The failure of R0

Empirical Estimation of R0 for Unknown Transmission Functions: The Case of Chronic Wasting Disease in Alberta

We consider the problem of estimating the basic reproduction number R₀ 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 R₀. We apply this method to chronic wasting disease spread in Alberta determining a range of possible R₀ and their sensitivity to the probability of deer annual survival.

Modeling the Effects of Multiple Intervention Strategies on Controlling Foot-and-Mouth Disease

Foot-and-mouth disease (FMD) is a threat to economic security and infrastructure as well as animal health, in both developed and developing countries. We propose and analyze an optimal control problem where the control system is a mathematical model for FMD that incorporates vaccination and culling of infectious animals. The control functions represent the fraction of animals that are vaccinated during an outbreak, infectious symptomatic animals that are detected and culled, and infectious nonsymptomatic animals that are detected and culled. Our aim was to study how these control measures should be implemented for a certain time period, in order to reduce or eliminate FMD in the community, while minimizing the interventions implementation costs. A cost-effectiveness analysis is carried out, to compare the application of each one of the control measures, separately or in combination.

Estimating the Basic Reproductive Number (R0) for African Swine Fever Virus (ASFV) Transmission between Pig Herds in Uganda

African swine fever (ASF) is a highly contagious, lethal and economically devastating haemorrhagic disease of domestic pigs. Insights into the dynamics and scale of virus transmission can be obtained from estimates of the basic reproduction number (R₀). We estimate R₀ for ASF virus in small holder, free-range pig production system in Gulu, Uganda. The estimation was based on data collected from outbreaks that affected 43 villages (out of the 289 villages with an overall pig population of 26,570) between April 2010 and November 2011. A total of 211 outbreaks met the criteria for inclusion in the study. Three methods were used, specifically; (i) GIS- based identification of the nearest infectious neighbour based on the Euclidean distance between outbreaks, (ii) epidemic doubling time, and (iii) a compartmental susceptible-infectious (SI) model. For implementation of the SI model, three approaches were used namely; curve fitting (CF), a linear regression model (LRM) and the SI/N proportion. The R₀ estimates from the nearest infectious neighbour and epidemic doubling time methods were 3.24 and 1.63 respectively. Estimates from the SI-based method were 1.58 for the CF approach, 1.90 for the LRM, and 1.77 for the SI/N proportion. Since all these values were above one, they predict the observed persistence of the virus in the population. We hypothesize that the observed variation in the estimates is a consequence of the data used. Higher resolution and temporally better defined data would likely reduce this variation. This is the first estimate of R₀ for ASFV in a free range smallholder pig keeping system in sub-Saharan Africa and highlights the requirement for more efficient application of available disease control measures.

The basic reproduction number as a predictor for epidemic outbreaks in temporal networks

The basic reproduction number R₀—the number of individuals directly infected by an infectious person in an otherwise susceptible population—is arguably the most widely used estimator of how severe an epidemic outbreak can be. This severity can be more directly measured as the fraction of people infected once the outbreak is over, Ω. In traditional mathematical epidemiology and common formulations of static network epidemiology, there is a deterministic relationship between R₀ and Ω. However, if one considers disease spreading on a temporal contact network—where one knows when contacts happen, not only between whom—then larger R₀ does not necessarily imply larger Ω. In this paper, we numerically investigate the relationship between R₀ and Ω for a set of empirical temporal networks of human contacts. Among 31 explanatory descriptors of temporal network structure, we identify those that make R₀ an imperfect predictor of Ω. We find that descriptors related to both temporal and topological aspects affect the relationship between R₀ and Ω, but in different ways.

Tactics and strategies for managing Ebola outbreaks and the salience of immunization

We present a stochastic transmission chain simulation model for Ebola viral disease (EVD) in West Africa, with the salutary result that the virus may be more controllable than previously suspected. The ongoing tactics to detect cases as rapidly as possible and isolate individuals as safely as practicable is essential to saving lives in the current outbreaks in Guinea, Liberia, and Sierra Leone. Equally important are educational campaigns that reduce contact rates between susceptible and infectious individuals in the community once an outbreak occurs. However, due to the relatively low R 0 of Ebola (around 1.5 to 2.5 next generation cases are produced per current generation case in naïve populations), rapid isolation of infectious individuals proves to be highly efficacious in containing outbreaks in new areas, while vaccination programs, even with low efficacy vaccines, can be decisive in curbing future outbreaks in areas where the Ebola virus is maintained in reservoir populations.

