Transmission dynamics of the great influenza pandemic of 1918 in Geneva, Switzerland: Assessing the effects of hypothetical interventions

Recurrent outbreaks of the avian H5N1 influenza virus in Asia represent a constant global pandemic threat. We characterize and evaluate hypothetical public health measures during the 1918 influenza pandemic in the Canton of Geneva, Switzerland. The transmission rate, the recovery rate, the diagnostic rate, the relative infectiousness of asymptomatic cases, and the proportion of clinical cases are estimated through least-squares fitting of the model to epidemic curve data of the cumulative number of hospital notifications. The latent period and the case fatality proportion are taken from published literature. We determine the variance and identifiability of model parameters via a simulation study. Our epidemic model agrees well with the observed epidemic data. We estimate the basic reproductive number for the spring wave R1;=1.49 (95% CI: 1.45-1.53) and the reproductive number for the fall wave R2;=3.75 (95% CI: 3.57-3.93). In addition, we estimate the clinical reporting for these two waves to be 59.7% (95% CI: 55.7-63.7) and 83% (95% CI: 79-87). We surmise that the lower reporting in the first wave can be explained by a lack of initial awareness of the epidemic and the relative higher severity of the symptoms experienced during the fall wave. We found that effective isolation measures in hospital clinics at best would only ensure control with probability 0.87 while reducing the transmission rate by >76.5% guarantees stopping an epidemic.     

Traveling waves in a model of influenza A drift

Between major pandemics, the influenza A virus changes its antigenic properties by accumulating point mutations (drift) mainly in the RNA genes that code for the surface proteins hemagglutinin (HA) and neuraminidase (NA). The successful strain (variant) that will cause the next epidemic is selected from a reduced number of progenies that possess relatively high transmissibility and the ability to escape from the immune surveillance of the host. In this paper, we analyse a one-dimensional model of influenza A drift (Z. Angew. Math. Mech. 76 (2) (1996) 421) that generalizes the classical SIR model by including mutation as a diffusion process in a phenotype space of variants. The model exhibits traveling wave solutions with an asymptotic wave speed that matches well those obtained from numerical simulations. As exact solutions for these waves are not available, asymptotic estimates for the amplitudes of infected and recovered classes are provided through an exponential approximation based on the smallness of the diffusion constant. Through this approximation, we find simple scaling properties to several parameters of relevance to the epidemiology of the disease.     

The 1918-1919 influenza pandemic in England and Wales: spatial patterns in transmissibility and mortality impact

Spatial variations in disease patterns of the 1918-1919 influenza pandemic remain poorly studied. We explored the association between influenza death rates, transmissibility and several geographical and demographic indicators for the autumn and winter waves of the 1918-1919 pandemic in cities, towns and rural areas of England and Wales. Average measures of transmissibility, estimated by the reproduction number, ranged between 1.3 and 1.9, depending on model assumptions and pandemic wave and showed little spatial variation. Death rates varied markedly with urbanization, with 30-40% higher rates in cities and towns compared with rural areas. In addition, death rates varied with population size across rural settings, where low population areas fared worse. By contrast, we found no association between transmissibility, death rates and indicators of population density and residential crowding. Further studies of the geographical mortality patterns associated with the 1918-1919 influenza pandemic may be useful for pandemic planning.     

Mechanistic modelling of the three waves of the 1918 influenza pandemic

Influenza pandemics through history have shown very different patterns of incidence, morbidity and mortality. In particular, pandemics in different times and places have shown anywhere from one to three “waves” of incidence. Understanding the factors that underlie variability in temporal patterns, as well as patterns of morbidity and mortality, is important for public health planning. We use a likelihood-based approach to explore different potential explanations for the three waves of incidence and mortality seen in the 1918 influenza pandemic in London, England. Our analysis suggests that temporal variation in transmission rate provides the best proximate explanation and that the variation in transmission required to generate these three epidemic waves is within biologically plausible values.     

Inferring the causes of the three waves of the 1918 influenza pandemic in England and Wales

Past influenza pandemics appear to be characterized by multiple waves of incidence, but the mechanisms that account for this phenomenon remain unclear. We propose a simple epidemic model, which incorporates three factors that might contribute to the generation of multiple waves: (i) schools opening and closing, (ii) temperature changes during the outbreak, and (iii) changes in human behaviour in response to the outbreak. We fit this model to the reported influenza mortality during the 1918 pandemic in 334 UK administrative units and estimate the epidemiological parameters. We then use information criteria to evaluate how well these three factors explain the observed patterns of mortality. Our results indicate that all three factors are important but that behavioural responses had the largest effect. The parameter values that produce the best fit are biologically reasonable and yield epidemiological dynamics that match the observed data well.     

