Coexistence of different serotypes of dengue virus

 We formulate a non–linear system of differential equations that models the dynamics of dengue fever. This disease is produced by any of the four serotypes of dengue arbovirus. Each serotype produces permanent immunity to it, but only a certain degree of cross–immunity to heterologous serotypes. In our model we consider the relation between two serotypes. Our interest is to analyze the factors that allow the invasion and persistence of different serotypes in the human population. Analysis of the model reveals the existence of four equilibrium points, which belong to the region of biological interest. One of the equilibrium points corresponds to the disease–free state, the other three equilibria correspond to the two states where just one serotype is present, and the state where both serotypes coexist, respectively. We discuss conditions for the asymptotic stability of equilibria, supported by analytical and numerical methods. We find that coexistence of both serotypes is possible for a large range of parameters. Received: 7 July 1998 / Revised version: 12 July 2002 / Published online: 26 September 2002 Keywords or phrases: Dengue fever – Primary and secondary infections – Serotype – Coexistence – Threshold – Basic reproduction number – Persistence     

The onset of oscillatory dynamics in models of multiple disease strains

 We examine a generalised SIR model for the infection dynamics of four competing disease strains. This model contains four previously-studied models as special cases. The different strains interact indirectly by the mechanism of cross-immunity; individuals in the host population may become immune to infection by a particular strain even if they have only been infected with different but closely related strains. Several different models of cross-immunity are compared in the limit where the death rate is much smaller than the rate of recovery from infection. In this limit an asymptotic analysis of the dynamics of the models is possible, and we are able to compute the location and nature of the Takens–Bogdanov bifurcation associated with the presence of oscillatory dynamics observed by previous authors. Received: 5 December 2001 / Revised version: 5 May 2002 / Published online: 17 October 2002 Keywords or phrases: Infection – Pathogen – Epidemiology – Multiple strains – Cross-immunity – Oscillations – Dynamics – Bifurcations     

Using dimension reduction to improve outbreak predictability of multistrain diseases

Multistrain diseases have multiple distinct coexisting serotypes (strains). For some diseases, such as dengue fever, the serotypes interact by antibody-dependent enhancement (ADE), in which infection with a single serotype is asymptomatic, but contact with a second serotype leads to higher viral load and greater infectivity. We present and analyze a dynamic compartmental model for multiple serotypes exhibiting ADE. Using center manifold techniques, we show how the dynamics rapidly collapses to a lower dimensional system. Using the constructed reduced model, we can explain previously observed synchrony between certain classes of primary and secondary infectives (Schwartz et al. in Phys Rev E 72:066201, 2005). Additionally, we show numerically that the center manifold equations apply even to noisy systems. Both deterministic and stochastic versions of the model enable prediction of asymptomatic individuals that are difficult to track during an epidemic. We also show how this technique may be applicable to other multistrain disease models, such as those with cross-immunity.     

Immunological serotype interactions and their effect on the epidemiological pattern of dengue

Long-term epidemiological data reveal multi-annual fluctuations in the incidence of dengue fever and dengue haemorrhagic fever, as well as complex cyclical behaviour in the dynamics of the four serotypes of the dengue virus. It has previously been proposed that these patterns are due to the phenomenon of the so-called antibody-dependent enhancement (ADE) among dengue serotypes, whereby viral replication is increased during secondary infection with a heterologous serotype; however, recent studies have implied that this positive reinforcement cannot account for the temporal patterns of dengue and that some form of cross-immunity or external forcing is necessary. Here, we show that ADE alone can produce the observed periodicities and desynchronized oscillations of individual serotypes if its effects are decomposed into its two possible manifestations: enhancement of susceptibility to secondary infections and increased transmissibility from individuals suffering from secondary infections. This decomposition not only lowers the level of enhancement necessary for realistic disease patterns but also reduces the risk of stochastic extinction. Furthermore, our analyses reveal a time-lagged correlation between serotype dynamics and disease incidence rates, which could have important implications for understanding the irregular pattern of dengue epidemics.     

How much complexity is needed to describe the fluctuations observed in dengue hemorrhagic fever incidence data?

