scales as a function of k and R₀² (see Box 2). As an example, if k = 0.16, as estimated for SARS infections [11], the number of infected individuals needed to guarantee an outbreak increases 4-fold compared to a homogeneous population (Material S.3 in Text S1).

Box 2. Heterogeneous Outbreak Threshold

In a heterogeneous host population (see the main text for the bases of this heterogeneity), it has been shown that the number of secondary infections generated per infected individual can be well described by a negative binomial distribution with mean R₀ and dispersion parameter k [11]. The dispersion parameter determines the level of variation in the number of secondary infections: if k = 1, we have a homogeneous outbreak, but heterogeneity increases as k drops below 1; that is, it enlarges the proportion of infected individuals that are either “super-spreaders” or “dead-ends” (those that do not transmit the pathogen). Lloyd-Smith et al. [11] showed how to estimate R₀ and k from previous epidemics through applying a maximum-likelihood model to individual transmission data.Although in this case it is not possible to find a strict analytical form for the outbreak threshold, progress can be made if we measure the ratio of the heterogeneous and homogeneous thresholds. This function yields values that are independent of a strict cutoff probability (Material S.3 in Text S1). By investigating this ratio, we first found that for a fixed R₀, a function of order 1/k fitted the numerical solutions very well. By measuring these curves for different R₀ values, we further found that a function of order 1/R₀ ² provided a good fit to the coefficients. By fitting a function of order 1/kR₀ ² to the numerical data using least-squares regression in Mathematica 8.0 [19], we obtained the following adjusted form for the outbreak threshold T₀ in a heterogeneous population:(1)As in the homogeneous case, T₀ only provides us with an order of magnitude and it can be multiplied by −Log(c) to find the number of infected hosts required for there to be a probability of outbreak equal to 1-c. A sensitivity analysis shows that Equation 1 tends to be more strongly affected by changes in R₀ than in k (Material S.3 in Text S1).The outbreak threshold T₀ of an epidemic, which we define as the number of infected hosts above which there is very likely to be a major outbreak, can be estimated using simple formulae. Currently, to declare that an outbreak has occurred, studies choose an arbitrary low or high threshold depending, for instance, on whether they are monitoring disease outbreaks or modeling probabilities of emergence. We show that the outbreak threshold can be defined without resorting to an arbitrary cutoff. Of course, the generality of this definition has a cost, which is that the corresponding value of T₀ is only an order of magnitude. Modifications are needed to set a specific cutoff value or to capture host heterogeneity in transmission or incomplete sampling.These results are valid if there are enough susceptible individuals present to maintain an epidemic in the initial stages, as assumed in most studies on emergence [6], [11]–[13], otherwise the pathogen may die out before the outbreak threshold is reached (Box 3 and Material S.2 in Text S1). Yet the key message generally holds that while the number of infections lies below the threshold, there is a strong chance that the pathogen will vanish without causing a major outbreak. From a biological viewpoint, unless R₀ is close to one, these thresholds tend to be small (on the order of 5 to 20 individuals). This contrasts with estimates of the Critical Community Size, which tend to lie in the hundreds of thousands of susceptible individuals [3], [7], [8]. Therefore, while only a small infected population is needed to trigger a full-scale epidemic, a much larger pool of individuals are required to maintain an epidemic, once it appears, and prevent it from fading out. This makes sense, since there tends to be more susceptible hosts early on in the outbreak than late on.

Box 3. Effect of Limiting Host Population Size

The basic result for the homogeneous population, T₀∼1/Log(R₀), assumes that during the time to pathogen outbreak, there are always enough susceptible individuals available to transmit to, so R₀ remains approximately constant during emergence. This assumption can be violated if R₀ is close to 1, or if the population size is small. More precisely, if the maximum outbreak size in a Susceptible-Infected-Recovered (SIR) epidemic, which is given byis less than 1/Log(R₀), then the threshold cannot be reached. Since this maximum is dependent on the population size, outbreaks in smaller populations are less likely to reach the outbreak threshold. For example, if N = 10,000 then R₀ needs to exceed 1.06 for 1/Log(R₀) to be reached; this increases to 1.34 if N decreases to 100. Further details are in Material S.2 in Text S1.Estimates of R₀ and k from previous outbreaks can be used to infer the approximate size of this threshold, to determine whether a handful or hundreds of infected individuals are needed for an outbreak to establish itself. Box 4 outlines two case studies (smallpox in England and SARS in Singapore), estimates of T₀ for these, and how knowledge of the threshold could have aided their control. These examples highlight how the simplicity and rigorousness of the definition of T₀ opens a wide range of applications, as it can be readily applied to specific situations in order to determine the most adequate policies to prevent pathogen outbreaks.

