The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda

Despite improved control measures, Ebola remains a serious public health risk in African regions where recurrent outbreaks have been observed since the initial epidemic in 1976. Using epidemic modeling and data from two well-documented Ebola outbreaks (Congo 1995 and Uganda 2000), we estimate the number of secondary cases generated by an index case in the absence of control interventions R0. Our estimate of R0 is 1.83 (SD 0.06) for Congo (1995) and 1.34 (SD 0.03) for Uganda (2000). We model the course of the outbreaks via an SEIR (susceptible-exposed-infectious-removed) epidemic model that includes a smooth transition in the transmission rate after control interventions are put in place. We perform an uncertainty analysis of the basic reproductive number R0 to quantify its sensitivity to other disease-related parameters. We also analyse the sensitivity of the final epidemic size to the time interventions begin and provide a distribution for the final epidemic size. The control measures implemented during these two outbreaks (including education and contact tracing followed by quarantine) reduce the final epidemic size by a factor of 2 relative the final size with a 2-week delay in their implementation.     

Variability order of the latent and the infectious periods in a deterministic SEIR epidemic model and evaluation of control effectiveness

We use distribution theory and ordering of non-negative random variables to study the Susceptible-Exposed-Infectious-Removed (SEIR) model with two control measures, quarantine and isolation, to reduce the spread of an infectious disease. We identify that the probability distributions of the latent period and the infectious period are primary features of the SEIR model to formulate the epidemic threshold and to evaluate the effectiveness of the intervention measures. If the primary features are changed, the conclusions will be altered in an importantly different way. For the latent and infectious periods with known mean values, it is the dilation, a generalization of variance, of their distributions that ranks the effectiveness of these control measures. We further propose ways to set quarantine and isolation targets to reduce the controlled reproduction number below the threshold using observed initial growth rate from outbreak data. If both quarantine and isolation are 100% effective, one can directly use the observed growth rate for setting control targets. If they are not 100% effective, some further knowledge of the distributions is required.     

Control of emerging infectious diseases using responsive imperfect vaccination and isolation

This paper is concerned with a stochastic model for the spread of an SEIR (susceptible --> exposed (= latent) --> infective --> removed) epidemic among a population partitioned into households, featuring different rates of infection for within and between households. The model incorporates responsive vaccination and isolation policies, based upon the appearance of diagnosed cases in households. Different models for imperfect vaccine response are considered. A threshold parameter R*, which determines whether or not a major epidemic can occur, and the probability of a major epidemic are obtained for different infectious and latent period distributions. Simpler expressions for these quantities are obtained in the limiting case of infinite within-household infection rate. Numerical studies suggest that the choice of infectious period distribution and whether or not latent individuals are vaccine-sensitive have a material influence on the spread of the epidemic, while, for given vaccine efficacy, the choice of vaccine action model is less influential. They also suggest that an effective isolation policy has a more significant impact than vaccination. The results show that R* alone is not sufficient to summarise the potential for an epidemic.     

一个具非单调传染率的SEIR传染病模型的全局稳定性

本文研究一类描述某种严重疾病的传染数目变大时在心理上产生影响的非单调传染率的SEIR传染病模型.研究表明模型的动力行为和疾病的爆发完全由基本再生数R0决定.当R0≤1时,无病平衡点是全局稳定的,疾病消亡;当R0〉1时,地方病平衡点是全局稳定的,疾病持续且发展成地方病.     

Global Stability for Delay SIR and SEIR Epidemic Models with Nonlinear Incidence Rate

In this paper, based on SIR and SEIR epidemic models with a general nonlinear incidence rate, we incorporate time delays into the ordinary differential equation models. In particular, we consider two delay differential equation models in which delays are caused (i) by the latency of the infection in a vector, and (ii) by the latent period in an infected host. By constructing suitable Lyapunov functionals and using the Lyapunov–LaSalle invariance principle, we prove the global stability of the endemic equilibrium and the disease-free equilibrium for time delays of any length in each model. Our results show that the global properties of equilibria also only depend on the basic reproductive number and that the latent period in a vector does not affect the stability, but the latent period in an infected host plays a positive role to control disease development.     

