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## Über dieses Buch

The contributions by epidemic modeling experts describe how mathematical models and statistical forecasting are created to capture the most important aspects of an emerging epidemic.Readers will discover a broad range of approaches to address questions, such as

Can we control Ebola via ring vaccination strategies?

How quickly should we detect Ebola cases to ensure epidemic control?

What is the likelihood that an Ebola epidemic in West Africa leads to secondary outbreaks in other parts of the world? When does it matter to incorporate the role of disease-induced mortality on epidemic models? What is the role of behavior changes on Ebola dynamics? How can we better understand the control of cholera or Ebola using optimal control theory?How should a population be structured in order to mimic the transmission dynamics of diseases such as chlamydia, Ebola, or cholera?How can we objectively determine the end of an epidemic?How can we use metapopulation models to understand the role of movement restrictions and migration patterns on the spread of infectious diseases?

How can we capture the impact of household transmission using compartmental epidemic models?

How could behavior-dependent vaccination affect the dynamical outcomes of epidemic models?

The derivation and analysis of the mathematical models addressing these questions provides a wide-ranging overview of the new approaches being created to better forecast and mitigate emerging epidemics.

This book will be of interest to researchers in the field of mathematical epidemiology, as well as public health workers.

## Inhaltsverzeichnis

### A Reality of Its Own

In September 2014, the CDC published a supplement to the MMWR that announced a worst-case estimate of 1.4 million cases of Ebola in Liberia and Sierra Leone (Meltzer et al., MMWR 63(3):1–14, 2014, [1]). The epidemic was then 6 months old and 8,000 cases had been reported. It was estimated that at least 2.5 times that many had occurred, and the 1.4 million was based on the then estimated incidence of 21,000 cases in 6 months. The method was mathematically simple—based primarily on mean incubation period, contact index, and specific sets of patient circumstances—but the details were complicated.
Richard Rothenberg

### Modeling the Impact of Behavior Change on the Spread of Ebola

We create a compartmental mathematical model to analyze the role of behavior change in slowing the spread of the Ebola virus disease (EVD) in the 2014–2015 Western Africa epidemic. Our model incorporates behavior change, modeled as decreased contact rates between susceptible and infectious individuals, the prevention of traditional funerals, and/or increased access to medical facilities. We derived the basic reproductive number for the model, and approximated the parameter values for the spread of the EVD in Monrovia. We used sensitivity analysis to quantify the relative importance of the timing, and magnitude, of the population reducing their contact rates, avoiding the traditional burial practices, and having access to medical treatment facilities. We found that reducing the number of contacts made by infectious individuals in the general population is the most effective intervention method for mitigating an EVD epidemic. While healthcare interventions delayed the onset of the epidemic, healthcare alone is insufficient to stop the epidemic in the model.
Jessica R. Conrad, Ling Xue, Jeremy Dewar, James M. Hyman

### A Model for Coupled Outbreaks Contained by Behavior Change

Large epidemics such as the recent Ebola crisis in West Africa occur when local efforts to contain outbreaks fail to overcome the probabilistic onward transmission to new locations. As a result, there may be large differences in total epidemic size from similar initial conditions. This work seeks to determine the extent to which the effects of behavior changes and metapopulation coupling on epidemic size can be characterized. While mathematical models have been developed to study local containment by social distancing, intervention and other behavior changes, their connection to larger-scale transmission is relatively underdeveloped. We make use of the assumption that behavior changes limit local transmission before susceptible depletion to develop a time-varying birth-death process capturing the dynamic decrease of the transmission rate associated with behavior changes. We derive an expression for the mean outbreak size of this model and show that the distribution of outbreak sizes is approximately geometric. This allows a probabilistic extension whereby infected individuals may initiate new outbreaks. From this model we characterize the overall epidemic size as a function of the behavior change rate and the probability that an infected individual starts a new outbreak. We find good agreement between the analytical results and stochastic simulations leading to novel findings including critical learning rates that demarcate large and small epidemic sizes.
John M. Drake, Andrew W. Park