Box 4. Two Case Studies: Smallpox in England and SARS in Singapore

A smallpox outbreak (Variola minor) was initiated in Birmingham, United Kingdom in 1966 due to laboratory release. We calculate a threshold such that the chance of extinction is less than 0.1%, which means that T₀ is equal to 7 times Equation 1. With an estimated R₀ of 1.6 and dispersion parameter k = 0.65 [11], T₀ is approximately equal to 9 infections. The transmission chain for this outbreak is now well-known [20]. Due to prior eradication of smallpox in the United Kingdom, the pathogen was not recognised until around the 45^th case was detected, by which point a full-scale epidemic was underway. A second laboratory outbreak arose in 1978, but the initial case (as well as a single secondary case) was quickly isolated, preventing a larger spread of the pathogen. Given the fairly low T₀ for the previous epidemic, early containment was probably essential in preventing a larger outbreak.The SARS outbreak in Singapore in 2003 is an example of an outbreak with known super-spreaders [21], with an estimated initial R₀ of 1.63 and a low k of 0.16 [11]. T₀ is estimated to be around 27 infections. The first cases were observed in late February, with patients being admitted for pneumonia. Strict control measures were invoked from March 22^nd onwards, including home quarantining of those exposed to SARS patients and closing down of a market where a SARS outbreak was observed. By this date, 57 cases were detected, although it is unclear how many of those cases were still ongoing on the date. This point is important, as it is the infected population size that determines T₀.Overall, very early measures were necessary to successfully prevent a smallpox outbreak due to its rapid spread. In theory, it should have been “easier” to contain the SARS outbreak, as its threshold is three times greater than that for smallpox due to higher host heterogeneity (k). However, the first reported infected individual was a super-spreader, who infected at least 21 others. This reflects that in heterogeneous outbreaks, although the emergence probability is lower, the disease spread is faster (compared to homogeneous infections) once it does appear [11]. Quick containment of the outbreak was difficult to achieve since SARS was not immediately recognised, as well as the fact that the incubation period is around 5 days, by which point it had easily caused more secondary cases. However, in subsequent outbreaks super-spreaders might not be infected early on, allowing more time to contain the spread.For newly-arising outbreaks, T₀ can be applied in several ways. If the epidemic initially spreads slowly, then R₀ and T₀ can be measured directly. Alternatively, estimates of T₀ can be calculated from previous outbreaks, as outlined above. In both cases, knowing what infected population size is needed to guarantee emergence can help to assess how critical a situation is. More generally, due to the difficulty in detecting real-world outbreaks that go extinct very quickly, experimental methods might be useful in determining to what extent different levels of T₀ capture the likelihood of full epidemic emergence.     

COVID-19 modeling and non-pharmaceutical interventions in an outpatient dialysis unit

This paper describes a data-driven simulation study that explores the relative impact of several low-cost and practical non-pharmaceutical interventions on the spread of COVID-19 in an outpatient hospital dialysis unit. The interventions considered include: (i) voluntary self-isolation of healthcare personnel (HCPs) with symptoms; (ii) a program of active syndromic surveillance and compulsory isolation of HCPs; (iii) the use of masks or respirators by patients and HCPs; (iv) improved social distancing among HCPs; (v) increased physical separation of dialysis stations; and (vi) patient isolation combined with preemptive isolation of exposed HCPs. Our simulations show that under conditions that existed prior to the COVID-19 outbreak, extremely high rates of COVID-19 infection can result in a dialysis unit. In simulations under worst-case modeling assumptions, a combination of relatively inexpensive interventions such as requiring surgical masks for everyone, encouraging social distancing between healthcare professionals (HCPs), slightly increasing the physical distance between dialysis stations, and—once the first symptomatic patient is detected—isolating that patient, replacing the HCP having had the most exposure to that patient, and relatively short-term use of N95 respirators by other HCPs can lead to a substantial reduction in both the attack rate and the likelihood of any spread beyond patient zero. For example, in a scenario with R₀ = 3.0, 60% presymptomatic viral shedding, and a dialysis patient being the infection source, the attack rate falls from 87.8% at baseline to 34.6% with this intervention bundle. Furthermore, the likelihood of having no additional infections increases from 6.2% at baseline to 32.4% with this intervention bundle.     