A qualitative analysis of a model for the transmission of varicella-zoster virus

Varicella-zoster virus (VZV) is a herpesvirus which is the known agent for causing varicella (chickenpox) in its initial manifestation and zoster (shingles) in a reactivated state. The standard SEIR compartmental model is modified to include the cycle of shingles acquisition, recovery, and possible reacquisition. The basic reproduction number R(0) shows the influence of the zoster cycle and how shingles can be important in maintaining VZV in populations. The model has the typical threshold behavior in the sense that when R(0)1, the virus persists over time and so chickenpox and shingles remain endemic.     

Epidemiological effects of seasonal oscillations in birth rates

Seasonal oscillations in birth rates are ubiquitous in human populations. These oscillations might play an important role in infectious disease dynamics because they induce seasonal variation in the number of susceptible individuals that enter populations. We incorporate seasonality of birth rate into the standard, deterministic susceptible-infectious-recovered (SIR) and susceptible-exposed-infectious-recovered (SEIR) epidemic models and identify parameter regions in which birth seasonality can be expected to have observable epidemiological effects. The SIR and SEIR models yield similar results if the infectious period in the SIR model is compared with the "infected period" (the sum of the latent and infectious periods) in the SEIR model. For extremely transmissible pathogens, large amplitude birth seasonality can induce resonant oscillations in disease incidence, bifurcations to stable multi-year epidemic cycles, and hysteresis. Typical childhood infectious diseases are not sufficiently transmissible for their asymptotic dynamics to be likely to exhibit such behaviour. However, we show that fold and period-doubling bifurcations generically occur within regions of parameter space where transients are phase-locked onto cycles resembling the limit cycles beyond the bifurcations, and that these phase-locking regions extend to arbitrarily small amplitude of seasonality of birth rates. Consequently, significant epidemiological effects of birth seasonality may occur in practice in the form of transient dynamics that are sustained by demographic stochasticity.     

A Network-based Analysis of the 1861 Hagelloch Measles Data

Summary In this article, we demonstrate a statistical method for fitting the parameters of a sophisticated network and epidemic model to disease data. The pattern of contacts between hosts is described by a class of dyadic independence exponential-family random graph models (ERGMs), whereas the transmission process that runs over the network is modeled as a stochastic susceptible-exposed-infectious-removed (SEIR) epidemic. We fit these models to very detailed data from the 1861 measles outbreak in Hagelloch, Germany. The network models include parameters for all recorded host covariates including age, sex, household, and classroom membership and household location whereas the SEIR epidemic model has exponentially distributed transmission times with gamma-distributed latent and infective periods. This approach allows us to make meaningful statements about the structure of the population-separate from the transmission process-as well as to provide estimates of various biological quantities of interest, such as the effective reproductive number, R. Using reversible jump Markov chain Monte Carlo, we produce samples from the joint posterior distribution of all the parameters of this model-the network, transmission tree, network parameters, and SEIR parameters-and perform Bayesian model selection to find the best-fitting network model. We compare our results with those of previous analyses and show that the ERGM network model better fits the data than a Bernoulli network model previously used. We also provide a software package, written in R, that performs this type of analysis.     

The evolutionary consequences of alternative types of imperfect vaccines

The emergence and spread of mutant pathogens that evade the effects of prophylactic interventions, including vaccines, threatens our ability to control infectious diseases globally. Imperfect vaccines (e.g. those used against influenza), while not providing life-long immunity, confer protection by reducing a range of pathogen life-history characteristics; conversely, mutant pathogens can gain an advantage by restoring the same range of traits in vaccinated hosts. Using an SEIR model motivated by equine influenza, we investigate the evolutionary consequences of alternative types of imperfect vaccination, by comparing the spread rate of three types of mutant pathogens, in response to three types of vaccines. All mutant types spread faster in response to a transmission-blocking vaccine, relative to vaccines that reduce the proportion of exposed vaccinated individuals becoming infectious, and to vaccines that reduce the length of the infectious period; this difference increases with increasing vaccine efficacy. We interpret our results using the first published Price equation formulation for an SEIR model, and find that our main result is explained by the effects of vaccines on the equilibrium host distribution across epidemiological classes. In particular, the proportion of vaccinated infectious individuals among all exposed and infectious hosts, which is relatively higher in the transmission-blocking vaccine scenario, is important in explaining the faster spread of mutant strains in response to that vaccine. Our work illustrates the connection between epidemiological and evolutionary dynamics, and the need to incorporate both in order to explain and interpret findings of complicated infectious disease dynamics.     

