Dengue is a viral mosquito-borne infection which in recent years has become a major international public health concern, a leading cause of illness and death in the tropics and subtropics. Dengue fever is caused by four antigenically distinct viruses, designated dengue types 1, 2, 3 and 4. Infection by one serotype confers life-long immunity to only that serotype and temporary cross-immunity to other serotypes exists. It lasts from three to nine months, when the antibody levels created during the response to that infection would be enough to protect against infection by a different but related serotype. Two variants of the disease exist: dengue fever (DF), a non-fatal form of illness, and dengue hemorrhagic fever (DHF), which may evolve toward a severe form known as dengue shock syndrome (DSS).
Epidemiological studies support the association of DHF with secondary dengue infection, and there is good evidence that sequential infection increases the risk of developing DHF, due to a process described as antibody-dependent enhancement (ADE), where the pre-existing antibodies to previous dengue infection cannot neutralize but rather enhance the new infection.
Treatment of uncomplicated dengue cases is only supportive, and severe dengue cases requires careful attention to fluid management and proactive treatment of hemorrhagic symptoms. A vaccine against dengue is not yet available, since it would have to simulate a protective immune response to all four serotypes, although several candidates of tetravalent vaccines are at various stages of development. So far, prevention of exposure and vector control remain the only alternatives to prevent dengue transmission.
In recent years, mathematical modeling became an interesting tool for the understanding of infectious diseases epidemiology and dynamics. A series of deterministic compartment models such as Susceptible-Infected (SI) and Susceptible-Infected-Recovered (SIR) e.g. have been proposed based on the flow patterns between compartments of hosts. The SIR epidemic model divides the population into three classes: susceptible (S), Infected (I) and Recovered (R). It can be applied to infectious diseases where waning immunity can happen, and assuming that the transmission of the disease is contagious from person to person, the susceptibles become infected and infectious, are cured and become recovered. After a waning immunity period, the recovered individual can become susceptible again. Multi-strain dynamics are generally modeled with extended SIR-type models.
Dengue epidemiology dynamics is well known to be particularly complex with large fluctuations of disease incidences. To capture differences in primary dengue infection by one strain and secondary dengue infection by another strain, a two-strain SIR-type model for the host population have to be considered. Dengue models including multi-strain interactions via ADE, but without temporary cross immunity period, have shown deterministic chaos when strong infectivity on secondary infection was assumed. The addition of the temporary cross immunity period in such models brings a new chaotic attractor in an unexpected and more biologically realistic parameter region of reduced infectivity on secondary infection.
In this thesis we present different extensions of the classical single-strain SIR model for the host population motivated by modeling dengue fever epidemiology with its peculiar ADE phenomenology. We focused in a minimal model, where the notion of at least two different strains is needed to describe differences between primary and secondary dengue infections. The models divide the host population into susceptible, infected and recovered individuals with subscripts for the respective strains. The individuals can be susceptibles without a previous dengue infection, infected and recovered for the first time, susceptible with an experienced previous infection and infected for the second time, now with a different strain, and more likely been hospitalized due to the ADE effect leading to severe disease. Our analysis showed a rich dynamic structure, including deterministic chaos, in wider and more biologically realistic parameter regions just by adding temporary cross-immunity to previously existing dengue models.
In Chapter 1 we presented the properties of the basic SIR epidemic model for infectious diseases with a summary of the analysis of the dynamics, identifying the thresholds and equilibrium points in order to introduce notation, terminology. These results were then generalized to more advanced models motivated by dengue fever epidemiology. In Chapter 2 we present the basic two-strain SIR-type model motivated by modeling dengue fever epidemiology. We focused on the multi-strain aspect and its effects on the host population, taking effects of the vector dynamics or seasonality only in account by the effective parameters in the SIR-type model, but not modeling these mechanisms explicitly. In Chapter 3 a detailed bifurcation analysis for the basic multi-strain dengue model, in terms of the ADE parameter phi and a parameter for the temporary cross immunity alpha, is presented.