Monitoring Linked Epidemics: The Case of Tuberculosis and HIV

Background

The tight epidemiological coupling between HIV and its associated opportunistic infections leads to challenges and opportunities for disease surveillance.

Methodology/Principal Findings

We review efforts of WHO and collaborating agencies to track and fight the TB/HIV co-epidemic, and discuss modeling—via mathematical, statistical, and computational approaches—as a means to identify disease indicators designed to integrate data from linked diseases in order to characterize how co-epidemics change in time and space. We present R _TB/HIV, an index comparing changes in TB incidence relative to HIV prevalence, and use it to identify those sub-Saharan African countries with outlier TB/HIV dynamics. R _TB/HIV can also be used to predict epidemiological trends, investigate the coherency of reported trends, and cross-check the anticipated impact of public health interventions. Identifying the cause(s) responsible for anomalous R _TB/HIV values can reveal information crucial to the management of public health.

Conclusions/Significance

We frame our suggestions for integrating and analyzing co-epidemic data within the context of global disease monitoring. Used routinely, joint disease indicators such as R _TB/HIV could greatly enhance the monitoring and evaluation of public health programs.     

Modelling optimal vaccination strategy for SARS-CoV-2 in the UK

The COVID-19 outbreak has highlighted our vulnerability to novel infections.Faced with this threat and no effective treatment, in line with many other countries, the UK adopted enforced social distancing (lockdown) to reduce transmission—successfully reducing the reproductive number R below one. However, given the large pool of susceptible individuals that remain, complete relaxation of controls is likely to generate a substantial further outbreak. Vaccination remains the only foreseeable means of both containing the infection and returning to normal interactions and behaviour. Here, we consider the optimal targeting of vaccination within the UK, with the aim of minimising future deaths or quality adjusted life year (QALY) losses. We show that, for a range of assumptions on the action and efficacy of the vaccine, targeting older age groups first is optimal and may be sufficient to stem the epidemic if the vaccine prevents transmission as well as disease.     

Estimation of the Basic Reproductive Number and Mean Serial Interval of a Novel Pathogen in a Small,Well-Observed Discrete Population

Background

Accurately assessing the transmissibility and serial interval of a novel human pathogen is public health priority so that the timing and required strength of interventions may be determined. Recent theoretical work has focused on making best use of data from the initial exponential phase of growth of incidence in large populations.

Methods

We measured generational transmissibility by the basic reproductive number R₀ and the serial interval by its mean T_g. First, we constructed a simulation algorithm for case data arising from a small population of known size with R₀ and T_g also known. We then developed an inferential model for the likelihood of these case data as a function of R₀ and T_g. The model was designed to capture a) any signal of the serial interval distribution in the initial stochastic phase b) the growth rate of the exponential phase and c) the unique combination of R₀ and T_g that generates a specific shape of peak incidence when the susceptible portion of a small population is depleted.

Findings

Extensive repeat simulation and parameter estimation revealed no bias in univariate estimates of either R₀ and T_g. We were also able to simultaneously estimate both R₀ and T_g. However, accurate final estimates could be obtained only much later in the outbreak. In particular, estimates of T_g were considerably less accurate in the bivariate case until the peak of incidence had passed.

Conclusions

The basic reproductive number and mean serial interval can be estimated simultaneously in real time during an outbreak of an emerging pathogen. Repeated application of these methods to small scale outbreaks at the start of an epidemic would permit accurate estimates of key parameters.     

