The dynamics of measles in sub-Saharan Africa.

Although vaccination has almost eliminated measles in parts of the world, the disease remains a major killer in some high birth rate countries of the Sahel. On the basis of measles dynamics for industrialized countries, high birth rate regions should experience regular annual epidemics. Here, however, we show that measles epidemics in Niger are highly episodic, particularly in the capital Niamey. Models demonstrate that this variability arises from powerful seasonality in transmission-generating high amplitude epidemics-within the chaotic domain of deterministic dynamics. In practice, this leads to frequent stochastic fadeouts, interspersed with irregular, large epidemics. A metapopulation model illustrates how increased vaccine coverage, but still below the local elimination threshold, could lead to increasingly variable major outbreaks in highly seasonally forced contexts. Such erratic dynamics emphasize the importance both of control strategies that address build-up of susceptible individuals and efforts to mitigate the impact of large outbreaks when they occur

Dynamical epidemic suppression using stochastic prediction and control

We consider the effects of noise on a model of epidemic outbreaks, where the outbreaks appear. randomly. Using a constructive transition approach that predicts large outbreaks, prior to their occurrence, we derive an adaptive control. scheme that prevents large outbreaks from occurring. The theory inapplicable to a wide range of stochastic processes with underlying deterministic structure.Comment: 14 pages, 6 figure

Rich dynamics in multi-strain models:non-linear dynamics and deterministic chaos in dengue fever epidemiology