Revisiting the basic reproductive number for malaria and its implications for malaria control

The prospects for the success of malaria control depend, in part, on the basic reproductive number for malaria, R0. Here, we estimate R0 in a novel way for 121 African populations, and thereby increase the number of R0 estimates for malaria by an order of magnitude. The estimates range from around one to more than 3,000. We also consider malaria transmission and control in finite human populations, of size H. We show that classic formulas approximate the expected number of mosquitoes that could trace infection back to one mosquito after one parasite generation, Z0(H), but they overestimate the expected number of infected humans per infected human, R0(H). Heterogeneous biting increases R0 and, as we show, Z0(H), but we also show that it sometimes reduces R0(H); those who are bitten most both infect many vectors and absorb infectious bites. The large range of R0 estimates strongly supports the long-held notion that malaria control presents variable challenges across its transmission spectrum. In populations where R0 is highest, malaria control will require multiple, integrated methods that target those who are bitten most. Therefore, strategic planning for malaria control should consider R0, the spatial scale of transmission, human population density, and heterogeneous biting.     

Theory versus data: how to calculate R0?

To predict the potential severity of outbreaks of infectious diseases such as SARS, HIV, TB and smallpox, a summary parameter, the basic reproduction number R(0), is generally calculated from a population-level model. R(0) specifies the average number of secondary infections caused by one infected individual during his/her entire infectious period at the start of an outbreak. R(0) is used to assess the severity of the outbreak, as well as the strength of the medical and/or behavioral interventions necessary for control. Conventionally, it is assumed that if R(0)>1 the outbreak generates an epidemic, and if R(0)<1 the outbreak becomes extinct. Here, we use computational and analytical methods to calculate the average number of secondary infections and to show that it does not necessarily represent an epidemic threshold parameter (as it has been generally assumed). Previously we have constructed a new type of individual-level model (ILM) and linked it with a population-level model. Our ILM generates the same temporal incidence and prevalence patterns as the population-level model; we use our ILM to directly calculate the average number of secondary infections (i.e., R(0)). Surprisingly, we find that this value of R(0) calculated from the ILM is very different from the epidemic threshold calculated from the population-level model. This occurs because many different individual-level processes can generate the same incidence and prevalence patterns. We show that obtaining R(0) from empirical contact tracing data collected by epidemiologists and using this R(0) as a threshold parameter for a population-level model could produce extremely misleading estimates of the infectiousness of the pathogen, the severity of an outbreak, and the strength of the medical and/or behavioral interventions necessary for control.     

Global dynamics of an SEIR epidemic model with saturating contact rate

Heesterbeek and Metz [J. Math. Biol. 31 (1993) 529] derived an expression for the saturating contact rate of individual contacts in an epidemiological model. In this paper, the SEIR model with this saturating contact rate is studied. The basic reproduction number R0 is proved to be a sharp threshold which completely determines the global dynamics and the outcome of the disease. If R0 < or =1, the disease-free equilibrium is globally stable and the disease always dies out. If R0 > 1, there exists a unique endemic equilibrium which is globally stable and the disease persists at an endemic equilibrium state if it initially exists. The contribution of the saturating contact rate to the basic reproduction number and the level of the endemic equilibrium is also analyzed.     

Assessing the role of HLA-linked and unlinked determinants of disease.

The relationship between increased risk in relatives over population prevalence (lambda R = KR/K) and probability of sharing zero marker alleles identical by descent (ibd) at a linked locus (such as HLA) by an affected relative pair is examined. For a model assuming a single disease-susceptibility locus or group of loci tightly linked to a marker locus, the relationship is remarkably simple and general. Namely, if phi R is the prior probability for the relative pair to share zero marker alleles identical by descent, then P (sharing 0 markers/both relatives are affected) is just phi R/lambda R. Alternatively, lambda AR, the increased risk over population prevalence to a relative R due to a disease locus tightly linked to marker locus A, equals the prior probability that the relative pair share zero A alleles ibd divided by the posterior probability that they share zero alleles ibd, given that they are both affected. For example, for affected sib pairs, P (sharing 0 markers/both sibs are affected) = .25/lambda S. This formula holds true for any number of alleles at the disease locus and for their frequencies, penetrances, and population prevalence. Similar formulas are derived for sharing one and two markers. Application of these formulas to several well-studied HLA-associated diseases yields the following results: For multiple sclerosis, insulin-dependent diabetes mellitus, and coeliac disease, a single-locus model of disease susceptibility is rejected, implying the existence of additional unlinked familial determinants. For all three diseases, the effect of the HLA-linked locus on familiality is minor: for multiple sclerosis, it accounts for only a 2.5-fold increased risk to sibs over the population prevalence, compared to an observed value of 20; for coeliac disease, it accounts for approximately a 5.25-fold increased risk to sibs, while the observed value is on the order of 60; for insulin-dependent diabetes mellitus, it accounts for a 3.42-fold increased risk in sibs, while the observed value is 15. In all cases, the secondary determinants must be outside the HLA region. For tuberculoid leprosy, an unlinked familial determinant is also implicated (increased risk to sibs due to HLA = 1.49; observed value = 2.38). For hemochromatosis and Hodgkin's disease, there is little evidence for HLA-unlinked familial determinants. With this formula, it is also possible to examine the hypothesis of pleiotropy versus linkage dis-equilibrium by comparing lambda AS with the increased risk to sibs due to the associated allele(s).(ABSTRACT TRUNCATED AT 400 WORDS)     