### Real-Time Assessment of the International Spreading Risk Associated with the 2014 West African Ebola Outbreak

The 2014 West African Ebola Outbreak is the largest Ebola virus disease (EVD) epidemic ever recorded, not only in number of cases but also in geographical extent. Unlike previous EVD outbreaks, the large number of cases observed in major cities with international airports raised the concern about the possibility of exportation of the infection in countries around the world. Starting in July 2014, we used the Global Epidemic and Mobility model to provide a real-time assessment of the potential international spread of the EVD epidemic. We modeled the unfolding of the outbreak in the most affected countries, considered different scenarios reflecting changes in the disease dynamic, and provided estimates for the probability of observing imported cases around the world for 220 countries. The model went through successive calibrations as more surveillance data were available, providing projections extending from a few weeks to several months. The results show that along the entire course of the epidemic the probability of observing cases outside of Africa was small, but not negligible, from September to November 2014. The inflection point of the epidemic occurred in late September and early October 2014 with a consistent longitudinal decrease in new cases, thus averting the status quo epidemic growth that could have seen hundreds of exported cases at the global scale in the following months.
Ana Pastore-Piontti, Qian Zhang, Marcelo F. C. Gomes, Luca Rossi, Chiara Poletto, Vittoria Colizza, Dennis L. Chao, Ira M. Longini, M. Elizabeth Halloran, Alessandro Vespignani

### Modeling the Case of Early Detection of Ebola Virus Disease

The most recent Ebola outbreak in West Africa highlighted critical weaknesses in the medical infrastructure of the affected countries, including effective diagnostics tools, sufficient isolation wards, and enough medical personnel. Here, we develop and analyze a mathematical model to assess the impact of early diagnosis of pre-symptomatic individuals on the transmission dynamics of Ebola virus disease in West Africa. Our findings highlight the importance of implementing integrated control measures of early diagnosis and isolation. The mathematical analysis shows a threshold where early diagnosis of pre-symptomatic individuals, combined with a sufficient level of effective isolation, can lead to an epidemic control of Ebola virus disease.
Diego Chowell, Muntaser Safan, Carlos Castillo-Chavez

### Modeling Ring-Vaccination Strategies to Control Ebola Virus Disease Epidemics

The 2013-15 Ebola epidemic that primarily affected Guinea, Sierra Leone and Liberia has become the most devastating Ebola epidemic in history [1]. This unprecedented epidemic appears to have stemmed from a single spillover event in South Guinea in December 2013 and rapidly spread to neighboring Sierra Leone and Guinea in a matter of weeks. Here we employ a network-based transmission model to evaluate the potential impact of reactive ring-vaccination strategies in the context of the Ebola epidemic in West Africa. We model ring-based vaccination strategies that incorporate the radius of contacts that are vaccinated for each infectious individual, the time elapsed from individual infectiousness to vaccinating susceptible and exposed contacts, and the number of available vaccine doses. Our baseline spatial transmission model in which the ring vaccination strategy is investigated has been previously shown to capture Ebola-like epidemics characterized by an initial phase of sub-exponential epidemic growth. Here we also extend this baseline model to account for heterogeneous community transmission rates that may be defined as a scalable function of the distance between an infectious individual and each member of that individual’s community. Overall, our findings indicate that reactive ring-vaccination strategies can effectively mitigate established Ebola epidemics. Importantly, we studied scenarios with varying number of weeks elapsed between the onset of symptoms and the day contacts are vaccinated and found that it is still beneficial to vaccinate contacts after the infectious period has elapsed. Our results indicate that while it is beneficial to vaccinate members of the community, the probability of extinction is not very sensitive to which contacts in the community are vaccinated unless transmission varies very steeply on the network distance between individuals. Both of these observations underscore the fact that vaccination can be effective by reducing transmission at the community level.
Gerardo Chowell, Maria Kiskowski

### Evaluating the Number of Sickbeds During Ebola Epidemics Using Optimal Control Theory

Optimal control (OC) theory is a powerful tool to guide the design and implementation of control intervention strategies against epidemics. This technique defined control measures under a predetermined objective while minimizing the costs associated with the implementation of the control strategy. Here we use optimal control and epidemic modeling to explore the uncertainty in hospital bed capacity that would be needed to control an Ebola epidemic under different initial conditions, variation in the basic reproduction number, and associated costs to implement control measures. In particular, we focus on assessing the impact of effective isolation of infectious individuals in the health care setting because one key factor that facilitated the development of the Ebola epidemic in West Africa was the lack of public health surveillance systems to detect new outbreaks and the healthcare capacity that is needed to enforce infection control practices.
Eunok Jung, Jonggul Lee, Gerardo Chowell