Tese de doutoramento (co-tutela), Biology (Population Biology), Vrije Uiversity Amesterdam, Universidade de Lisboa, 2012Throughout human history, infectious diseases have caused debilitation and pre- mature death to large portions of the human population, leading to serious social- economic concerns. Many factors have contributed to the persistence and increa- se in the occurrence of infectious disease (such as demographic factors, political, social and economic changes, environmental change, public health care and infra- structure, microbial adaptation, etc.). According to the World Health Organiza- tion (WHO), are the second leading cause of death globally after cardiovascular diseases (WHO, 2010). In recent years, mathematical modeling became an im- portant tool for the understanding of infectious disease epidemiology and has led to great advances in conceiving disease control strategies, including vaccination programs. One of the most important infectious diseases is dengue, a major international public health concern with more than 55% of world population at risk of acquiring the infection. Dengue is a viral mosquito-borne infection, 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 related serotypes. The temporary cross-immunity period lasts from three to nine months and it is related to antibody levels created during the immune response to a previous dengue infection. It is stated that such high antibody levels would be enough to protect the individual against an immediately new dengue infection caused 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. 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 den- gue 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 de- velopment. 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 un- derstanding of infectious diseases epidemiology and dynamics. A series of deter- ministic compartment models such as Susceptible-Infected (SI) and Susceptible- Infected-Recovered (SIR) for example, have been proposed based on the flow patterns between compartments of hosts. The SIR epidemic model divides the population into three classes concerning the disease stages: susceptible (S), In- fected (I) and Recovered (R). This model framework can represent infectious diseases where waning immunity can happen. 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 to reinfection. Multi-strain dynamics, such as dengue epidemiology, are generally modeled with extended SIR-type models. Dengue fever dynamic is well known to be particularly complex with large fluctuations of disease incidences. To capture differences in primary and secondary dengue infections, a two-strain SIR-type model for the host population has to be considered. Dengue models including multi-strain interactions via ADE, but without temporary cross-immunity, have shown already 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 wider and unexpected parameter region. In this thesis we present different extensions of the classical single-strain SIR model motivated by modeling dengue fever epidemiology with its peculiar ADE phenomenology. We focus on a minimalistic model, where the notion of at least two different strains is needed to describe differences between primary and se- condary dengue infections. The models divide the host population into suscepti- ble, infected and recovered individuals with subscripts for the respective strains. The individuals can be (1) susceptibles without a previous dengue infection; (2) infected and recovered for the first time; (3) susceptible with an experienced pre- vious dengue infection and (4) infected for the second time with a different strain, more likely to be hospitalized due to the ADE effect leading to severe disease. Our analysis shows a rich dynamic structure, including deterministic chaos in wi- der and more biologically realistic parameter regions, just by adding temporary cross-immunity to previously existing dengue models. In Chapter 1 we present the properties of the basic SIR epidemic model applied to infectious diseases. A summary of the analysis of the dynamics identifying the thresholds and equilibrium points in order to introduce notation and terminology are presented. These results were then generalized to a more advanced models motivated by dengue fever epidemiology. In Chapter 2 the basic two-strain SIR- type model motivated by modeling dengue fever epidemiology is presented. In this chapter we focused on the multi-strain aspect and its effects on the host population. The effects of the vector dynamics or seasonality is taken in account only by the effective parameters of the SIR-type model, but these mechanisms are not modeled explicitly. In Chapter 3 a detailed bifurcation analysis for the basic multi-strain dengue model is presented where the ADE parameter A and the temporary cross-immunity parameter R are studied. In Chapter 4 the seasonally forced system with temporary cross-immunity and possible secondary infection is analyzed. This study was motivated by dengue hemorrhagic fever monitoring data. The role of seasonality and import of infected individuals are now considered as biologically relevant effects to determine 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. At the moment only such minimalistic models have a chance to be qualitati- vely 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 due to the long term unpredictability versus short term predictability. Recent- ly, this short term predictability has been used for temporally local approaches in statistical inference on the cost of difficulty in obtaining a final definite best answer to the parameter estimation problem. 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 epidemic generate a low number of cases, would make the statistical tests of vaccine efficacy much more difficult compared with the information provided by a vaccine trial performed in a epidemic year with much higher numbers of cases. Thus predictability of the next season’s height of the dengue peak, on the basis of deterministic balance of infected and susceptible, would be of major practical use. Although the fact that disease propagation is an inherently stochastic phe- nomenon, dengue models are mainly expressed mathematically as a set of de- terministic differential equations, which are easier to analyze. The mean field approximation, an approximation of stochastic processes leading to deterministic dynamics, is a good approximation to be used in order to understand better the behavior of the stochastic systems in certain parameter regions, where the dyna- mics of the mean quantities are approximated by neglecting correlations. Howe- ver, 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 the stochastic version of the minimalistic multi-strain model is presented. In this chapter we investigate the interplay between stochasticity, seasonality and impor- ted cases of the disease. The introduction of stochasticity reveal a scenario where noise and complex deterministic skeleton strongly interact. For large enough population size, the stochastic system could be well described by the determinis- tic 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 statistical inference can still attempt to estimate all initial conditions as well as the few model parameters. Concerning data availability, long term epidemiologi- cal 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 shown a good quali- tative agreement between empirical data and model output (see Chapter 4 and Chapter 5). These results were obtained just by combining the ADE effect, ge- nerating difference in transmissibility on primary and secondary infections, with the temporary cross-immunity aspect. Differently from the minimalistic dengue model, the four-strain model is mathematically represented by a system of 26 ODE’ s. It becomes a very high dimensional system and obviously very difficult to be used for parameter inference due to the high number of initial conditions. In Chapter 6 we present the multi-strain dengue model for the four existing se- rotypes. For four different strains, 1, 2, 3 and 4, we now label the SIR classes for the hosts that have seen each one of the possible strains. Again, without epidemi- ological asymmetry between strains, once the serotype data are recent and very short to give any realistic information concerning difference in biological parame- ters (such as infection and recovery rates) for a given strain. In this chapter we present the bifurcation diagram comparison for both two-strain and four-strain model. In the relevant parameter region of < 1, when dengue patients in a secondary infection evolving to severe disease due to the ADE phenomenon con- tribute less to the force of infection, the bifurcation points appear to happen at similar parameter regions, well below the region of interest 1. We 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. Statistical inference to estimate the basic parameters of transmission, infectivity, disease severity (ADE parameter) and temporary cross-immunity period using empirical data of incidence of severe disease is needed to identify eventual deviations from the simplest symmetric case investigated here. Further work on the parameter estimation using the minimalistic dengue model is in progress. The vector dynamics might also play a role in understanding the final picture when comparing