Physiological analysis on oscillatory behavior of glucose-insulin regulation by model with delays

Despite temporally forced transmission driving many infectious diseases, analytical insight into its role when combined with stochastic disease processes and non-linear transmission has received little attention. During disease outbreaks, however, the absence of saturation effects early on in well-mixed populations mean that epidemic models may be linearised and we can calculate outbreak properties, including the effects of temporal forcing on fade-out, disease emergence and system dynamics, via analysis of the associated master equations. The approach is illustrated for the unforced and forced SIR and SEIR epidemic models. We demonstrate that in unforced models, initial conditions (and any uncertainty therein) play a stronger role in driving outbreak properties than the basic reproduction number R₀, while the same properties are highly sensitive to small amplitude temporal forcing, particularly when R₀ is small. Although illustrated for the SIR and SEIR models, the master equation framework may be applied to more realistic models, although analytical intractability scales rapidly with increasing system dimensionality. One application of these methods is obtaining a better understanding of the rate at which vector-borne and waterborne infectious diseases invade new regions given variability in environmental drivers, a particularly important question when addressing potential shifts in the global distribution and intensity of infectious diseases under climate change.     

Latent Coinfection and the Maintenance of Strain Diversity

Technologies for strain differentiation and typing have made it possible to detect genetic diversity of pathogens, both within individual hosts and within communities. Coinfection of a host by more than one pathogen strain may affect the relative frequency of these strains at the population level through complex within- and between-host interactions; in infectious diseases that have a long latent period, interstrain competition during latency is likely to play an important role in disease dynamics. We show that SEIR models that include a class of latently coinfected individuals can have markedly different long-term dynamics than models without coinfection, and that coinfection can greatly facilitate the stable coexistence of strains. We demonstrate these dynamics using a model relevant to tuberculosis in which people may experience latent coinfection with both drug sensitive and drug resistant strains. Using this model, we show that the existence of a latent coinfected state allows the possibility that disease control interventions that target latency may facilitate the emergence of drug resistance.     

Allometric scaling and seasonality in the epidemics of wildlife diseases

We present a susceptibles-exposed-infectives (SEI) model to analyze the effects of seasonality on epidemics, mainly of rabies, in a wide range of wildlife species. Model parameters are cast as simple allometric functions of host body size. Via nonlinear analysis, we investigate the dynamical behavior of the disease for different levels of seasonality in the transmission rate and for different values of the pathogen basic reproduction number (R(0)) over a broad range of body sizes. While the unforced SEI model exhibits long-term epizootic cycles only for large values of R(0), the seasonal model exhibits multiyear periodicity for small values of R(0). The oscillation period predicted by the seasonal model is consistent with those observed in the field for different host species. These conclusions are not affected by alternative assumptions for the shape of seasonality or for the parameters that exhibit seasonal variations. However, the introduction of host immunity (which occurs for rabies in some species and is typical of many other wildlife diseases) significantly modifies the epidemic dynamics; in this case, multiyear cycling requires a large level of seasonal forcing. Our analysis suggests that the explicit inclusion of periodic forcing in models of wildlife disease may be crucial to correctly describe the epidemics of wildlife that live in strongly seasonal environments.     