### Inverse Problems and Ebola Virus Disease Using an Age of Infection Model

Parameter estimation problems in ordinary and partial differential equations constitute a large class of models described by ill-posed operator equations. A considerable number of such problems come from epidemiology and infectious disease modeling, with Ebola Virus Disease (EVD) being a very important example. While it is not difficult to find a solution of an SEIJCR ODE constrained least squares problem, this problem is extremely unstable and a number of different parameter combinations produce essentially the same case curve. This is a serious obstacle in the study of the Ebola virus epidemics, since reliable approximations of system parameters are important for the proper assessment of existing control measures as well as for the forward projections aimed at testing a variety of contact tracing policies. In this paper, we attempt a stable estimation of system parameters with the use of iterative regularization along with a special algorithm for computing initial values. The numerical study is illustrated by data fitting and forward projections for the most recent EVD outbreak in Sierra Leone and Liberia.
Alexandra Smirnova, Linda DeCamp, Hui Liu

### Assessing the Efficiency of Movement Restriction as a Control Strategy of Ebola

We formulate a two-patch mathematical model for Ebola Virus Disease dynamics in order to evaluate the effectiveness of travel restriction (cordons sanitaires), mandatory movement restrictions between communities while exploring their role on disease dynamics and final epidemic size. Simulations show that strict restrictions in movement between high and low risk areas of closely linked communities may have a deleterious impact on the overall levels of infection in the total population.
Baltazar Espinoza, Victor Moreno, Derdei Bichara, Carlos Castillo-Chavez

### Patch Models of EVD Transmission Dynamics

Mathematical models have the potential to be useful to forecast the course of epidemics. In this chapter, a family of logistic patch models are preliminarily evaluated for use in disease modeling and forecasting. Here we also derive the logistic equation in an infectious disease transmission context based on population behavior and used it for forecasting the trajectories of the 2013–2015 Ebola epidemic in West Africa. The logistic model is then extended to include spatial population heterogeneity by using multi-patch models that incorporate migration between patches and logistic growth within each patch. Each model’s ability to forecast epidemic data was assessed by comparing model forecasting error, parameter distributions and parameter confidence intervals as functions of the number of data points used to calibrate the models. The patch models show an improvement over the logistic model in short-term forecasting, but naturally require the estimation of more parameters from limited data.
Bruce Pell, Javier Baez, Tin Phan, Daozhou Gao, Gerardo Chowell, Yang Kuang

### From Bee Species Aggregation to Models of Disease Avoidance: The Ben-Hur effect

The movie Ben-Hur highlights the dynamics of contagion associated with leprosy, a pattern of forced aggregation driven by the emergence of symptoms and the fear of contagion. The 2014 Ebola outbreaks reaffirmed the dynamics of redistribution among symptomatic and asymptomatic or non-infected individuals as a way to avoid contagion. In this manuscript, we explore the establishment of clusters of infection via density-dependence avoidance (diffusive instability). We illustrate this possibility in two ways: using a phenomenological driven model where disease incidence is assumed to be a decreasing function of the size of the symptomatic population and with a model that accounts for the deliberate movement of individuals in response to a gradient of symptomatic infectious individuals. The results in this manuscript are preliminary but indicative of the role that behavior, here modeled in crude simplistic ways, may have on disease dynamics, particularly on the spatial redistribution of epidemiological classes.
K. E. Yong, E. Díaz Herrera, C. Castillo-Chavez

### Designing Public Health Policies to Mitigate the Adverse Consequences of Rural-Urban Migration via Meta-Population Modeling

This study extends the model considered in [3] (Chap. 8 in this volume) by incorporating spatially explicit migration of individuals. A three-patch meta-population model is used to explore vaccination strategies for a vaccine-preventable disease. Spatial movements of individuals between patches are mainly migration from rural to urban and peri-urban for greater economic opportunities. Stochastic simulations evaluate the effects of alternative vaccination strategies on preventing disease outbreaks, examine the distribution of possible outcomes, and compare the likelihood of outbreak mitigation and prevention across immunization policies. Two types of vaccine coverage are compared. One is homogeneous coverage, in which relevant sub-populations receive vaccination with equal probability; and the other is heterogeneous coverage, in which sub-populations can receive vaccination with different probabilities. Results suggest that when sub-populations differ in density (which may affect contact rates), heterogeneous vaccination coverage among migrants is most effective according to measures such as final epidemic size, peak size, number of vaccine doses needed to prevent outbreaks, and likelihood of containing an outbreak. This suggests that public health efforts to mitigate vaccine-preventable diseases must consider migration.
Zhilan Feng, Yiqiang Zheng, Nancy Hernandez-Ceron, Henry Zhao