Exploring the Impact of Prostitution on HIV/AIDS Transmission

HIV/AIDS has been somehow linked to prostitution for decades now. A mathematical model is presented to assess the link between prostitution and HIV transmission. The epidemic thresholds known as the reproduction numbers and equilibria for the model are determined and stabilities analyzed. Analysis of the reproduction numbers suggests that HIV/AIDS control using antiretroviral therapy is more effective in the absence of prostitution. Numerical simulations further show high levels of HIV/AIDS when percentage of prostitutes in the community is high. Results from this study suggest that effectively controlling HIV/AIDS requires strategies that address both prostitution and HIV/AIDS transmission. Addressing HIV/AIDS through condom use and antiretroviral therapy may not be enough to stem HIV/AIDS in the community as some drug/alcohol misusing prostitutes may not be able to negotiate for safe sex while they are in drunken stupor. Furthermore, prostitutes are likely to get infected by different HIV strains some of which may be resistant to the antiretroviral therapy regimen in use.

The basic reproduction number obtained from Jacobian and next generation matrices - A case study of dengue transmission modelling

The basic reproduction number is a key parameter in mathematical modelling of transmissible diseases. From the stability analysis of the disease free equilibrium, by applying Routh-Hurwitz criteria, a threshold is obtained, which is called the basic reproduction number. However, the application of spectral radius theory on the next generation matrix provides a different expression for the basic reproduction number, that is, the square root of the previously found formula. If the spectral radius of the next generation matrix is defined as the geometric mean of partial reproduction numbers, however the product of these partial numbers is the basic reproduction number, then both methods provide the same expression. In order to show this statement, dengue transmission modelling incorporating or not the transovarian transmission is considered as a case study. Also tuberculosis transmission and sexually transmitted infection modellings are taken as further examples.

Stability, bifurcation and chaos analysis of vector-borne disease model with application to Rift Valley fever

This paper investigates a RVF epidemic model by qualitative analysis and numerical simulations. Qualitative analysis have been used to explore the stability dynamics of the equilibrium points while visualization techniques such as bifurcation diagrams, Poincaré maps, maxima return maps and largest Lyapunov exponents are numerically computed to confirm further complexity of these dynamics induced by the seasonal forcing on the mosquitoes oviposition rates. The obtained results show that ordinary differential equation models with external forcing can have rich dynamic behaviour, ranging from bifurcation to strange attractors which may explain the observed fluctuations found in RVF empiric outbreak data, as well as the non deterministic nature of RVF inter-epidemic activities. Furthermore, the coexistence of the endemic equilibrium is subjected to existence of certain number of infected Aedes mosquitoes, suggesting that Aedes have potential to initiate RVF epidemics through transovarial transmission and to sustain low levels of the disease during post epidemic periods. Therefore we argue that locations that may serve as RVF virus reservoirs should be eliminated or kept under control to prevent multi-periodic outbreaks and consequent chains of infections. The epidemiological significance of this study is: (1) low levels of birth rate (in both Aedes and Culex) can trigger unpredictable outbreaks; (2) Aedes mosquitoes are more likely capable of inducing unpredictable behaviour compared to the Culex; (3) higher oviposition rates on mosquitoes do not in general imply manifestation of irregular behaviour on the dynamics of the disease. Finally, our model with external seasonal forcing on vector oviposition rates is able to mimic the linear increase in livestock seroprevalence during inter-epidemic period showing a constant exposure and presence of active transmission foci. This suggests that RVF outbreaks partly build upon RVF inter-epidemic activities. Therefore, active RVF surveillance in livestock is recommended.

Estimates of the reproduction number for seasonal, pandemic, and zoonotic influenza: a systematic review of the literature

BACKGROUND

The potential impact of an influenza pandemic can be assessed by calculating a set of transmissibility parameters, the most important being the reproduction number (R), which is defined as the average number of secondary cases generated per typical infectious case.