Different extensions of the classical single-strain SIR model for the host population, motivated by modeling dengue fever epidemiology, have reported a rich dynamic structure including deterministic chaos which was able to mimic the large fluctuations of disease incidences. A comparison between the basic two-strain dengue model, which already captures differences between primary and secondary infections including temporary cross-immunity, with the four-strain dengue model, that introduces the idea of competition of multiple strains in dengue epidemics shows that the difference between first and secondary infections drives the rich dynamics more than the detailed number of strains to be considered in the model structure. Chaotic dynamics were found to happen in the same parameter region of interest, for both the two and the four-strain models, able to describe the fluctuations observed in empirical data and shows a qualitatively good agreement between empirical data and model simulation. The predictability of the system does not change significantly when considering two or four strains, i.e. both models present a positive dominant Lyapunov exponent giving approximately the same prediction horizon in time series. Since the law of parsimony favors the simplest of two competing models, the two-strain model would be the better candidate to be analyzed, as well the best option for estimating all initial conditions and the few model parameters based on the available incidence data.     

Theoretical considerations of cross-immunity, recombination and the evolution of new parasitic strains.

We explore the dynamics of multiple strains of a parasite in order to assess the conditions under which a novel strain, perhaps a mutant or migrant, may invade a population that already carries an endemic strain. Multiple strain dynamics can be modeled through coinfection or complete cross-immunity. We examine these three modes to discuss the relationships among cross-immunity, the basic reproductive rates of each strain, and the invasion of the new strain. Superinfection is more restrictive than coinfection in the proportion of parameters that allows invasion. The coinfection model is extended to allow haploid strains to undergo recombination within the host. We investigate the effects of recombination and cross-immunity on the invasion of new strains. Interestingly, although recombination is understood to generate diversity, it is not always advantageous.     

Getting a free ride on poultry farms: how highly pathogenic avian influenza may persist in spite of its virulence

It is well-known that highly pathogenic avian influenza (HPAI) strains can arise from low pathogenic strains (LPAI) during epidemics in poultry farms. Despite this, the possibility that partial cross-immunity triggered by previous exposure to LPAI viruses may reduce the pathogenicity of HPAI and thus enhance its persistence has been generally overlooked in both empirical and theoretical work on avian influenza. We propose a simple mathematical model to investigate the interacting dynamics of HPAI and LPAI strains of avian influenza in small-scale poultry farms. Through the analysis of a deterministic ordinary differential equations model, we show that: (1) for a wide range of realistic model parameters, the reduction in pathogenicity yielded by previous LPAI infection might allow an HPAI strain that would not be able to persist in a host population when alone (ℜ₀ < 1) to invade and co-exist in the host population along with the LPAI strain and (2) the coexistence between the HPAI and LPAI strains may be characterized by multiyear periodicity. Because simulations showed that troughs between epidemics can be deep, with only a fraction of existing flocks infected by the HPAI strain, we also ran an individual-based stochastic version of the dynamical model to analyze the potential for natural fade-out of the HPAI strain. The analysis of the stochastic model confirms the prediction that previous exposure to a LPAI strain can significantly increase the duration of the epidemics by an HPAI strain before it fades from the population.     

Mathematical models for the <Emphasis Type="Italic">Aedes aegypti</Emphasis> dispersal dynamics: Travelling waves by wing and wind

Biological invasion is an important area of research in mathematical biology and more so if it concerns species which are vectors for diseases threatening the public health of large populations. That is certainly the case for Aedes aegypti and the dengue epidemics in South America. Without the prospect of an effective and cheap vaccine in the near future, any feasible public policy for controlling the dengue epidemics in tropical climates must necessarily include appropriate strategies for minimizing the mosquito population factor. The present paper discusses some mathematical models designed to describe A. aegypti’s vital and dispersal dynamics, aiming to highlight practical procedures for the minimization of its impact as a dengue vector. A continuous model including diffusion and advection shows the existence of a stable travelling wave in many situations and a numerical study relates the wavefront speed to a few crucial parameters. Strategies for invasion containment and its prediction based on measurable parameters are analysed.     