the model output with the available empirical data. Following the investigations described in this thesis, a number of research directions could be addressed, involving the minimalistic dengue model. Future work would be to investigate extensions of the multi-strain model to address the following questi- ons and issues: (1) How much (more or less than first infection) does secondary infection contribute to the force of infection? (2) Does there exist a difference between the forces of infection for the different strains and to what extent can the bifurcation structure explain the viral diversity contribution? (3) Formulate hy- potheses using the mechanism of temporary cross-immunity suitable to recurrent infections protection. (4) Model the vaccine trials based on short term predic- tability of chaotic systems to be applied when tetravalent vaccines will become available. And (5) propose targets for intervention and control design according to the expected impact of the disease. My special interest would be to get the model fully parametrized on data referring to incidence of severe disease and pre- valence of infection. With such a model framework we would be able to give an insight into the predictability of upcoming dengue outbreaks. This epidemiolo- gical tool would help to understand the effects of control measures and therefore to guide the policies of prevention and control of the dengue virus transmissionAo longo da história, as doenças infecciosas veem causado o enfraquecimento e morte prematura de grandes parte da população humana, gerando sérias preocupações sociais e económicas. Muitos são os fatores quem têm contribuído para a persistência e o aumento na ocorrência de doenças infecciosas, tais como factores demográficos, mudanças políticas, sociais e económicas, mudanças ambientais, adaptação microbiana, etc. Segundo a Organizão Mundial de Saúde (OMS), as doenças infecciosas são a segunda principal causa de morte no mundo, depois das doenças cardiovasculares (WHO, 2010). Dentre as doenças transmissíveis mais preocupantes, o dengue ´e, de acordo com a OMS, um problema de saúde pública internacional, com mais de 55% da população mundial vivendo em áreas com risco de transmissão da infecção. O dengue, uma infecção ao viral transmitida por mosquitos, é uma das principais causas de doença e morte nos trópicos e subtrópicos. A infecção pelo vírus do dengue pode ser causada por qualquer uma das quatro cepas existentes, designadas por serotipos DEN − 1, DEN − 2, DEN − 3 e DEN − 4. Estes serotipos são distintos, porém, antigenicamente relacionados. A infecção gerada por um determinado serotipo confere imunidade total e permanente (ao longo da vida) para apenas aquele serotipo, e também imunidade cruzada temporária para os outros serotipos. A imunidade cruzada temporária tem uma duração estimada que varia de três a nove meses, e está relacionada com os n´níveis de anticorpos gerados durante a resposta imune a uma primeira infecção pelo vírus do dengue. Afirma-se que o alto nível destes anticorpos seria suficiente para a proteção contra outras infecções causadas por patogenos antigenicamente relacionados. O dengue pode se manifestar em duas formas clínicas: dengue clássico (DC), uma forma não-fatal da doença, e dengue hemorrágica (DH), que pode evoluir para uma forma muito grave conhecida como síndrome do choque do dengue (DSS). Estudos epidemiológicos associam os casos graves da doença (DH) com a segunda infecção do dengue. Existem boas evidências relacionando as infecções sequenciais pelos vírus do dengue e o aumento para os riscos do desenvolvimento do dengue hemorrágico. Esta associação se deve a um processo imunológico chamado de antibody-dependent enhancemet (ADE). O antibody-dependent enhan- cement ocorre quando os anticorpos pré-existentes, provenientes de uma primeira infecção do dengue, não neutralizam mas sim realçam a nova infecção pelo vírus do dengue. Não existe uma medicação específica para a infecção do dengue. O tratamento dos casos de dengue clássico é apenas de suporte e para os casos de dengue hemorrágico a hospitalização é frequentemente necessária para obtenção de um tratamento adequado. A vacina contra o dengue ainda não está disponível, uma vez que terá que simular proteccão para todos os quatro serotipos existentes. Atu- almente, algumas vacinas candidatas encontram-se em diversos estágios de desenvolvimento. Até o presente momento, a prevenção na exposição e o controle dos vetores são as únicas alternativas para a prevenção da transmissão do dengue. A modelação matemática tornou-se uma ferramenta importante para a compreensão da epidemiologia e da dinâmica das doenças infecciosas. Uma série de modelos deterministicos, tais como o modelo Susceptível-Infectado (SI) e o modelo Susceptível-Infectado-Recuperado (SIR), por exemplo, têm sido propostos com base nos padrões de fluxo para cada um dos compartimentos representando os estágios da doença. O modelo epidemiológico SIR divide a população de indivíduos em três classes: Susceptíveis (S), Infectados (I) e Recuperados (R). Este tipo de modelo pode ser utilizado para representar, por exemplo, as doenças infecciosas que não conferem imunidade permanente, possibilitando a reinfecçã. Assumindo que a transmissão da doença se faz de pessoa para pessoa, os indivíduos susceptíveis tornam-se infectados e infecciosos (capazes de transmitir a doença), se curam e se tornam recuperados (com imunidade temporária ao patógeno causador da doença). Depois de um determinado período tempo, acontece a perda desta imunidade e o indivíduo tornar-se novamente susceptível, podendo se reinfectar. A dinâmica multi-estirpe é geralmente modelada utilizando extensões dos modelos do tipo SIR. Para capturar as diferenças entre a primeira e a segunda infecção é preciso considerar pelo menos dois serotipos diferentes na composição do modelo do tipo SIR. A dinâmica da epidemiologia do dengue é particularmente complexa, com grandes flutuações (variações em quantidade ao longo do tempo) na incidência da doença. Modelos matemáticos recentes para a transmissão do vírus do den- gue se concentram no efeito ADE e na imunidade cruzada temporária. Estes modelos apresentam resultados de flutuações críticas com distribuição em lei de potência para os casos da doença, caos determinístico e dessincronização caótica, devido a sua estrutura multi-estirpe. O comportamento caótico é obtido quando assumindo infectividade muito alta para a segunda infecção do dengue, isto é, assumindo que os indivíduos na segunda infecção pelo vírus do dengue transmitem a doença com uma taxa muita mais elevada do que os indivíduos na primeira infecção. Considerações da imunidade cruzada temporária associada ao efeito ADE gera uma nova janela caótica inesperada e biologicamente mais realistas, onde a infectividade dos indivíduos na segunda infecção do dengue é reduzida devido a severidade da doença e a provável hospitalização causada pelo processo imunológico do ADE. Nesta tese apresentamos a análise e os resultados obtidos em diferentes extensões do modelo clássico SIR. Estes modelos foram motivado pela epidemiologia do dengue e a sua peculiar característica imunológica causada pelo antibody- dependent enhancement. O nosso estudo se concentra em um modelo minimalístico, em que pelo menos dois serotipos diferentes são necessários para descrever as diferenças entre as infecções primária e secundária causadas pelas diferentes cepas do vírus do dengue. Os modelos dividem a população humana em susceptíveis, infectados e recuperados, e utiliza índices para diferenciar cada um dos serotipos. Os indivíduos podem ser: (1) susceptíveis sem nenhuma infecção prévia pelo vírus do dengue, (2) infectados e recuperados pela primeira vez, (3) susceptíveis com um histórico de infecção prévia e (4) infectados pela segunda vez (por uma cepa diferente da primeira infecção) e, provavelmente hospitalizados devido ao processo de ADE. O modelo minimalístico apresenta uma dinâmica estrutural rica ao incorporar aos modelos já existentes para a transmissão do dengue, o período de imunidade cruzada temporária associada ao processo de antibody-dependent enhancement capaz de gerar diferenças nas taxas de transmissão para as infecçoes primárias e secundárias da doença. No Capítulo 1 apresentamos as propriedades do modelo básico SIR aplicado ao estudo das doenças transmissíveis. A análise da dinâmica apresentada, identificando os limites e os pontos de equilíbrio, com o objectivo de introduzir a notação e a terminologia utilizada. Estes resultados são posteriormente generalizados para os modelos motivados pela epidemiologia do dengue. No Capítulo 2, o modelo básico do tipo SIR para dois serotipos diferentes é apresentado e analisado. Este capítulo enfatiza o aspecto multi-estirpe e seus efeitos sobre a população humana. Os efeitos da dinâmica dos vetores e ou da sazonalidade não são modelados explicitamente, sendo levados em conta apenas pelos parâmetros efetivos do modelo. No Capítulo 3 apresentamos uma análise detalhada dos pontos de bifurcações encontrados para os parâmetros de ADE () e de imunidade cruzada temporária (). No Capítulo 4, o modelo sazonal do dengue é apresentado. Com base nos dados disponíveis de monitoramento do dengue, o papel da força sazonal e os casos importados da doença foram considerados como efeitos biologicamente relevantes para a de