METHODS

We conducted a systematic review to summarize published estimates of R for pandemic or seasonal influenza and for novel influenza viruses (e.g. H5N1). We retained and summarized papers that estimated R for pandemic or seasonal influenza or for human infections with novel influenza viruses.

RESULTS

The search yielded 567 papers. Ninety-one papers were retained, and an additional twenty papers were identified from the references of the retained papers. Twenty-four studies reported 51 R values for the 1918 pandemic. The median R value for 1918 was 1.80 (interquartile range [IQR]: 1.47-2.27). Six studies reported seven 1957 pandemic R values. The median R value for 1957 was 1.65 (IQR: 1.53-1.70). Four studies reported seven 1968 pandemic R values. The median R value for 1968 was 1.80 (IQR: 1.56-1.85). Fifty-seven studies reported 78 2009 pandemic R values. The median R value for 2009 was 1.46 (IQR: 1.30-1.70) and was similar across the two waves of illness: 1.46 for the first wave and 1.48 for the second wave. Twenty-four studies reported 47 seasonal epidemic R values. The median R value for seasonal influenza was 1.28 (IQR: 1.19-1.37). Four studies reported six novel influenza R values. Four out of six R values were <1.

CONCLUSIONS

These R values represent the difference between epidemics that are controllable and cause moderate illness and those causing a significant number of illnesses and requiring intensive mitigation strategies to control. Continued monitoring of R during seasonal and novel influenza outbreaks is needed to document its variation before the next pandemic.

From regional pulse vaccination to global disease eradication: insights from a mathematical model of poliomyelitis

Mass-vaccination campaigns are an important strategy in the global fight against poliomyelitis and measles. The large-scale logistics required for these mass immunisation campaigns magnifies the need for research into the effectiveness and optimal deployment of pulse vaccination. In order to better understand this control strategy, we propose a mathematical model accounting for the disease dynamics in connected regions, incorporating seasonality, environmental reservoirs and independent periodic pulse vaccination schedules in each region. The effective reproduction number, Re, is defined and proved to be a global threshold for persistence of the disease. Analytical and numerical calculations show the importance of synchronising the pulse vaccinations in connected regions and the timing of the pulses with respect to the pathogen circulation seasonality. Our results indicate that it may be crucial for mass-vaccination programs, such as national immunisation days, to be synchronised across different regions. In addition, simulations show that a migration imbalance can increase Re and alter how pulse vaccination should be optimally distributed among the patches, similar to results found with constant-rate vaccination. Furthermore, contrary to the case of constant-rate vaccination, the fraction of environmental transmission affects the value of Re when pulse vaccination is present.

Spatial interactions and cooperation can change the speed of evolution of complex phenotypes

Complex traits arise from the interactions among multiple gene products. In the case where the complex phenotype is separated from the wild type by a fitness valley or a fitness plateau, the generation of a complex phenotype may take a very long evolutionary time. Interestingly, the rate of evolution depends in nontrivial ways on various properties of the underlying stochastic process, such as the spatial organization of the population and social interactions among cells. Here we review some of our recent work that investigates these phenomena in asexual populations. The role of spatial constraints is quite complex: there are realistic cases where spatial constrains can accelerate or delay evolution, or even influence it in a nonmonotonic fashion, where evolution works fastest for intermediate-range constraints. Social interactions among cells can be studied in the context of the division-of-labor games. Under a range of circumstances, cooperation among cells can lead to a relatively fast creation of a complex phenotype as an emerging (distributed) property. If we further assume the presence of cheaters, we observe the emergence of a fully mutated population of cells possessing the complex phenotype. Applications of these ideas to cancer initiation and biofilm formation in bacteria are discussed.

Estimated effects of projected climate change on the basic reproductive number of the Lyme disease vector Ixodes scapularis

BACKGROUND

The extent to which climate change may affect human health by increasing risk from vector-borne diseases has been under considerable debate.

OBJECTIVES

We quantified potential effects of future climate change on the basic reproduction number (R0) of the tick vector of Lyme disease, Ixodes scapularis, and explored their importance for Lyme disease risk, and for vector-borne diseases in general.

METHODS

We applied observed temperature data for North America and projected temperatures using regional climate models to drive an I. scapularis population model to hindcast recent, and project future, effects of climate warming on R0. Modeled R0 increases were compared with R0 ranges for pathogens and parasites associated with variations in key ecological and epidemiological factors (obtained by literature review) to assess their epidemiological importance.