### Age of Infection Epidemic Models

The age of infection model, first introduced by Kermack and McKendrick in 1927, is a general structure for compartmental epidemic models, including models with heterogeneous mixing. It is possible to estimate the basic reproduction number if the initial exponential growth rate and the infectivity as a function of time since being infected are known, and this is also possible for models with heterogeneous mixing.
Fred Brauer

### Optimal Control of Vaccination in an Age-Structured Cholera Model

A cholera model with continuous age structure is given as a system of hyperbolic (first-order) partial differential equations (PDEs) in combination with ordinary differential equations. Asymptomatic infected and susceptibles with partial immunity are included in this epidemiology model with vaccination rate as a control; minimizing the symptomatic infecteds while minimizing the cost of the vaccinations represents the goal. With the method of characteristics and a fixed point argument, the existence of a solution to our nonlinear state system is achieved. The representation and existence of a unique optimal control are derived. The steps to justify the optimal control results for such a system with first order PDEs are given. Numerical results illustrate the effect of age structure on optimal vaccination rates.
K. Renee Fister, Holly Gaff, Suzanne Lenhart, Eric Numfor, Elsa Schaefer, Jin Wang

### A Multi-risk Model for Understanding the Spread of Chlamydia

Chlamydia trachomatis, CT, infection is the most frequently reported sexually transmitted infection in the United States. To better understand the recent increase in disease prevalence, and help guide in mitigation efforts, we created and analyzed a multi-risk model for the spread of chlamydia in the heterosexual community. The model incorporates the heterogeneous mixing between men and women with different number of partners and the parameters are defined to approximate the disease transmission in the 15–25 year-old New Orleans African American community. We use sensitivity analysis to assess the relative impact of different levels of screening interventions and behavior changes on the basic reproduction number. Our results quantify, and validate, the impact that reducing the probability of transmission per sexual contact, such as using prophylactic condoms, can have on CT prevalence.
Asma Azizi, Ling Xue, James M. Hyman

### The 1997 Measles Outbreak in Metropolitan São Paulo, Brazil: Strategic Implications of Increasing Urbanization

Background: Despite a routine two-dose measles vaccination program, mass campaigns in 1987 and 1992 and low subsequent incidence, São Paulo experienced an outbreak between May and October of 1997 with over 42,000 confirmed cases, mostly young adults, and 42 measles-associated deaths, mostly infants. To eliminate measles, the Pan American Health Organization (PAHO) recommended supplementing routine childhood vaccination (keep-up) via mass campaigns, initially to reduce (catch-up) and periodically to maintain (follow-up) susceptible numbers below the epidemic threshold. Methods: To determine if a follow-up campaign during 1996, when due in São Paulo State, might have prevented or mitigated this outbreak, we modeled measles in metropolitan São Paulo. We also evaluated the actual impact of emergency outbreak-control efforts and hypothetical impact of vaccinating adolescent and young adult immigrants. Results: A mass campaign targeting children aged 6–59 months reduced cases as much as 77 %, but a follow-up campaign among children aged 1–4 years during 1996 might have been even more effective. Susceptible adolescents would have escaped, however, setting the stage for future outbreaks. Vaccinating people in the immigrant age range mitigated this potential. Conclusions: As the immunity required to prevent outbreaks depends on population density, rural people are less likely to be immune than urban ones the same age. Thus, when there is rural-urban migration, births are not the sole demographic process eroding urban population immunity. Vaccinating immigrants in bus stations, peripheral shantytowns, or sites of employment for unskilled laborers is more efficient than increasing rural immunity.
José Cassio de Moraes, Maria Claudia Corrêa Camargo, Maria Lúcia Rocha de Mello, Bradley S. Hersh, John W. Glasser