Dynamics of annual influenza A epidemics with immuno-selection

 The persistence of Influenza A in the human population relies on continual changes in the viral surface antigens allowing the virus to reinfect the same hosts every few years. The epidemiology of such a drifting virus is modeled by a discrete season-to-season map. During the epidemic season only one strain is present and its transmission dynamics follows a standard epidemic model. After the season, cross-immunity to next year's virus is determined from the proportion of hosts that were infected during the season. A partial analysis of this map shows the existence of oscillations where epidemics occur at regular or irregular intervals. Received: 16 February 2001 / Revised version: 11 June 2002 / Published online: 28 February 2003 Key words or phrases: Infectious disease – Influenza drift – Cross-immunity – Seasonal epidemics – Iterated map     

Host switching vs. host sharing in overlapping sylvatic Trypanosoma cruzi transmission cycles

ABSTRACT

The principle of competitive exclusion is well established for multiple populations competing for the same resource, and simple models for multistrain infection exhibit it as well when cross-immunity precludes coinfections. However, multiple hosts provide niches for different pathogens to occupy simultaneously. This is the case for the vector-borne parasite Trypanosoma cruzi in overlapping sylvatic transmission cycles in the Americas, where it is enzootic. This study uses cycles in the USA involving two different hosts but the same vector species as a context for the study of the mechanisms behind the communication between the two cycles. Vectors dispersing in search of new hosts may be considered to move between the two cycles (host switching) or, more simply, to divide their time between the two host types (host sharing). Analysis considers host switching as an intermediate case between isolated cycles and intermingled cycles (host sharing) in order to examine the role played by the host-switching rate in permitting coexistence of multiple strains in a single-host population. Results show that although the population dynamics (demographic equilibria) in host-switching models align well with those in the limiting models (host sharing or isolated cycles), infection dynamics differ significantly, in ways that sometimes illuminate the underlying epidemiology (such as differing host susceptibilities to infection) and sometimes reveal model limitations (such as host switching dominating the infection dynamics). Numerical work suggests that the model explains the trace presence of TcI in raccoons but not the more significant co-persistence observed in woodrats.     

Controlling Dengue with Vaccines in Thailand

Background

Dengue is a mosquito-borne infectious disease that constitutes a growing global threat with the habitat expansion of its vectors Aedes aegyti and A. albopictus and increasing urbanization. With no effective treatment and limited success of vector control, dengue vaccines constitute the best control measure for the foreseeable future. With four interacting dengue serotypes, the development of an effective vaccine has been a challenge. Several dengue vaccine candidates are currently being tested in clinical trials. Before the widespread introduction of a new dengue vaccine, one needs to consider how best to use limited supplies of vaccine given the complex dengue transmission dynamics and the immunological interaction among the four dengue serotypes.

Methodology/Principal Findings

We developed an individual-level (including both humans and mosquitoes), stochastic simulation model for dengue transmission and control in a semi-rural area in Thailand. We calibrated the model to dengue serotype-specific infection, illness and hospitalization data from Thailand. Our simulations show that a realistic roll-out plan, starting with young children then covering progressively older individuals in following seasons, could reduce local transmission of dengue to low levels. Simulations indicate that this strategy could avert about 7,700 uncomplicated dengue fever cases and 220 dengue hospitalizations per 100,000 people at risk over a ten-year period.

Conclusions/Significance

Vaccination will have an important role in controlling dengue. According to our modeling results, children should be prioritized to receive vaccine, but adults will also need to be vaccinated if one wants to reduce community-wide dengue transmission to low levels.     

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.     

The impact of cross-immunity, mutation and stochastic extinction on pathogen diversity

We examine the dynamics of antigenically diverse infectious agents using a mathematical model describing the transmission dynamics of arbitrary numbers of pathogen strains, interacting via cross-immunity, and in the presence of mutations generating new strains and stochastic extinctions of existing ones. Equilibrium dynamics fall into three classes depending on cross-immunity, transmissibility and host population size: systems where global extinction is likely, stable single-strain persistence, and multiple-strain persistence with stable diversity. Where multi-strain dynamics are stable, a diversity threshold region separates a low-prevalence, low-diversity region of parameter space from a high-diversity, high-prevalence region. The location of the threshold region is determined by the reproduction number of the pathogen and the intensity of cross-immunity, with the sharpness of the transition being determined by the manner in which immunity accrues with repeated infections. Host population size and cross-immunity are found to be the most decisive factors in determining pathogen diversity. While the model framework developed is simplified, we show that it can capture essential aspects of the complex evolutionary dynamics of pathogens such as influenza.     