RESULTS

R0 for I. scapularis in North America increased during the years 1971-2010 in spatio-temporal patterns consistent with observations. Increased temperatures due to projected climate change increased R0 by factors (2-5 times in Canada and 1.5-2 times in the United States), comparable to observed ranges of R0 for pathogens and parasites due to variations in strains, geographic locations, epidemics, host and vector densities, and control efforts.

CONCLUSIONS

Climate warming may have co-driven the emergence of Lyme disease in northeastern North America, and in the future may drive substantial disease spread into new geographic regions and increase tick-borne disease risk where climate is currently suitable. Our findings highlight the potential for climate change to have profound effects on vectors and vector-borne diseases, and the need to refocus efforts to understand these effects.

Identifying the Conditions Under Which Antibodies Protect Against Infection by Equine Infectious Anemia Virus

The ability to predict the conditions under which antibodies protect against viral infection would transform our approach to vaccine development. A more complete understanding is needed of antibody protection against lentivirus infection, as well as the role of mutation in resistance to an antibody vaccine. Recently, an example of antibody-mediated vaccine protection has been shown via passive transfer of neutralizing antibodies before equine infectious anemia virus (EIAV) infection of horses with severe combined immunodeficiency (SCID). Viral dynamic modeling of antibody protection from EIAV infection in SCID horses may lead to insights into the mechanisms of control of infection by antibody vaccination. In this work, such a model is constructed in conjunction with data from EIAV infection of SCID horses to gain insights into multiple strain competition in the presence of antibody control. Conditions are determined under which wild-type infection is eradicated with the antibody vaccine. In addition, a three-strain competition model is considered in which a second mutant strain may coexist with the first mutant strain. The conditions that permit viral escape by the mutant strains are determined, as are the effects of variation in the model parameters. This work extends the current understanding of competition and antibody control in lentiviral infection, which may provide insights into the development of vaccines that stimulate the immune system to control infection effectively.

Dynamics and control of foot-and-mouth disease in endemic countries: a pair approximation model

Previous mathematical models of spatial farm-to-farm transmission of foot and mouth disease (FMD) have explored the impacts of control measures such as culling and vaccination during a single outbreak in a country normally free of FMD. As a result, these models do not include factors that are relevant to countries where FMD is endemic in some regions, like long-term waning natural and vaccine immunity, use of prophylactic vaccination and disease re-importations. These factors may have implications for disease dynamics and control, yet few models have been developed for FMD-endemic settings. Here we develop and study an SEIRV (susceptible-exposed-infectious-recovered-vaccinated) pair approximation model of FMD. We focus on long term dynamics by exploring characteristics of repeated outbreaks of FMD and their dependence on disease re-importation, loss of natural immunity, and vaccine waning. We find that the effectiveness of ring and prophylactic vaccination strongly depends on duration of natural immunity, rate of vaccine waning, and disease re-introduction rate. However, the number and magnitude of FMD outbreaks are generally more sensitive to the duration of natural immunity than the duration of vaccine immunity. If loss of natural immunity and/or vaccine waning happen rapidly, then multiple epidemic outbreaks result, making it difficult to eliminate the disease. Prophylactic vaccination is more effective than ring vaccination, at the same per capita vaccination rate. Finally, more frequent disease re-importation causes a higher cumulative number of infections, although a lower average epidemic peak. Our analysis demonstrates significant differences between dynamics in FMD-free settings versus FMD-endemic settings, and that dynamics in FMD-endemic settings can vary widely depending on factors such as the duration of natural and vaccine immunity and the rate of disease re-importations. We conclude that more mathematical models tailored to FMD-endemic countries should be developed that include these factors.