Modeling the Transmission of Middle East Respirator Syndrome Corona Virus in the Republic of Korea

The 2015 epidemic of Middle East respiratory syndrome (MERS) in the Republic of Korea has been the largest outbreak outside Middle East. This epidemic had caused 185 laboratory-confirmed cases and 36 deaths in the Republic of Korea until September 2, 2015, which attracted public’s attention. Based on the detailed data of patients released by World Health Organization (WHO) and actual propagation of the epidemic, we construct two dynamical models to simulate the propagation processes from May 20 to June 8 and from June 9 to July 10, 2015, respectively and find that the basic reproduction number R ₀ reaches up to 4.422. The numerical analysis shows that the reasons of the outbreak spread quickly are lack of self-protection sense and targeted control measures. Through partial correction analysis, the parameters β ₁ and γ have strong correlations with R ₀, i.e., the infectivity and proportion of the asymptomatic infected cases have much influence on the spread of disease. By sensitivity analysis, strengthening self-protection ability of susceptible and quickly isolating or monitoring close contacts are effective measures to control the disease.     

An SIS patch model with variable transmission coefficients

In this paper, an SIS patch model with non-constant transmission coefficients is formulated to investigate the effect of media coverage and human movement on the spread of infectious diseases among patches. The basic reproduction number R₀ is determined. It is shown that the disease-free equilibrium is globally asymptotically stable if R₀?1, and the disease is uniformly persistent and there exists at least one endemic equilibrium if R₀>1. In particular, when the disease is non-fatal and the travel rates of susceptible and infectious individuals in each patch are the same, the endemic equilibrium is unique and is globally asymptotically stable as R₀>1. Numerical calculations are performed to illustrate some results for the case with two patches.     

Social Network Sensors for Early Detection of Contagious Outbreaks

Current methods for the detection of contagious outbreaks give contemporaneous information about the course of an epidemic at best. It is known that individuals near the center of a social network are likely to be infected sooner during the course of an outbreak, on average, than those at the periphery. Unfortunately, mapping a whole network to identify central individuals who might be monitored for infection is typically very difficult. We propose an alternative strategy that does not require ascertainment of global network structure, namely, simply monitoring the friends of randomly selected individuals. Such individuals are known to be more central. To evaluate whether such a friend group could indeed provide early detection, we studied a flu outbreak at Harvard College in late 2009. We followed 744 students who were either members of a group of randomly chosen individuals or a group of their friends. Based on clinical diagnoses, the progression of the epidemic in the friend group occurred 13.9 days (95% C.I. 9.9–16.6) in advance of the randomly chosen group (i.e., the population as a whole). The friend group also showed a significant lead time (p<0.05) on day 16 of the epidemic, a full 46 days before the peak in daily incidence in the population as a whole. This sensor method could provide significant additional time to react to epidemics in small or large populations under surveillance. The amount of lead time will depend on features of the outbreak and the network at hand. The method could in principle be generalized to other biological, psychological, informational, or behavioral contagions that spread in networks.     

Modeling Dynamics of Culex pipiens Complex Populations and Assessing Abatement Strategies for West Nile Virus

The primary mosquito species associated with underground stormwater systems in the United States are the Culex pipiens complex species. This group represents important vectors of West Nile virus (WNV) throughout regions of the continental U.S. In this study, we designed a mathematical model and compared it with surveillance data for the Cx. pipiens complex collected in Beaufort County, South Carolina. Based on the best fit of the model to the data, we estimated parameters associated with the effectiveness of public health insecticide (adulticide) treatments (primarily pyrethrin products) as well as the birth, maturation, and death rates of immature and adult Cx. pipiens complex mosquitoes. We used these estimates for modeling the spread of WNV to obtain more reliable disease outbreak predictions and performed numerical simulations to test various mosquito abatement strategies. We demonstrated that insecticide treatments produced significant reductions in the Cx. pipiens complex populations. However, abatement efforts were effective for approximately one day and the vector mosquitoes rebounded until the next treatment. These results suggest that frequent insecticide applications are necessary to control these mosquitoes. We derived the basic reproductive number (ℜ₀) to predict the conditions under which disease outbreaks are likely to occur and to evaluate mosquito abatement strategies. We concluded that enhancing the mosquito death rate results in lower values of ℜ₀, and if ℜ₀<1, then an epidemic will not occur. Our modeling results provide insights about control strategies of the vector populations and, consequently, a potential decrease in the risk of a WNV outbreak.     