Estimating the magnitude and direction of altered arbovirus transmission due to viral phenotype

Vectorial capacity is a measure of the transmission potential of a vector borne pathogen within a susceptible population. Vector competence, a component of the vectorial capacity equation, is the ability of an arthropod to transmit an infectious agent following exposure to that agent. Comparisons of arbovirus strain-specific vector competence estimates have been used to support observed or hypothesized differences in transmission capability. Typically, such comparisons are made at a single time point during the extrinsic incubation period, the time in days it takes for the virus to replicate and disseminate to the salivary glands. However, vectorial capacity includes crucial parameters needed to effectively evaluate transmission capability, though often this is based on the discrete vector competence values. Utilization of the rate of change of vector competence over a range of days gives a more accurate measurement of the transmission potential. Accordingly, we investigated the rate of change in vector competence of dengue virus in Aedes aegypti mosquitoes and the resulting vectorial capacity curves. The areas under the curves represent the effective vector competence and the cumulative transmission potentials of arboviruses within a population of mosquitoes. We used the calculated area under the curve for each virus strain and the corresponding variance estimates to test for differences in cumulative transmission potentials between strains of dengue virus based on our dynamic model. To further characterize differences between dengue strains, we devised a displacement index interpreted as the capability of a newly introduced strain to displace the established, dominant circulating strain. The displacement index can be used to better understand the transmission dynamics in systems where multiple strains/serotypes circulate or even multiple arbovirus species. The use of a rate of a rate of change based model of vectorial capacity and the informative calculations of the displacement index will lead to better measurements of the differences in transmission potential of arboviruses.     

The dynamics of cocirculating influenza strains conferring partial cross-immunity

 We develop a model that describes the dynamics of a finite number of strains that confer partial cross-protection among strains. The immunity structure of the host population is captured by an index-set notation where the index specifies the set of strains to which the host has been exposed. This notation allows us to derive threshold conditions for the invasion of a new strain and to show the existence of an endemic multi-strain equilibrium in a special case. The dynamics of systems consisting of more than two strains can exhibit sustained oscillations caused by an overshoot in the immunity to a specific strain if cross-protection is sufficiently strong. Received 15 January 1996; received in revised form 24 June 1996     

Dynamic Epidemiological Models for Dengue Transmission: A Systematic Review of Structural Approaches

Dengue is a vector-borne disease recognized as the major arbovirose with four immunologically distant dengue serotypes coexisting in many endemic areas. Several mathematical models have been developed to understand the transmission dynamics of dengue, including the role of cross-reactive antibodies for the four different dengue serotypes. We aimed to review deterministic models of dengue transmission, in order to summarize the evolution of insights for, and provided by, such models, and to identify important characteristics for future model development. We identified relevant publications using PubMed and ISI Web of Knowledge, focusing on mathematical deterministic models of dengue transmission. Model assumptions were systematically extracted from each reviewed model structure, and were linked with their underlying epidemiological concepts. After defining common terms in vector-borne disease modelling, we generally categorised fourty-two published models of interest into single serotype and multiserotype models. The multi-serotype models assumed either vector-host or direct host-to-host transmission (ignoring the vector component). For each approach, we discussed the underlying structural and parameter assumptions, threshold behaviour and the projected impact of interventions. In view of the expected availability of dengue vaccines, modelling approaches will increasingly focus on the effectiveness and cost-effectiveness of vaccination options. For this purpose, the level of representation of the vector and host populations seems pivotal. Since vector-host transmission models would be required for projections of combined vaccination and vector control interventions, we advocate their use as most relevant to advice health policy in the future. The limited understanding of the factors which influence dengue transmission as well as limited data availability remain important concerns when applying dengue models to real-world decision problems.     