Dynamics of the force of infection: insights from Echinococcus multilocularis infection in foxes

Characterizing the force of infection (FOI) is an essential part of planning cost effective control strategies for zoonotic diseases. Echinococcus multilocularis is the causative agent of alveolar echinococcosis in humans, a serious disease with a high fatality rate and an increasing global spread. Red foxes are high prevalence hosts of E. multilocularis. Through a mathematical modelling approach, using field data collected from in and around the city of Zurich, Switzerland, we find compelling evidence that the FOI is periodic with highly variable amplitude, and, while this amplitude is similar across habitat types, the mean FOI differs markedly between urban and periurban habitats suggesting a considerable risk differential. The FOI, during an annual cycle, ranges from (0.1,0.8) insults (95% CI) in urban habitat in the summer to (9.4, 9.7) (95% CI) in periurban (rural) habitat in winter. Such large temporal and spatial variations in FOI suggest that control strategies are optimal when tailored to local FOI dynamics.

Human alveolar echinococcosis (AE) is caused by the fox tapeworm E. multilocularis and has a high fatality rate if untreated. The frequency of the tapeworm in foxes can be reduced through the regular distribution of anthelmintic baits and thus decrease the risk of zoonotic transmission. Here, we estimate the force of infection to foxes using a mathematical model and data from necropsied foxes. The results suggest that the frequency of anthelmintic baiting of foxes can be optimised to local variations in transmission that depend upon season and type of fox habitat.

Data-driven modeling to assess receptivity for Rift Valley Fever virus

Rift Valley Fever virus (RVFV) is an enzootic virus that causes extensive morbidity and mortality in domestic ruminants in Africa, and it has shown the potential to invade other areas such as the Arabian Peninsula. Here, we develop methods for linking mathematical models to real-world data that could be used for continent-scale risk assessment given adequate data on local host and vector populations. We have applied the methods to a well-studied agricultural region of California with [Formula: see text]1 million dairy cattle, abundant and competent mosquito vectors, and a permissive climate that has enabled consistent transmission of West Nile virus and historically other arboviruses. Our results suggest that RVFV outbreaks could occur from February-November, but would progress slowly during winter-early spring or early fall and be limited spatially to areas with early increases in vector abundance. Risk was greatest in summer, when the areas at risk broadened to include most of the dairy farms in the study region, indicating the potential for considerable economic losses if an introduction were to occur. To assess the threat that RVFV poses to North America, including what-if scenarios for introduction and control strategies, models such as this one should be an integral part of the process; however, modeling must be paralleled by efforts to address the numerous remaining gaps in data and knowledge for this system.

A mathematical model of avian influenza with half-saturated incidence

The widespread impact of avian influenza viruses not only poses risks to birds, but also to humans. The viruses spread from birds to humans and from human to human In addition, mutation in the primary strain will increase the infectiousness of avian influenza. We developed a mathematical model of avian influenza for both bird and human populations. The effect of half-saturated incidence on transmission dynamics of the disease is investigated. The half-saturation constants determine the levels at which birds and humans contract avian influenza. To prevent the spread of avian influenza, the associated half-saturation constants must be increased, especially the half-saturation constant H m for humans with mutant strain. The quantity H m plays an essential role in determining the basic reproduction number of this model. Furthermore, by decreasing the rate β m at which human-to-human mutant influenza is contracted, an outbreak can be controlled more effectively. To combat the outbreak, we propose both pharmaceutical (vaccination) and non-pharmaceutical (personal protection and isolation) control methods to reduce the transmission of avian influenza. Vaccination and personal protection will decrease β m, while isolation will increase H m. Numerical simulations demonstrate that all proposed control strategies will lead to disease eradication; however, if we only employ vaccination, it will require slightly longer to eradicate the disease than only applying non-pharmaceutical or a combination of pharmaceutical and non-pharmaceutical control methods. In conclusion, it is important to adopt a combination of control methods to fight an avian influenza outbreak.

Multiple estimates of transmissibility for the 2009 influenza pandemic based on influenza-like-illness data from small US military populations

Rapidly characterizing the amplitude and variability in transmissibility of novel human influenza strains as they emerge is a key public health priority. However, comparison of early estimates of the basic reproduction number during the 2009 pandemic were challenging because of inconsistent data sources and methods. Here, we define and analyze influenza-like-illness (ILI) case data from 2009-2010 for the 50 largest spatially distinct US military installations (military population defined by zip code, MPZ). We used publicly available data from non-military sources to show that patterns of ILI incidence in many of these MPZs closely followed the pattern of their enclosing civilian population. After characterizing the broad patterns of incidence (e.g. single-peak, double-peak), we defined a parsimonious SIR-like model with two possible values for intrinsic transmissibility across three epochs. We fitted the parameters of this model to data from all 50 MPZs, finding them to be reasonably well clustered with a median (mean) value of 1.39 (1.57) and standard deviation of 0.41. An increasing temporal trend in transmissibility ([Formula: see text], p-value: 0.013) during the period of our study was robust to the removal of high transmissibility outliers and to the removal of the smaller 20 MPZs. Our results demonstrate the utility of rapidly available - and consistent - data from multiple populations.