Using Combined Diagnostic Test Results to Hindcast Trends of Infection from Cross-Sectional Data

Infectious disease surveillance is key to limiting the consequences from infectious pathogens and maintaining animal and public health. Following the detection of a disease outbreak, a response in proportion to the severity of the outbreak is required. It is thus critical to obtain accurate information concerning the origin of the outbreak and its forward trajectory. However, there is often a lack of situational awareness that may lead to over- or under-reaction. There is a widening range of tests available for detecting pathogens, with typically different temporal characteristics, e.g. in terms of when peak test response occurs relative to time of exposure. We have developed a statistical framework that combines response level data from multiple diagnostic tests and is able to ‘hindcast’ (infer the historical trend of) an infectious disease epidemic. Assuming diagnostic test data from a cross-sectional sample of individuals infected with a pathogen during an outbreak, we use a Bayesian Markov Chain Monte Carlo (MCMC) approach to estimate time of exposure, and the overall epidemic trend in the population prior to the time of sampling. We evaluate the performance of this statistical framework on simulated data from epidemic trend curves and show that we can recover the parameter values of those trends. We also apply the framework to epidemic trend curves taken from two historical outbreaks: a bluetongue outbreak in cattle, and a whooping cough outbreak in humans. Together, these results show that hindcasting can estimate the time since infection for individuals and provide accurate estimates of epidemic trends, and can be used to distinguish whether an outbreak is increasing or past its peak. We conclude that if temporal characteristics of diagnostics are known, it is possible to recover epidemic trends of both human and animal pathogens from cross-sectional data collected at a single point in time.     

Empirical Estimation of R 0 for Unknown Transmission Functions: The Case of Chronic Wasting Disease in Alberta

We consider the problem of estimating the basic reproduction number R ₀ from data on prevalence dynamics at the beginning of a disease outbreak. We derive discrete and continuous time models, some coefficients of which are to be fitted from data. We show that prevalence of the disease is sufficient to determine R ₀. We apply this method to chronic wasting disease spread in Alberta determining a range of possible R ₀ and their sensitivity to the probability of deer annual survival.     

On the Origin and the Signal-Shaping Mechanism of the Fast Photosignal in the Vertebrate Retina

Fast photosignals (FPS) with R₁ and R₂ components were measured in retinas of cattle, rat, and frog within a temperature range of 0° to 60°C. Except for temperatures near 0°C the signal rise of the R₁ component was determined by the duration of the exciting flash. The kinetics of the R₂ component and the meta transition of rhodopsin in the cattle and rat retina were compared. For the analysis of the FPS it is presupposed that the signal is produced by light-induced charges on the outer segment envelope membrane that spread onto the whole plasma membrane of the photoreceptor cell. To a good approximation, this mechanism can be described by a model circuit with two distinct capacitors. In this model, the charging capacitance of the pigmented outer segment envelope membrane and the capacitance of the receptor's nonpigmented plasma membrane are connected via the extra- and intracellular electrolyte resistances. The active charging is explained by two independent processes, both with exponential rise (R₁ and R₂), that are due to charge displacements within the pigmented envelope membrane. The time constant τ₂ of the R₂ membrane charging process shows a strong temperature dependence that of the charge redistribution, τ_r, a weak one. In frog and cattle retinas the active charging is much slower within a large temperature range than the passive charge redistribution. From the two-capacitor model it follows for τ_r « τ₂ that the rise of the R₂ component is determined by τ_r, whereas the decay is given by τ₂. For the rat retina, however, τ₂ approaches τ_r at physiological temperatures and becomes <τ_r above 45°C. In this temperature range where τ₂ ≈ τ_r, both processes affect rise and decay of the photosignal. The absolute values of τ_r are in good accordance with the known electric parameters of the photoreceptors. At least in the cattle retina, the time constant τ₂ is identical with that of the slow component of the meta II formation. The strong temperature dependence of the meta transition time gives rise to the marked decrease of the R₂ amplitude with falling temperature. As the R₁ rise could not be fully time resolved the signal analysis does not yield the time constant τ₁ of the R₁ generating process. It could be established, however, within the whole temperature range that the decay of the R₁ component is determined by τ_r. Using an extended model that allows for membrane leakage, we show that in normal ringer solution the membrane time constant does not influence the signal time-course and amplitude.     