In Chapter 4 our study focused on the seasonally forced system with temporary cross immunity and possible secondary infection, motivated by dengue fever monitoring data, where the role of seasonality and import of infected individuals in such systems were considered as biologically relevant effects determining the dynamical behavior of the system. A comparative study between three different scenarios (non-seasonal, low seasonal and high seasonal with a low import of infected individuals) is presented. The extended models show complex dynamics and qualitatively a good agreement between empirical DHF monitoring data and the obtained model simulation. We discuss the role of seasonal forcing and the import of infected individuals in such systems, the biological relevance and its implications for the analysis of the available dengue data.
At the moment only such minimalistic models have a chance to be qualitatively understood well and eventually tested against existing data. The simplicity of the model (low number of parameters and state variables) offer a promising perspective on parameter values inference from the DHF case notifications. Such a technical parameter estimation is notoriously difficult for chaotic time series but short term approaches are possible.
Being able to predict future outbreaks of dengue in the absence of human interventions is a major goal if one wants to understand the effects of control measures. Even after a dengue virus vaccine has become accessible or available, this holds true for the implementation of a vaccination program. For example, to perform a vaccine trial in a year where the disease affect low numbers of individuals, generating low number of cases, would make statistical tests of vaccine efficacy much more difficult than when a vaccine trial would be performed in a year where the disease would affect much more individuals, generating high numbers of cases. Thus predictability of the next season's hight of the dengue peak on the basis of deterministic balance of infected and susceptible would be of major practical use.
Besides the fact that disease propagation is an inherently stochastic phenomenon, dengue models are mainly expressed mathematically as a set of deterministic differential equations which are easier to analyze. The mean field approximation is a good approximation to be used in order to understand better the behavior of the stochastic systems in certain parameter regions, where the dynamics of the mean quantities are approximated by neglecting correlations. However, it is only stochastic, as opposed to deterministic, models that can capture the fluctuations observed in some of the available time series data. In Chapter 5 we present the stochastic version of a minimalistic multi-strain model, which captures essential differences between primary and secondary infections in dengue fever epidemiology, and investigate the interplay between stochasticity, seasonality and import. The introduction of stochasticity was needed to explain the fluctuations observed in some of the available data sets, revealing a scenario where noise and complex deterministic skeleton strongly interact. For large enough population size, the stochastic system could be well described by the deterministic skeleton, where the essential dynamics are captured, gaining insight into the relevant parameter values purely on topological information of the dynamics.
The two-strain dengue model is a $9$ dimensional system and therefore, future parameter estimation can still attempt to estimate all initial conditions as well as the few model parameters. Concerning data availability, long term epidemiological data consist on monthly incidences of hospitalized DHF cases. For such a data scenario, models that are able to generate both primary and secondary infection cases (with a different strain), without the need of considering differences on the dynamics of different co-circulating dengue serotypes, have show a good qualitative agreement between empirical data and model output (see Chapter 4 and Chapter 5), just by incorporating ADE and temporary cross-immunity. Differently from the minimalistic dengue model, the four-strain model is a 25 dimensional system, a very high dimensional system and difficult to estimate due to the number of initials conditions. For four different strains, 1, 2, 3 and 4, we now label the SIR classes for the hosts that have seen the individual strains, again without epidemiological asymmetry between strains, once the serotype data are recent and very short to give any realistic information concerning difference in biological parameters such as infection and recovery rates for a given strain. In Chapter 6 we present the bifurcation diagram comparison for both two-strain and four-strain model, in the relevant parameter region of phi<1, when dengue patients in a secondary infection evolving to severe disease because of the ADE phenomenon contribute less to the force of infection, and not more, as previous models suggested and show that qualitatively, the bifurcation points appear to happen at similar parameter regions, well below the region of interest phi\approx 1.
We therefore conclude that the two-strain model in its simplicity is a good model to be analyzed giving the expected complex behavior to explain the fluctuations observed in empirical data. For future parameter estimation, the two-strain model can be used to attempt to estimate all initial conditions as well as the few model parameters. Further work on the parameter estimation using the minimalistic dengue model is in progress.