Epidemics

Table of Contents

1. Frontmatter

2. Chapter 1. Introduction

   Ottar Bjørnstad

   Abstract

   The use of mathematical models to understand infectious disease dynamics has a very rich history in epidemiology. Kermack and McKendrick (1927) is the seminal paper that introduced the equations for the general Susceptible–Infected–Removed model and showed how a set of restrictive assumptions lead to the standard SIR model of ordinary differential equations.

3. Time

   1. Frontmatter

   2. Chapter 2. SIR

      Ottar Bjørnstad

      Abstract

      The following 10 chapters are devoted to the study of patterns of infection over time and age. The current chapter introduces the basics of compartmental modeling of transmission dynamics. This is followed by a chapter with in-depth discussion of the reproduction number, R ₀, which is the most important quantity for understanding epidemics of infectious agents. The subsequent chapters detail the importance of age structure and seasonality in shaping epidemics and pandemics as well as several important time series methods for characterizing and understanding temporal recurrence patterns of infection. The last two chapters explore how ideas from dynamical systems theory can help explain several very curious aspects of the waxing and waning of infection through time.

   3. Chapter 3. R 0

      Ottar Bjørnstad

      Abstract

      For directly transmitted pathogens, R ₀ is per definition the expected number of secondary cases that arise from a typical infectious index case in a completely susceptible host population. R ₀ plays a critical role for a number of aspects of disease dynamics and is therefore the focus of much study in historical and contemporary infectious disease dynamics (Heesterbeek & Dietz, 1996).

   4. Chapter 4. FoI and Age-Dependence

      Ottar Bjørnstad

   5. Chapter 5. The Catalytic Model

      Ottar Bjørnstad

      Abstract

      While immunobiology is not the focus of this text, some basic underpinnings are useful for motivating the so-called catalytic model to study how immunity may build up with age, how age-specific heterogeneities may affect this, and how we can use immune data to back-calculate key dynamic quantities. There are two main branches of the adaptive immune system—the part of the immune system that helps to build a repertoire for protection against reinfection.

   6. Chapter 6. Seasonality

      Ottar Bjørnstad

      Abstract

      Host behavior and environmental factors influence disease dynamics in a variety of ways through affecting the pathogen such as the survival of infective stages outside the host and via host demographies from changing birth rates, carrying capacitities, social organization, etc. Sometimes such influences have relatively subtle consequences (e.g., slight changes in R ₀) as is likely the effect of absolute humidity on influenza transmission (Lowen et al., 2007; Bjørnstad & Viboud, 2016).

   7. Chapter 7. Time Series Analysis

      Ottar Bjørnstad

      Abstract

      Analysis of epidemic time series is a large endeavor because of the richness of dynamical patterns and a plentitude of historical data (Rohani & King, 2010). A wide range of tools are used, some of which are borrowed from mainstream statistics and other of which are custom-made. The classic “mainstream” methods belong to two categories: the so-called time-domain and frequency-domain methods. The autocorrelation function and ARIMA models belong to the former class and spectral analysis, and the periodogram belongs to the latter. Hybrid time/frequency methods have become increasingly prominent in the form of wavelet analysis because it allows the study of changes in disease dynamics through time (Grenfell et al., 2001).

   8. Chapter 8. TSIR

      Ottar Bjørnstad

      Abstract

      There are many strategies for estimating the parameters of dynamic models from time series data. They differ conceptually in the way they handle demographic and environmental stochasticity (sometimes referred to jointly as “process error”), observation error, and partial (missing) observation. The strategies also often vary by whether the underlying dynamics is thought to be best approximated in continuous time (differential models) or discrete time (difference models).

   9. Chapter 9. Stochastics

      Ottar Bjørnstad

      Abstract

      When fitting mechanistic models to data, we have to consider carefully the relationship between the nature of the data versus the nature of the model state variables. For example, when working with continuous-time S(E)IR models, it is important to keep in mind that incidence is not prevalence. The results from integrating the compartmental models represent prevalence over time (i.e., the number or fraction of a population that is infected). Most public health data, in contrast, tracks incidence—the number of new cases in any given time interval. We thus need to do something more than trying to match simulated prevalence with observed incidence. We therefore start with a toy example in which the simulated data actually represents prevalence.

   10. Chapter 10. Stability and Resonant Periodicity

       Ottar Bjørnstad

       Abstract

       The rabies virus infects a wide range of mammalian carnivores across the world with spillovers to non-competent hosts including humans. While not always classified as rabies, there are a wide range of related lyssaviruses of bats that can also spill over to humans. These viruses are transmitted from saliva during aggressive encounters involving biting.

   11. Chapter 11. Exotica

       Ottar Bjørnstad

       Abstract

       Chapter 10 discussed how a linear approximation to the perennially nonlinear dynamics of infectious disease can provide important insights on invasion, stability, and resonant periodicity. As remarked by Nisbet and Gurney (1982) more generally, linear approximation can often provide remarkably useful insights for nonlinear ecological systems as long as they are not too nonlinear.