Serotype cycles in cholera dynamics

Interest in understanding strain diversity and its impact on disease dynamics has grown over the past decade. Theoretical disease models of several co-circulating strains indicate that incomplete cross-immunity generates conditions for strain-cycling behaviour at the population level. However, there have been no quantitative analyses of disease time-series that are clear examples of theoretically expected strain cycling. Here, we analyse a 40-year (1966-2005) cholera time-series from Bangladesh to determine whether patterns evident in these data are compatible with serotype-cycling behaviour. A mathematical two-serotype model is capable of explaining the oscillations in case patterns when cross-immunity between the two serotypes, Inaba and Ogawa, is high. Further support that cholera's serotype-cycling arises from population-level immunity patterns is provided by calculations of time-varying effective reproductive rates. These results shed light on historically observed serotype dominance shifts and have important implications for cholera early warning systems.     

Phenotypic differences in viral immune escape explained by linking within-host dynamics to host-population immunity

Viruses that do not cause life-long immunity persist by evolving rapidly in response to prevailing host immunity. The immune-escape mutants emerge frequently, displacing or co-circulating with native strains even though mutations conferring immune evasion are often detrimental to viral replication. The epidemiological dynamics of immune-escape in acute-infection viruses with high transmissibility have been interpreted mainly through immunity dynamics at the host population level, despite the fact that immune-escape evolution involves dynamical processes that feedback across the within- and between-host scales. To address this gap, we use a nested model of within- and between-host infection dynamics to examine how the interaction of viral replication rate and cross-immunity imprint host population immunity, which in turn determines viral immune escape. Our explicit consideration of direct and immune-mediated competitive interactions between strains within-hosts revealed three insights pertaining to risk and control of viral immune-escape: (1) replication rate and immune-stimulation deficiencies (i.e., original antigenic sin) act synergistically to increase immune escape, (2) immune-escape mutants with replication deficiencies relative to their wildtype progenitor are most successful under moderate cross-immunity and frequent re-infections, and (3) the immunity profile along short host-transmission chains (local host-network structure) is a key determinant of immune escape.     

Deterministic extinction effect of parasites on host populations

 Experimental studies have shown that parasites can reduce host density and even drive host population to extinction. Conventional mathematical models for parasite-host interactions, while can address the host density reduction scenario, fail to explain such deterministic extinction phenomena. In order to understand the parasite induced host extinction, Ebert et al. (2000) formulated a plausible but ad hoc epidemiological microparasite model and its stochastic variation. The deterministic model, resembles a simple SI type model, predicts the existence of a globally attractive positive steady state. Their simulation of the stochastic model indicates that extinction of host is a likely outcome in some parameter regions. A careful examination of their ad hoc model suggests an alternative and plausible model assumption. With this modification, we show that the revised parasite-host model can exhibit the observed parasite induced host extinction. This finding strengthens and complements that of Ebert et al. (2000), since all continuous models are likely break down when all population densities are small. This extinction dynamics resembles that of ratio-dependent predator-prey models. We report here a complete global study of the revised parasite-host model. Biological implications and limitations of our findings are also presented. Received: 30 October 2001 / Revised version: 11 February 2002 / Published online: 17 October 2002 Work is partially supported by NSF grant DMS-0077790 Mathematics Subject Classification (2000): 34C25, 34C35, 92D25. Keywords or phrases: Microparasite model – Ratio-dependent predator-prey model – Host extinction – Global stability – Biological control     

Asymmetric competitive suppression between strains of dengue virus

Background  

Within-host competition between strains of a vector-borne pathogen can affect strain frequencies in both the host and vector, thereby affecting viral population dynamics. However little is known about inter-strain competition in one of the most genetically diverse and epidemiologically important mosquito-borne RNA virus: dengue virus (DENV). To assess the strength and symmetry of intra-host competition among different strains of DENV, the effect of mixed infection of two DENV serotypes, DENV2 and DENV4, on the replication of each in cultured mosquito cells was tested. The number of infectious particles produced by each DENV strain in mixed infections was compared to that in single infections to determine whether replication of each strain was decreased in the presence of the other strain (i.e., competition). The two DENV strains were added to cells either simultaneously (coinfection) or with a 1 or 6-hour time lag between first and second serotype (superinfection).