Influenza drift and epidemic size: the race between generating and escaping immunity

Influenza in humans is characterised by strongly annual dynamics and antigenic evolution leading to partial escape from prior host immunity. The variability of new epidemic strains depends on the amount of virus currently circulating. In this paper, the amount of antigenic variation produced each year is dependent on the epidemic size. Our model reduces to a one-dimensional map and a full mathematical analysis is presented. This simple system suggests some basic principles which may be more generally applicable. In particular, for diseases with antigenic drift, vaccination may be doubly beneficial. Not only does it protect the population through classical herd immunity, but the overall case reduction reduces the chance of new variants being produced; hence, subsequent epidemics may be milder as a result of this positive feedback. Also, a disease with a high innate rate of antigenic variation will always be able to invade a susceptible population, whereas a disease with less potential for variation may require several introduction events to become endemic.     

Age profile of immunity to influenza: effect of original antigenic sin

When multiple infections are possible during an individual’s lifetime, as with influenza, a host’s history of infection and immunity will determine the result of future exposures. In turn, the suite of varying individual infection histories will shape the population level dynamics of the disease. Exploring the consequences of precisely how immunity is acquired using mathematical models has proven challenging though: if n strains have circulated previously, there are 2ⁿ combinations of past infection to consider. However, by using an age-structured mathematical model of a disease with multiple strains, we can examine the population immune profile without explicitly keeping track of all possible infection histories. This framework allows previously unknown consequences of assumptions about immune acquisition to be observed. In particular, we see that ‘original antigenic sin’ can reduce immunity in some age groups: these immune blind spots could be responsible for the unexpectedly high severity of certain past influenza epidemics.     

Estimation of the reproduction number of dengue fever from spatial epidemic data

Dengue, a vector-borne disease, thrives in tropical and subtropical regions worldwide. A retrospective analysis of the 2002 dengue epidemic in Colima located on the Mexican central Pacific coast is carried out. We estimate the reproduction number from spatial epidemic data at the level of municipalities using two different methods: (1) Using a standard dengue epidemic model and assuming pure exponential initial epidemic growth and (2) Fitting a more realistic epidemic model to the initial phase of the dengue epidemic curve. Using Method I, we estimate an overall mean reproduction number of 3.09 (95% CI: 2.34,3.84) as well as local reproduction numbers whose values range from 1.24 (1.15,1.33) to 4.22 (2.90,5.54). Using Method II, the overall mean reproduction number is estimated to be 2.0 (1.75,2.23) and local reproduction numbers ranging from 0.49 (0.0,1.0) to 3.30 (1.63,4.97). Method I systematically overestimates the reproduction number relative to the refined Method II, and hence it would overestimate the intensity of interventions required for containment. Moreover, optimal intervention with defined resources demands different levels of locally tailored mitigation. Local epidemic peaks occur between the 24th and 35th week of the year, and correlate positively with the final local epidemic sizes (rho=0.92, P-value<0.001). Moreover, final local epidemic sizes are found to be linearly related to the local population size (P-value<0.001). This observation supports a roughly constant number of female mosquitoes per person across urban and rural regions.     

Qualitative analysis of models with different treatment protocols to prevent antibiotic resistance

This paper is concerned with the qualitative analysis of two models [S. Bonhoeffer, M. Lipsitch, B.R. Levin, Evaluating treatment protocols to prevent antibiotic resistance, Proc. Natl. Acad. Sci. USA 94 (1997) 12106] for different treatment protocols to prevent antibiotic resistance. Detailed qualitative analysis about the local or global stability of the equilibria of both models is carried out in term of the basic reproduction number R₀. For the model with a single antibiotic therapy, we show that if R₀ < 1, then the disease-free equilibrium is globally asymptotically stable; if R₀ > 1, then the disease-endemic equilibrium is globally asymptotically stable. For the model with multiple antibiotic therapies, stabilities of various equilibria are analyzed and combining treatment is shown better than cycling treatment. Numerical simulations are performed to show that the dynamical properties depend intimately upon the parameters.     

The role of health care workers and antiviral drugs in the control of pandemic influenza

Until a vaccine against the new strain becomes available, the response to newly emerged pandemic influenza will consist of the use of antiviral drugs and measures that limit exposure to infectious individuals. These first-line defence measures include isolating cases upon diagnosis, reducing close contacts, the use of personal protective equipment and hygiene, and using antiviral drugs for treatment and prophylaxis. There are significant 'costs' associated with control measures, so to justify such interventions it is important to assess their potential to reduce transmission. In this paper, we determine the effect that a number of different antiviral interventions have on the reproduction number of infectives and the probability that an imported infection fades out, and determine parameter scenarios for which these interventions are able to eliminate an emerging pandemic of influenza. We also assess the role that health care workers play in transmission and the extent to which providing them with antiviral prophylaxis and personal protective equipment modifies this role. Our results indicate that this class requires protection to avoid a greatly disproportionate contribution to early infective numbers, and for the maintenance of a stable health care system. Further, we show that the role children play in increasing transmission is moderate, in spite of closer mixing with other children.     