Predicting and controlling infectious disease epidemics using temporal networks

Infectious diseases can be considered to spread over social networks of people or animals. Mainly owing to the development of data recording and analysis techniques, an increasing amount of social contact data with time stamps has been collected in the last decade. Such temporal data capture the dynamics of social networks on a timescale relevant to epidemic spreading and can potentially lead to better ways to analyze, forecast, and prevent epidemics. However, they also call for extended analysis tools for network epidemiology, which has, to date, mostly viewed networks as static entities. We review recent results of network epidemiology for such temporal network data and discuss future developments.

The network level reproduction number for infectious diseases with both vertical and horizontal transmission

A wide range of infectious diseases are both vertically and horizontally transmitted. Such diseases are spatially transmitted via multiple species in heterogeneous environments, typically described by complex meta-population models. The reproduction number, R0, is a critical metric predicting whether the disease can invade the meta-population system. This paper presents the reproduction number for a generic disease vertically and horizontally transmitted among multiple species in heterogeneous networks, where nodes are locations, and links reflect outgoing or incoming movement flows. The metapopulation model for vertically and horizontally transmitted diseases is gradually formulated from two species, two-node network models. We derived an explicit expression of R0, which is the spectral radius of a matrix reduced in size with respect to the original next generation matrix. The reproduction number is shown to be a function of vertical and horizontal transmission parameters, and the lower bound is the reproduction number for horizontal transmission. As an application, the reproduction number and its bounds for the Rift Valley fever zoonosis, where livestock, mosquitoes, and humans are the involved species are derived. By computing the reproduction number for different scenarios through numerical simulations, we found the reproduction number is affected by livestock movement rates only when parameters are heterogeneous across nodes. To summarize, our study contributes the reproduction number for vertically and horizontally transmitted diseases in heterogeneous networks. This explicit expression is easily adaptable to specific infectious diseases, affording insights into disease evolution.

Modeling Gender-Structured Wildlife Diseases with Harvesting: Chronic Wasting Disease as an Example

Chronic wasting disease (CWD) is a prion infectious disease that affects members of the deer family in North America. Concerns about the economic consequences of the presence of CWD have led management agencies to seek effective strategies to control CWD distribution and prevalence. Current mathematical models are either based on complex simulations or overly simplified compartmental models. We develop a mathematical model that includes gender structure to describe CWD in a logistically growing population. The model includes harvesting as a management strategy for the disease. We determine the stability conditions of the disease-free equilibrium for the model and calculate the basic reproduction number. We find an optimum interval of harvesting: with too little harvesting, the disease persists, whereas too much harvesting results in extinction of the population. A sensitivity analysis shows that the disease threshold is more sensitive to female than male harvesting and that harvesting has the greatest effect on the basic reproduction number. However, while harvesting may be a way to control CWD, the range of admissible harvesting rates may be very narrow, depending on other parameters.

Increased burst size in multiply infected cells can alter basic virus dynamics

BACKGROUND

The dynamics of viral infections have been studied extensively in a variety of settings, both experimentally and with mathematical models. The majority of mathematical models assumes that only one virus can infect a given cell at a time. It is, however, clear that especially in the context of high viral load, cells can become infected with multiple copies of a virus, a process called coinfection. This has been best demonstrated experimentally for human immunodeficiency virus (HIV), although it is thought to be equally relevant for a number of other viral infections. In a previously explored mathematical model, the viral output from an infected cell does not depend on the number of viruses that reside in the cell, i.e. viral replication is limited by cellular rather than viral factors. In this case, basic virus dynamics properties are not altered by coinfection.

