Mathematical Epidemiology: Infectious disease modeling across time, space, and scale

Rosemary Aogo

National Institutes of Health (Viral Epidemiology and Immunity Unit, Laboratory of Infectious Diseases, National Institute of Allergy and Infectious Diseases)

"A new model framework offers insights into the role of immune boosting and waning in shaping dengue epidemic dynamics."

Infection with any of the four dengue virus serotypes (DENV1-4) induces serotype-specific and cross-reactive antibodies that may increase disease severity during secondary infection with a different serotype. However, following secondary infection, individuals are at significantly reduced risk of subsequent severe disease with even unexposed serotypes. Previous dengue modeling studies with two or four serotypes have shown that periodicity in dengue incidence can be explained by enhancement between serotypes, transient cross-protective immunity, as well as vector distribution, population size, geography, and seasonality. However, all models have assumed complete protection against previous infecting serotypes and most models have assume complete protection against all serotypes after two sequential infections. Conversely, a recent longitudinal cohort study showed that antibodies wane for many years after secondary DENV infection, at times to the level observed following first DENV infection, suggesting immunity after two infections may not be life-long. In this study, we use a dataset of antibody titers to DENV and ZIKV measured annually in Nicaraguan family and pediatric cohorts from 2017-2021 and developed an immunity-structured SIR-type model that tracks immunity by titer rather than number of prior infections. We show that boosting and waning occur following major dengue and Zika outbreaks in highly immune Nicaraguan adult populations. Using our framework, we show that boosts in highly immune individuals contribute to herd immunity, delaying their contribution to the susceptible population and lowering the rate of dengue cases in future epidemics. However, as their immunity wanes due to lower transmission intensity, the susceptible fraction builds up until a major epidemic that includes re-infection of those with high titers once again depletes the susceptible pool. Comparatively, lifelong immunity in highly immune individuals as previously assumed in most studies results in a complete disease eradication after disease introduction and subsequent epidemic bouts are only sustained with a constant influx of infected individuals into the population by migration. Our model validation shows the interaction of immune boosting and waning in highly exposed adults better explains observed dengue epidemic dynamics than models assuming transient immunity or lifelong immunity in highly immune adult populations. Moreover, we show that ZIKV exposure modulates dengue immunity and create further delays between dengue epidemics. These findings suggest boosting and waning in highly immune individuals contributes to shaping epidemic dynamics and moreover, our study may inform vaccine strategies to maintain immunity over the life-course.
Additional authors: Jose Victor Zambrana, Sustainable Sciences Institute, Managua, 14007, Nicaragua, Department of Epidemiology, School of Public Health, University of Michigan, Ann Arbor, MI, 48109-2029, USA; Guillermina Kuan, Sustainable Sciences Institute, Managua, 14007, Nicaragua, Centro de Salud Sócrates Flores Vivas, Ministry of Health, Managua, 12014, Nicaragua; Angel Balmaseda, Sustainable Sciences Institute, Managua, 14007, Nicaragua, Laboratorio Nacional de Virología, Centro Nacional de Diagnóstico y Referencia, Ministry of Health, Managua, 16064, Nicaragua; Nery Sanchez, Sustainable Sciences Institute, Managua, 14007, Nicaragua; Sergio Ojeda, Sustainable Sciences Institute, Managua, 14007, Nicaragua; Aubree Gordon, Department of Epidemiology, School of Public Health, University of Michigan, Ann Arbor, MI, 48109-2029, USA; Eva Harris, Division of Infectious Diseases and Vaccinology, School of Public Health, University of California, Berkeley, Berkeley, CA, 94720-3370, USA; Leah Katzelnick, Viral Epidemiology and Immunity Unit, Laboratory of Infectious Diseases, National Institute of Allergy and Infectious Diseases

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Derdei M. Bichara

California State University, Fullerton (Mathematics)

"Effects of Heterogeneity in a Class of Bio-systems"

The role of heterogeneity in populations has long been recognized as a driving force in the spread of infectious diseases. Indeed, populations differ in their propensity to transmit or acquire infectious agents in terms of activities, socio-economic or genetic groups. Oftentimes, mathematical models in population dynamics that incorporate such heterogeneities use groups or classes as units and networks to describe the interactions between these units of the model. For many models that describe such phenomena, the complete global behavior of these systems have been open questions. In this talk, I provide a complete characterization of the some these problems.

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Paul Hurtado

University of Nevada-Reno (Mathematics & Statistics)

"Finding Reproduction Numbers for ODE Models of Arbitrary Finite Dimension Using The Generalized Linear Chain Trick"

Reproduction numbers, like the basic reproduction number R0, play an important role in the analysis and application of dynamic models of contagion spread (and parallels exist elsewhere, e.g., in multispecies ecological models). One difficulty in deriving these quantities is that they must be computed on a model-by-model basis, since it is typically impractical to obtain general reproduction number expressions applicable to a family of related models, especially if these models are of different dimensions (i.e., differing numbers of state variables). For example, this is typically the case for SIR-type infectious disease models derived using the classical linear chain trick (LCT). In this talk, I will provide an overview of how to find general reproduction number expressions for such model families using the next generation operator approach in conjunction with the generalized linear chain trick (GLCT). This shows how the GLCT enables modelers to draw insights from these results by leveraging theory and intuition from continuous time Markov chains (CTMCs) and their absorption time distributions (i.e., phase-type probability distributions). I will show an example application of this technique to find reproduction numbers for a family of generalized SEIRS models with an arbitrary number of state variables. These results highlight the utility of the GLCT for the derivation and analysis of mean field ODE models, especially when used in conjunction with theory from CTMCs and their associated phase-type distributions.
Additional authors: Cameron Richards, University of Nevada-Reno

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Zhuolin Qu

University of Texas at San Antonio (Department of Mathematics)

"Multistage Spatial Model for Informing Release of Wolbachia-Infected Mosquitoes as Disease Control"

Wolbachia is a natural bacterium that can infect Aedes mosquitoes and block the transmission of mosquito-borne diseases, including dengue fever, Zika, and chikungunya. Field trials have been conducted worldwide to suppress local epidemics. We present a new partial differential equation model for the spread of Wolbachia infection in mosquitoes. The model accounts for both the complex Wolbachia vertical transmission cycle and detailed life stages in the mosquitoes, and it also incorporates the spatial heterogeneity created by mosquito dispersion in the two-dimensional release domain. Field trials and previous modeling studies have shown that the fraction of infection among mosquitoes must exceed a threshold level for the infection to persist. We use the spatial model to identify a threshold condition for having a self-sustainable Wolbachia infection in the field. When above this threshold, the model gives rise to a spatial wave of Wolbachia infection. We quantify how the threshold condition and invasion velocity depend on the diffusion process and other model parameters, and we study different intervention scenarios to inform the efficient releases.
Additional authors: Tong Wu, Department of Mathematics, University of Texas at San Antonio

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