Origin of the 1918 pandemic H1N1 influenza A virus as studied by codon usage patterns and phylogenetic analysis

The pandemic of 1918 was caused by an H1N1 influenza A virus, which is a negative strand RNA virus; however, little is known about the nature of its direct ancestral strains. Here we applied a broad genetic and phylogenetic analysis of a wide range of influenza virus genes, in particular the PB1 gene, to gain information about the phylogenetic relatedness of the 1918 H1N1 virus. We compared the RNA genome of the 1918 strain to many other influenza strains of different origin by several means, including relative synonymous codon usage (RSCU), effective number of codons (ENC), and phylogenetic relationship. We found that the PB1 gene of the 1918 pandemic virus had ENC values similar to the H1N1 classical swine and human viruses, but different ENC values from avian as well as H2N2 and H3N2 human viruses. Also, according to the RSCU of the PB1 gene, the 1918 virus grouped with all human isolates and "classical" swine H1N1 viruses. The phylogenetic studies of all eight RNA gene segments of influenza A viruses may indicate that the 1918 pandemic strain originated from a H1N1 swine virus, which itself might be derived from a H1N1 avian precursor, which was separated from the bulk of other avian viruses in toto a long time ago. The high stability of the RSCU pattern of the PB1 gene indicated that the integrity of RNA structure is more important for influenza virus evolution than previously thought.     

Molecular events leading to the creation of a pandemic influenza virus

Influenza A virus is a potent pathogen of annual respiratory illness with huge potential of causing occasional pandemics of catastrophic consequences. In April 2009, a novel, swine-origin influenza A H1N1/09 virus was identified in Mexico which continued to spread globally. This unique virus emerged from an avian, human, Eurasian swine viral strain and a North American swine strain belonging to the lineage of the 1930 swine virus. Till date H1N1/09 pandemic has been relatively mild and lacks the previously described molecular markers of influenza A pathogenicity and transmissibility. In this review, we will discuss the molecular and antigenic determinants of this virus and its designation as a low pathogenic strain, which carries the potential to develop into a devastating strain with subsequent mutations and reassortments.     

Competition of pathogen strains leading to infection with variable infectivity and the effect of treatment

A model for the spread of two strains of a pathogen leading to an infection with variable infectivity is considered. The course of infection is described by two stages with different infectivity levels. The model is extended to account for treatment by including a third stage with different infectivity and survival for those treated. The contribution of each stage to incidence and prevalence is investigated and the effect of infectivity and survival on the basic reproduction ratio is examined. Standard equilibrium analysis is performed for both models, revealing that the successful strain is the one with highest reproduction ratio. If therapy, however, is more effective against the strain that wins in the absence of treatment and its reproduction ratio is sufficiently reduced, it might be outcompeted by the other strain after treatment becomes widely available. In this case, early introduction of treatment can prevent a major outbreak.     

Extension and verification of the SEIR model on the 2009 influenza A (H1N1) pandemic in Japan

In order to understand the evolution of the 2009 influenza A (H1N1) pandemic within local regions of Japan, we studied the significance of regional migration between these regions. For this purpose, we have employed an extended SEIR model to describe the immigration of infected people and the stochastic variation of the infectious efficiency. We then applied a data assimilation technique in order to study how the agreement of the simulation results with the observed data depends on the presence/absence of immigration and the degree of variation of the infectious efficiency. Reproducibility is evaluated by log-likelihood values. The log-likelihood does not indicate the significance of immigration. Although there are multiple waves in the time course of the number of reported infected individuals, these waves could be explained by the stochastic nature of infectious events.     

Comparative analysis of hemagglutinin of 2009 H1N1 influenza A pandemic indicates its evolution to 1918 H1N1 pandemic

To gain insight into the possible origin of the hemagglutinin of 2009 outbreak, we performed its comparative analysis with hemagglutinin of influenza viral strains from 2005 to 2008 and the past pandemics of 1977, 1968, 1957 and 1918. This insilico analysis showed a maximum sequence similarity between 2009 and 1918 pandemics. Primary structure analysis, antigenic and glycosylation site analyses revealed that this protein has evolved from 1918 pandemic. Phylogenetic analysis of HA amino acid sequence of 2009 influenza A(H1N1) viruses indicated that this virus possesses a distinctive evolutionary trait with 1918 influenza A virus. Although the disordered sequences are different among all the isolates, the disordered positions and sequences between 2009 and 1918 isolates show a greater similarity. Thus these analyses contribute to the evidence of the evolution of 2009 pandemic from 1918 influenza pandemic. This is the first computational evolutionary analysis of HA protein of 2009 H1N1 pandemic.     