RESULTS

Here, we explore the alternative assumption that multiply infected cells are characterized by an increased burst size and find that this can fundamentally alter model predictions. Under this scenario, establishment of infection may not be solely determined by the basic reproductive ratio of the virus, but can depend on the initial virus load. Upon infection, the virus population need not follow straight exponential growth. Instead, the exponential rate of growth can increase over time as virus load becomes larger. Moreover, the model suggests that the ability of anti-viral drugs to suppress the virus population can depend on the virus load upon initiation of therapy. This is because more coinfected cells, which produce more virus, are present at higher virus loads. Hence, the degree of drug resistance is not only determined by the viral genotype, but also by the prevalence of coinfected cells.

CONCLUSIONS

Our work shows how an increased burst size in multiply infected cells can alter basic infection dynamics. This forms the basis for future experimental testing of model assumptions and predictions that can distinguish between the different scenarios.

On the definition and the computation of the type-reproduction number T for structured populations in heterogeneous environments

In the context of mathematical epidemiology, the type-reproduction number (TRN) for a specific host type is interpreted as the average number of secondary cases of that type produced by the primary cases of the same host type during the entire course of infection. Here, it must be noted that T takes into account not only the secondary cases directly transmitted from the specific host but also the cases indirectly transmitted by way of other types, who were infected from the primary cases of the specific host with no intermediate cases of the target host. Roberts and Heesterbeek (Proc R Soc Lond B 270:1359-1364, 2003) have shown that T is a useful measure when a particular single host type is targeted in the disease control effort in a community with various types of host, based on the fact that the sign relation sign(R₀-1) = sign(T-1) holds between the basic reproduction number R₀ and T. In fact, T can be seen as an extension of R₀ in a sense that the threshold condition of the total population growth can be formulated by the reproduction process of the target type only. However, the original formulation is limited to populations with discrete state space in constant environments. In this paper, based on a new perspective of R₀ in heterogeneous environments (Inaba in J Math Biol 2011), we give a general definition of the TRN for continuously structured populations in heterogeneous environments and show some examples of its computation and applications.

Epidemic threshold and control in a dynamic network

In this paper we present a model describing susceptible-infected-susceptible-type epidemics spreading on a dynamic contact network with random link activation and deletion where link activation can be locally constrained. We use and adapt an improved effective degree compartmental modeling framework recently proposed by Lindquist et al. [J. Math Biol. 62, 143 (2010)] and Marceau et al. [Phys. Rev. E 82, 036116 (2010)]. The resulting set of ordinary differential equations (ODEs) is solved numerically, and results are compared to those obtained using individual-based stochastic network simulation. We show that the ODEs display excellent agreement with simulation for the evolution of both the disease and the network and are able to accurately capture the epidemic threshold for a wide range of parameters. We also present an analytical R0 calculation for the dynamic network model and show that, depending on the relative time scales of the network evolution and disease transmission, two limiting cases are recovered: (i) the static network case when network evolution is slow and (ii) homogeneous random mixing when the network evolution is rapid. We also use our threshold calculation to highlight the dangers of relying on local stability analysis when predicting epidemic outbreaks on evolving networks.

Estimating the transmission potential of supercritical processes based on the final size distribution of minor outbreaks

Use of the final size distribution of minor outbreaks for the estimation of the reproduction numbers of supercritical epidemic processes has yet to be considered. We used a branching process model to derive the final size distribution of minor outbreaks, assuming a reproduction number above unity, and applying the method to final size data for pneumonic plague. Pneumonic plague is a rare disease with only one documented major epidemic in a spatially limited setting. Because the final size distribution of a minor outbreak needs to be normalized by the probability of extinction, we assume that the dispersion parameter (k) of the negative-binomial offspring distribution is known, and examine the sensitivity of the reproduction number to variation in dispersion. Assuming a geometric offspring distribution with k=1, the reproduction number was estimated at 1.16 (95% confidence interval: 0.97–1.38). When less dispersed with k=2, the maximum likelihood estimate of the reproduction number was 1.14. These estimates agreed with those published from transmission network analysis, indicating that the human-to-human transmission potential of the pneumonic plague is not very high. Given only minor outbreaks, transmission potential is not sufficiently assessed by directly counting the number of offspring. Since the absence of a major epidemic does not guarantee a subcritical process, the proposed method allows us to conservatively regard epidemic data from minor outbreaks as supercritical, and yield estimates of threshold values above unity.