### Methods to Determine the End of an Infectious Disease Epidemic: A Short Review

Deciding the end of an epidemic is frequently associated with forthcoming changes in infectious disease control activities, including downgrading alert level in surveillance and restoring healthcare workers’ working shift back to normal. Despite the practical importance, there have been little epidemiological and laboratory methods that were proposed to determine the end of an epidemic. This short review was aimed to systematically discuss methodological principles of a small number of existing techniques and understand their advantages and disadvantages. Existing epidemiological methods have been mostly limited to a single-and-brief exposure setting, while the application to human-to-human transmissible disease epidemic with stochastic dependence structure in the observed case data has remained to be a statistical challenge. In veterinary applications, a large-scale sampling for laboratory testing has been commonly adapted to substantiate a freedom from disease, but such study has only accounted for binomial sampling process in estimating the error probability of elimination. Surveillance and mathematical modeling are two complementary instruments in the toolbox of epidemiologists. Combining their strengths would be highly beneficial to better define the end of an epidemic.
Hiroshi Nishiura

### Statistical Considerations in Infectious Disease Randomized Controlled Trials

Randomized controlled trials (RCT) provide the highest standard of evidence available for assessing treatment efficacy. Causal inferences are enabled and effects may be directly attributed to a treatment. The nature of infectious disease presents challenges to the design, conduct, and analysis of a trial for a new drug or therapy. Many of these challenges are statistical in nature and can be addressed with modern methods for planning and analyzing RCT data. In this chapter, some of these challenges are described and reviewed. Modern statistical modeling methods for analysis of correlated data are covered. Some challenges with sample size determination are outlined and updated methods for data monitoring, interim, and subgroup analyses detailed. Also, discernment is made between multisite and cluster randomized trials. Recommendations for best practices are included.
Matthew J. Hayat

### Epidemic Models With and Without Mortality: When Does It Matter?

We use an agent-based computer simulation designed to model the spread of the 1918 influenza pandemic to address the question of whether, and if so, when disease-related mortality should be included in an epidemic model. Simulation outcomes from identical models that differ only in the inclusion or exclusion of disease-related mortality are compared. Results suggest that unless mortality is very high (above a case fatality rate of about 18 % for influenza), mortality has a minimal impact on simulation outcomes. High levels of mortality, however, lower the percentage infected at the epidemic peak and reduce the overall number of cases because epidemic chains are shortened overall, and so a smaller proportion of the population becomes infected. Analyses also indicate that high levels of mortality can increase the chance of oscillations in disease incidence. The decision about whether to include disease-related mortality in a model should, however, take into account the fact that diseases such as influenza, that sicken a high proportion of a population, may nonetheless lead to high numbers of deaths. These deaths can affect a real population’s perception of and response to an epidemic, even when objective measures suggest the impact of mortality on epidemic outcomes is relatively low. Thus, careful attention should be paid to the possibility of such responses when developing epidemic control strategies.
Lisa Sattenspiel, Erin Miller, Jessica Dimka, Carolyn Orbann, Amy Warren

### Capturing Household Transmission in Compartmental Models of Infectious Disease

Social distancing policies may mitigate transmission of infectious disease by shifting individuals time spent in public into household environments. However, the efficacy of such a policy depends on the transmission differential between public and household environments. We extend the standard compartmental model of infectious disease with heterogeneous mixing to explicitly account for the health state of households. Our model highlights the fact that only households with an infectious individual pose a transmission risk to other household members. Moreover, susceptible households become infectious at a rate that depends on household size and the health status of the household members. We demonstrate our model by simulating an epidemic similar to the A/H1N1 2009 outbreak using empirical mixing patterns derived from time-use data in the United States. We find that household transmission accounts for 12–23 % of total cases. These results suggest that while social distancing policies encourage individuals to spend more time at home, the reduction of time in public improves public health outcomes on balance.
Jude Bayham, Eli P. Fenichel

### Bistable Endemic States in a Susceptible-Infectious-Susceptible Model with Behavior-Dependent Vaccination

Several new vaccines have the characteristic of being “imperfect” that is their protection wanes over time and supplies only partial protection from infection. On the other hand recent research has shown that the agents’ behavioral responses have the potential to dramatically affect the dynamics and control of infections. In this paper we investigate, for a simple susceptible-infective-susceptible (SIS) infection, the dynamic interplay between human behavior, in the form of an increasing prevalence-dependent vaccine uptake function, and vaccine imperfections. The mathematical analysis of the ensuing SISV model shows a complexly articulated bifurcation structure. First, the inclusion of the simplest possible hypothesis about vaccination behavior is capable to trigger, in appropriate windows of the key parameters, phenomena of multistability of endemic states. Second, as far as the stability of the disease-free equilibrium is concerned, the model preserves the backward bifurcation which is characteristic of SIS-type infections controlled by imperfect vaccines.
Alberto d’Onofrio, Piero Manfredi

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