On the relationship between mean antibody level, seroprotection and clinical protection from influenza

For many vaccines the amount of antibodies induced has a positive correlation with the likelihood of clinical protection from disease. Mean antibody level is therefore frequently used as a serological surrogate endpoint for vaccine efficacy. In addition, a dichotomous surrogate endpoint is often defined: seroprotection. We explore the relationship between mean antibody level, seroprotection and clinical protection from influenza, using a simple statistical model. The model reveals that the relationship depends not only on the mean but also on the dispersion of the antibody levels, the threshold for clinical protection and the clinical protection curve. The consequences for the interpretation of mean antibody levels and seroprotection rates in terms of clinical protection from influenza are discussed.     

On the role of cross-immunity and vaccines on the survival of less fit flu-strains

A pathogen's route to survival involves various mechanisms including its ability to invade (host's susceptibility) and its reproductive success within an invaded host ("infectiousness"). The immunological history of an individual often plays an important role in reducing host susceptibility or it helps the host mount a faster immunological response de facto reducing infectiousness. The cross-immunity generated by prior infections to influenza A strains from the same subtype provide a significant example. The results of this paper are based on the analytical study of a two-strain epidemic model that incorporates host isolation (during primary infection) and cross-immunity to study the role of invasion mediated cross-immunity in a population where a precursor related strain (within the same subtype, i.e. H3N2, H1N1) has already become established. An uncertainty and sensitivity analysis is carried out on the ability of the invading strain to survive for given cross-immunity levels. Our findings indicate that it is possible to support coexistence even in the case when invading strains are "unfit", that is, when the basic reproduction number of the invading strain is less than one. However, such scenarios are possible only in the presence of isolation. That is, appropriate increments in isolation rates and weak cross-immunity can facilitate the survival of less fit strains. The development of "flu" vaccines that minimally enhance herd cross-immunity levels may, by increasing genotype diversity, help facilitate the generation and survival of novel strains.     

On the dynamics of SEIRS epidemic model with transport-related infection

Transportation amongst cities is found as one of the main factors which affect the outbreak of diseases. To understand the effect of transport-related infection on disease spread, an SEIRS (Susceptible, Exposed, Infectious, Recovered) epidemic model for two cities is formulated and analyzed. The epidemiological threshold, known as the basic reproduction number, of the model is derived. If the basic reproduction number is below unity, the disease-free equilibrium is locally asymptotically stable. Thus, the disease can be eradicated from the community. There exists an endemic equilibrium which is locally asymptotically stable if the reproduction number is larger than unity. This means that the disease will persist within the community. The results show that transportation among regions will change the disease dynamics and break infection out even if infectious diseases will go to extinction in each isolated region without transport-related infection. In addition, the result shows that transport-related infection intensifies the disease spread if infectious diseases break out to cause an endemic situation in each region, in the sense of that both the absolute and relative size of patients increase. Further, the formulated model is applied to the real data of SARS outbreak in 2003 to study the transmission of disease during the movement between two regions. The results show that the transport-related infection is effected to the number of infected individuals and the duration of outbreak in such the way that the disease becomes more endemic due to the movement between two cities. This study can be helpful in providing the information to public health authorities and policy maker to reduce spreading disease when its occurs.     

Modelling the effects of pre-exposure and post-exposure vaccines in tuberculosis control

Epidemic control strategies alter the spread of the disease in the host population. In this paper, we describe and discuss mathematical models that can be used to explore the potential of pre-exposure and post-exposure vaccines currently under development in the control of tuberculosis. A model with bacille Calmette-Guerin (BCG) vaccination for the susceptibles and treatment for the infectives is first presented. The epidemic thresholds known as the basic reproduction numbers and equilibria for the models are determined and stabilities are investigated. The reproduction numbers for the models are compared to assess the impact of the vaccines currently under development. The centre manifold theory is used to show the existence of backward bifurcation when the associated reproduction number is less than unity and that the unique endemic equilibrium is locally asymptotically stable when the associated reproduction number is greater than unity. From the study we conclude that the pre-exposure vaccine currently under development coupled with chemoprophylaxis for the latently infected and treatment of infectives is more effective when compared to the post-exposure vaccine currently under development for the latently infected coupled with treatment of the infectives.