Impact of group mixing on disease dynamics

A general mathematical model is proposed to study the impact of group mixing in a heterogeneous host population on the spread of a disease that confers temporary immunity upon recovery. The model contains general distribution functions that account for the probabilities that individuals remain in the recovered class after recovery. For this model, the basic reproduction number R₀ is identified. It is shown that if R₀<1, then the disease dies out in the sense that the disease free equilibrium is globally asymptotically stable; whereas if R₀>1, this equilibrium becomes unstable. In this latter case, depending on the distribution functions and the group mixing strengths, the disease either persists at a constant endemic level or exhibits sustained oscillatory behavior.     

A Simulation Study to Assess Indicators of Antimicrobial Use as Predictors of Resistance: Does It Matter Which Indicator Is Used?

Objective

Indicators of antimicrobial use have been described previously, but few studies have compared their accuracy in prediction of antimicrobial resistance in hospital settings. This study aimed to identify conditions under which significant differences would be observed in the predictive accuracy of indicators in the context of surveillance of intensive care units (ICUs).

Methods

Ten resistance / antimicrobial use combinations were studied. We used simulation to determine if Québec’s network of 81 ICUs or the National Healthcare Safety Network (NHSN) of 2952 ICUs are large enough to allow the detection of predetermined differences between the most accurate and 1) the second most accurate indicator, and 2) the least accurate indicator, in more than 80% of simulations. For each indicator, we simulated absolute errors in prediction for each ICU and each 4-week period, for surveillance lasting up to 5 years. Absolute errors were generated following a binomial distribution, using mean absolute errors (MAEs) observed in 9 ICUs as the average proportion; simulated MAEs were compared using t-tests. This was repeated 1000 times per scenario.

Results

When comparing the two most accurate indicators, 80% power was reached less often with the Québec network versus the NHSN (0/20 versus 2/20 scenarios, with 5 years of surveillance data), a finding reinforced when comparing the most and least accurate indicators (3/20 versus 20/20 scenarios). When simulating 1 year of data, scenarios reaching an 80% power dropped to 0/20, comparing the two most accurate indicators with the larger network, and to 1/20, comparing the most and least accurate indicators with the smaller network.

Conclusion

Most of the time (72%), identifying an indicator of antimicrobial use predicting antimicrobial resistance with a better accuracy was not possible. The choice of an indicator for an eventual surveillance system should rely on criteria other that predictive accuracy.     

On the definition and utilization of heritable variation among hosts in reproduction ratio R0 for infectious diseases

Infectious diseases have a major role in evolution by natural selection and pose a worldwide concern in livestock. Understanding quantitative genetics of infectious diseases, therefore, is essential both for understanding the consequences of natural selection and for designing artificial selection schemes in agriculture. The basic reproduction ratio, R₀, is the key parameter determining risk and severity of infectious diseases. Genetic improvement for control of infectious diseases in host populations should therefore aim at reducing R₀. This requires definitions of breeding value and heritable variation for R₀, and understanding of mechanisms determining response to selection. This is challenging, as R₀ is an emergent trait arising from interactions among individuals in the population. Here we show how to define breeding value and heritable variation for R₀ for genetically heterogeneous host populations. Furthermore, we identify mechanisms determining utilization of heritable variation for R₀. Using indirect genetic effects, next-generation matrices and a SIR (Susceptible, Infected and Recovered) model, we show that an individual''s breeding value for R₀ is a function of its own allele frequencies for susceptibility and infectivity and of population average susceptibility and infectivity. When interacting individuals are unrelated, selection for individual disease status captures heritable variation in susceptibility only, yielding limited response in R₀. With related individuals, however, there is a secondary selection process, which also captures heritable variation in infectivity and additional variation in susceptibility, yielding substantially greater response. This shows that genetic variation in susceptibility represents an indirect genetic effect. As a consequence, response in R₀ increased substantially when interacting individuals were genetically related.     