Steady state analysis of influx and transbilayer distribution of ergosterol in the yeast plasma membrane

Mathematical modeling is now frequently used in outbreak investigations to understand underlying mechanisms of infectious disease dynamics, assess patterns in epidemiological data, and forecast the trajectory of epidemics. However, the successful application of mathematical models to guide public health interventions lies in the ability to reliably estimate model parameters and their corresponding uncertainty. Here, we present and illustrate a simple computational method for assessing parameter identifiability in compartmental epidemic models. We describe a parametric bootstrap approach to generate simulated data from dynamical systems to quantify parameter uncertainty and identifiability. We calculate confidence intervals and mean squared error of estimated parameter distributions to assess parameter identifiability. To demonstrate this approach, we begin with a low-complexity SEIR model and work through examples of increasingly more complex compartmental models that correspond with applications to pandemic influenza, Ebola, and Zika. Overall, parameter identifiability issues are more likely to arise with more complex models (based on number of equations/states and parameters). As the number of parameters being jointly estimated increases, the uncertainty surrounding estimated parameters tends to increase, on average, as well. We found that, in most cases, R0 is often robust to parameter identifiability issues affecting individual parameters in the model. Despite large confidence intervals and higher mean squared error of other individual model parameters, R0 can still be estimated with precision and accuracy. Because public health policies can be influenced by results of mathematical modeling studies, it is important to conduct parameter identifiability analyses prior to fitting the models to available data and to report parameter estimates with quantified uncertainty. The method described is helpful in these regards and enhances the essential toolkit for conducting model-based inferences using compartmental dynamic models.     

Stability properties of pulse vaccination strategy in SEIR epidemic model

The problem of the applicability of the pulse vaccination strategy (PVS) for the stable eradication of some relevant general class of infectious diseases is analyzed in terms of study of local asymptotic stability (LAS) and global asymptotic stability (GAS) of the periodic eradication solution for the SEIR epidemic model in which is included the PVS. Demographic variations due or not to diseased-related fatalities are also considered. Due to the non-triviality of the Floquet's matrix associate to the studied model, the LAS is studied numerically and in this way it is found a simple approximate (but analytical) sufficient criterion which is an extension of the LAS constraint for the stability of the trivial equilibrium in SEIR model without vaccination. The numerical simulations also seem to suggest that the PVS is slightly more efficient than the continuous vaccination strategy. Analytically, the GAS of the eradication solutions is studied and it is demonstrated that the above criteria for the LAS guarantee also the GAS.     

Identification and Characterization of ZEL-H16 as a Novel Agonist of the Histamine H(3) Receptor

The histamine H3 receptor (H3R) has been recognized as a promising target for the treatment of various central and peripheral nervous system diseases. In this study, a non-imidazole compound, ZEL-H16, was identified as a novel histamine H3 receptor agonist. ZEL-H16 was found to bind to human H3R with a Ki value of approximately 2.07 nM and 4.36 nM to rat H3R. Further characterization indicated that ZEL-H16 behaved as a partial agonist on the inhibition of forskolin-stimulated cAMP accumulation (the efficacy was 60% of that of histamine) and activation of ERK1/2 signaling (the efficacy was 50% of that of histamine) at H3 receptors, but acted as a full agonist just like histamin in the guinea-pig ileum contraction assay. These effects were blocked by pertussis toxin and H3 receptor specific antagonist thioperamide. ZEL-H16 showed no agonist or antagonist activities at the cloned human histamine H1, H2, and H4 receptors and other biogenic amine GPCRs in the CRE-driven reporter assay. Furthermore, our present data demonstrated that treatment of ZEL-H16 resulted in intensive H3 receptor internalization and delayed recycling to the cell surface as compared to that of control with treatment of histamine. Thus, ZEL-H16 is a novel and potent nonimidazole agonist of H3R, which might serve as a pharmacological tool for future investigations or as possible therapeutic agent of H3R.     

A net reproductive number for periodic matrix models

We give a definition of a net reproductive number R (0) for periodic matrix models of the type used to describe the dynamics of a structured population with periodic parameters. The definition is based on the familiar method of studying a periodic map by means of its (period-length) composite. This composite has an additive decomposition that permits a generalization of the Cushing-Zhou definition of R (0) in the autonomous case. The value of R (0) determines whether the population goes extinct (R (0)<1) or persists (R (0)>1). We discuss the biological interpretation of this definition and derive formulas for R (0) for two cases: scalar periodic maps of arbitrary period and periodic Leslie models of period 2. We illustrate the use of the definition by means of several examples and by applications to case studies found in the literature. We also make some comparisons of this definition of R (0) with another definition given recently by Baca?r.