Scaling of stochasticity in dengue hemorrhagic fever epidemics

In this paper we analyze 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 is 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 can be well described by the deterministic skeleton gaining insight on the relevant parameter values purely on topological information of the dynamics, rather than classical parameter estimation of which application is in general restricted to fairly simple dynamical scenarios

Dynamical epidemic suppression using stochastic prediction and control

We consider the effects of noise on a model of epidemic outbreaks, where the outbreaks appear randomly. Using a constructive transition approach that predicts large outbreaks prior to their occurrence, we derive an adaptive control scheme that prevents large outbreaks from occurring. The theory is applicable to a wide range of stochastic processes with underlying deterministic structure

Applications of dynamical systems in ecology

This thesis consists of five original pieces of work contained in chapters 2, 4, 6, 7 and 8. These cover four topics within the subject area of theoretical ecology: epidemiology, chaos in ecology, evolution and spatially extended ecological systems. Chapter 2 puts forward a new mechanism for producing chaos in ecology. We show that near extinctions in the SEIR model stabilise a chaotic repeller. This mechanism works for a wide-range of parameter values and so resolves the debate about which dynamic regime is associated with realistic values. It also highlights the problem of treating fluctuations as being either deterministically or stochastically produced. Chapter 4 describes a new technique for identifying chaos based on measuring the divergence of trajectories over a range of spatial scales. It correctly identifies noise scales and chaos in model systems and is also applied to some real ecological data sets. In chapters 4 and 5 we set evolutionary game theory in a nonlinear dynamical framework. We introduce a powerful new tool, the selective pressure, for analysing ecological models and identifying evolutionary stable states. It allows analysis of systems where complex attractors exist. We also study the evolution of phenotypic distributions and provide a new mechanism for evolutionary discontinuities. In chapter 6 we look at an individually-based spatially extended system. This model is spatially heterogeneous and stochastic. However we show that the dynamics on a certain scale are deterministic and low-dimensional. We show how to identify the most efficient spatial scale at which to monitor the system