The Role of Aquaporins in pH-Dependent Germination of Rhizopus delemar Spores

Rhizopus delemar and associated species attack a wide range of fruit and vegetables after harvest. Host nutrients and acidic pH are required for optimal germination of R. delemar, and we studied how this process is triggered. Glucose induced spore swelling in an acidic environment, expressed by an up to 3-fold increase in spore diameter, whereas spore diameter was smaller in a neutral environment. When suspended in an acidic environment, the spores started to float, indicating a change in their density. Treatment of the spores with HgCl₂, an aquaporin blocker, prevented floating and inhibited spore swelling and germ-tube emergence, indicating the importance of water uptake at the early stages of germination. Two putative candidate aquaporin-encoding genes—RdAQP1 and RdAQP2—were identified in the R. delemar genome. Both presented the conserved NPA motif and six-transmembrane domain topology. Expressing RdAQP1 and RdAQP2 in Arabidopsis protoplasts increased the cells'' osmotic water permeability coefficient (P_f) compared to controls, indicating their role as water channels. A decrease in R. delemar aquaporin activity with increasing external pH suggested pH regulation of these proteins. Substitution of two histidine (His) residues, positioned on two loops facing the outer side of the cell, with alanine eliminated the pH sensing resulting in similar P_f values under acidic and basic conditions. Since hydration is critical for spore switching from the resting to activate state, we suggest that pH regulation of the aquaporins can regulate the initial phase of R. delemar spore germination, followed by germ-tube elongation and host-tissue infection.     

Improved age determination of blood and teeth samples using a selected set of DNA methylation markers

Age estimation from DNA methylation markers has seen an exponential growth of interest, not in the least from forensic scientists. The current published assays, however, can still be improved by lowering the number of markers in the assay and by providing more accurate models to predict chronological age. From the published literature we selected 4 age-associated genes (ASPA, PDE4C, ELOVL2, and EDARADD) and determined CpG methylation levels from 206 blood samples of both deceased and living individuals (age range: 0–91 years). This data was subsequently used to compare prediction accuracy with both linear and non-linear regression models. A quadratic regression model in which the methylation levels of ELOVL2 were squared showed the highest accuracy with a Mean Absolute Deviation (MAD) between chronological age and predicted age of 3.75 years and an adjusted R² of 0.95. No difference in accuracy was observed for samples obtained either from living and deceased individuals or between the 2 genders. In addition, 29 teeth from different individuals (age range: 19–70 years) were analyzed using the same set of markers resulting in a MAD of 4.86 years and an adjusted R² of 0.74. Cross validation of the results obtained from blood samples demonstrated the robustness and reproducibility of the assay. In conclusion, the set of 4 CpG DNA methylation markers is capable of producing highly accurate age predictions for blood samples from deceased and living individuals     

Relative transmissibility of shigellosis among different age groups: A modeling study in Hubei Province,China

Shigellosis is a heavy disease burden in China especially in children aged under 5 years. However, the age-related factors involved in transmission of shigellosis are unclear. An age-specific Susceptible–Exposed–Infectious/Asymptomatic–Recovered (SEIAR) model was applied to shigellosis surveillance data maintained by Hubei Province Centers for Disease Control and Prevention from 2005 to 2017. The individuals were divided into four age groups (≤ 5 years, 6–24 years, 25–59 years, and ≥ 60 years). The effective reproduction number (R_eff), including infectivity (R_I) and susceptibility (R_S) was calculated to assess the transmissibility of different age groups. From 2005 to 2017, 130,768 shigellosis cases were reported in Hubei Province. The SEIAR model fitted well with the reported data (P < 0.001). The highest transmissibility (R_eff) was from ≤ 5 years to the 25–59 years (mean: 0.76, 95% confidence interval [CI]: 0.34–1.17), followed by from the 6–24 years to the 25–59 years (mean: 0.69, 95% CI: 0.35–1.02), from the ≥ 60 years to the 25–59 years (mean: 0.58, 95% CI: 0.29–0.86), and from the 25–59 years to 25–59 years (mean: 0.50, 95% CI: 0.21–0.78). The highest infectivity was in ≤ 5 years (R_I = 1.71), and was most commonly transmitted to the 25–59 years (45.11%). The highest susceptibility was in the 25–59 years (R_S = 2.51), and their most common source was the ≤ 5 years (30.15%). Furthermore, “knock out” simulation predicted the greatest reduction in the number of cases occurred by when cutting off transmission routes among ≤ 5 years and from 25–59 years to ≤ 5 years. Transmission in ≤ 5 years occurred mainly within the group, but infections were most commonly introduced by individuals in the 25–59 years. Infectivity was highest in the ≤ 5 years and susceptibility was highest in the 25–59 years. Interventions to stop transmission should be directed at these age groups.