4. Space

   1. Frontmatter

   2. Chapter 12. Spatial Dynamics

      Ottar Bjørnstad

      Abstract

      Space adds an additional axis to the richness of infectious disease dynamics. For example, Gog et al. (2014) detailed the diffusive nature of the spread of influenza A/H1N1pdv and Lau et al. (2017) characterized the geographic spread of the West African 2014–2015 Ebola outbreak. Walsh et al. (2005) calculated that Ebola was spreading through gorilla and chimpanzee populations at 50 km/year. Moreover, Grenfell and Harwood (1997) and Keeling et al. (2004) outlined how spatial spread may permit long-term persistence through metapopulation dynamics.

   3. Chapter 13. Spatial and Spatiotemporal Patterns

      Ottar Bjørnstad

      Abstract

      Spatial and spatiotemporal data analysis is of great importance in disease dynamics for a number of reasons such as looking for space-time clustering, hotspot detection, characterizing invasion waves, and quantifying spatial synchrony. Spatial synchrony—the level of correlation in outbreak dynamics at different locations—is of particular significance to acute immunizing infections, because asynchrony may permit regional persistence of infections despite local chains-of-transmission breaking during post-epidemic troughs (Keeling et al., 2004, see Sect. 15.7). Conversely, spatial synchrony can exacerbate the economic and public health burden because the resulting regionalized outbreaks can overwhelm logistical capabilities as was evident in the early part of the 2013–2014 West African Ebola outbreak and the 2020–2021 SARS-CoV-2 pandemic.

   4. Chapter 14. Transmission on Networks

      Ottar Bjørnstad

      Abstract

      Following the initial exploration of the simplest SIR model in Chap. 2, various chapters have explored a number of elaborations that are important in order to understand many aspects of infectious disease dynamics such as age-structure, seasonality, and more complex compartmental flows among individuals within a community. In addition to such heterogeneities which can be categorized according to covariates/cofactors (age, month, hospital versus community, etc.), there is often substantial variation that defies such classification. Lloyd-Smith et al. (2005), for example, identified significant heterogeneities from superspreading events during the 2003 SARS outbreak. Woolhouse et al. (1997) suggested a 80/20 rule-of-thumb: for many infections a core of 20% of infected accounts for 80% of onwards transmission.

   5. Chapter 15. Invasion and Eradication

      Ottar Bjørnstad

      Abstract

      Pathogens invade new host niches all the time. The global invasion of the human niche by SARS-CoV-2 during the 2020–22 pandemic is the most recent example, but cross-species transmission is ubiquitous. In 2009 Influenza A/H1N1pdm09 emerged and spread globally most likely after a triple recombination of human, avian, and porcine viral segments (Smith et al., 2009a). The HIV-1 pandemic started in the mid-twentieth century probably from bushmeat spillover of chimpanzee simian immunodeficiency virus, which itself is thought to have originated from spillovers from other primates, to go global in the 1970s (Hemelaar, 2012). Cross-species transmission is not just an issue of zoonotic spillover or anthropogenic spillback, it is equally important as spillover among animal species.

5. Miscellany

   1. Frontmatter

   2. Chapter 16. Parasitoids

      Ottar Bjørnstad

      Abstract

      This third part visits on a number of topics that are somewhat tangential to the main narrative of the monograph but that I have found useful for thinking on and analyzing data pertaining to infectious spread. The current chapter outlines how many of the ideas with regards to dynamics, persistence, and control carries over to other host/enemy systems of concern. Chapter 17 visits on multivariate methods to better characterize the in-host interactions among pathogens and the immune system that are ultimately responsible for shaping onwards transmission and epidemic flows. Chapter 18 is a brief sampler of how infectious disease processes in space and time generally lead to autocorrelated data that breach the classic statistical adage of “identically distributed, independent data” but for which a battery of modern methods can provide correct inference and additional insights.

   3. Chapter 17. Quantifying In-Host Patterns

      Ottar Bjørnstad

      Abstract

      This chapter is somewhat tangential to the main text but it does loosely loop back to the discussion in Sect. 1.2 on how patterns of in-host persistence are important determinants of population-level dynamics. In-host dynamics results from replication rates of pathogens as molded by the innate and adaptive branches of the host immune system. For example, using the TSIR as a tool for understanding plasmodium replication rates, Metcalf et al. (2011b, see also Sect. 8.9) documented a strong dose-response effect whereby the innate branch seemingly is able to slow the growth from low inocula but not subsequent anemia in the infected mice. Kamiya et al. (2020) provide further discussion of such dose-response effects and consequences for onwards transmission.

   4. Chapter 18. Non-Independent Data

      Ottar Bjørnstad

      Abstract

      Many infectious disease experiments result in non-independent data because of spatial autocorrelation across fields (such as discussed in Chap. 13), repeated measures on experimental animals (such as the in-host Plasmodium data discussed in Sect. 8.9), or other sources of correlated experimental responses among experimental units (such as the possibility of correlated infection fates among the rabbit littermates discussed in Sect. 5.2). Statistical methods that assume independence of observations are not strictly valid and/or fully effective on such data (e.g., Legendre, 1993; Keitt et al., 2002). Mixed-effects models and generalized linear mix-effects models (GLMMs) have been/are being developed to optimize the analysis of such data (Pinheiro & Bates, 2006; Bolker et al., 2009; Bates et al., 2015).

6. Backmatter