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# Modeling the Impact of Environment on Infectious Diseases

## Summary and Keywords

The introduction of pasteurization, antibiotics, and vaccinations, as well as improved sanitation, hygiene, and education, were critical in reducing the burden of infectious diseases and associated mortality during the 19th and 20th centuries and were driven by an improved understanding of disease transmission. This advance has led to longer average lifespans and the expectation that, at least in the developed world, infectious diseases were a problem of the past. Unfortunately this is not the case; infectious diseases still have a significant impact on morbidity and mortality worldwide. Moreover, the world is witnessing the emergence of new pathogens, the reemergence of old ones, and the spread of antibiotic resistance. Furthermore, effective control of infectious diseases is challenged by many factors, including natural disasters, extreme weather, poverty, international trade and travel, mass and seasonal migration, rural–urban encroachment, human demographics and behavior, deforestation and replacement with farming, and climate change.

The importance of environmental factors as drivers of disease has been hypothesized since ancient times; and until the late 19th century, miasma theory (i.e., the belief that diseases were caused by evil exhalations from unhealthy environments originating from decaying organic matter) was a dominant scientific paradigm. This thinking changed with the microbiology era, when scientists correctly identified microscopic living organisms as the pathogenic agents and developed evidence for transmission routes. Still, many complex patterns of diseases cannot be explained by the microbiological argument alone, and it is becoming increasingly clear that an understanding of the ecology of the pathogen, host, and potential vectors is required.

There is increasing evidence that the environment, including climate, can affect pathogen abundance, survival, and virulence, as well as host susceptibility to infection. Measuring and predicting the impact of the environment on infectious diseases, however, can be extremely challenging. Mathematical modeling is a powerful tool to elucidate the mechanisms linking environmental factors and infectious diseases, and to disentangle their individual effects. A common mathematical approach used in epidemiology consists in partitioning the population of interest into relevant epidemiological compartments, typically individuals unexposed to the disease (susceptible), infected individuals, and individuals who have cleared the infection and become immune (recovered). The typical task is to model the transitions from one compartment to another and to estimate how these populations change in time. There are different ways to incorporate the impact of the environment into this class of models. Two interesting examples are water-borne diseases and vector-borne diseases. For water-borne diseases, the environment can be represented by an additional compartment describing the dynamics of the pathogen population in the environment—for example, by modeling the concentration of bacteria in a water reservoir (with potential dependence on temperature, pH, etc.). For vector-borne diseases, the impact of the environment can be incorporated by using explicit relationships between temperature and key vector parameters (such as mortality, developmental rates, biting rate, as well as the time required for the development of the pathogen in the vector).

Despite the tremendous advancements, understanding and mapping the impact of the environment on infectious diseases is still a work in progress. Some fundamental aspects, for instance, the impact of biodiversity on disease prevalence, are still a matter of (occasionally fierce) debate. There are other important challenges ahead for the research exploring the potential connections between infectious diseases and the environment. Examples of these challenges are studying the evolution of pathogens in response to climate and other environmental changes; disentangling multiple transmission pathways and the associated temporal lags; developing quantitative frameworks to study the potential effect on infectious diseases due to anthropogenic climate change; and investigating the effect of seasonality. Ultimately, there is an increasing need to develop models for a truly “One Health” approach, that is, an integrated, holistic approach to understand intersections between disease dynamics, environmental drivers, economic systems, and veterinary, ecological, and public health responses.

# Introduction

During the 19th and 20th centuries, the burden of infectious diseases and their associated mortality greatly decreased, especially in developed countries (Brachman, 2003; Dye, 2014; Satcher, 1995). This was the natural consequence of the introduction of pasteurization, improved sanitation and hygiene, better education, and vaccination campaigns, all followed by the antibiotics era. This headway led to the perception that infectious diseases would progressively become a marginal problem for humanity (Brachman, 2003). Despite the tremendous advances of medicine and public health, infectious diseases still have a significant impact on morbidity and mortality in the 21st century globally. For instance, according to one estimation, in 2011 infectious diseases in the United Kingdom, a highly developed country, accounted for 7% of deaths and annual costs of £30bn (Davies, 2013).

Infectious diseases have not declined as expected. There are several reasons (Dye, 2014), including the urgent problems of the emergence and reemergence of new pathogens (Morse, 1995) and the spread of antibiotic resistance (CDC, 2015). Other important factors, however, are challenging the control of established, emerging and reemerging infectious diseases, including natural disasters, extreme weather, economic changes and poverty, commerce, international travel, and mass and seasonal migration; as well as the displacement of refugees owing to conflict and violence; rural–urban encroachment on human demographics and behavior; technology and industry globalization; breakdown in public health measures and infrastructures; climate change; deforestation and replacement with crop farms; disruption and contamination of water bodies; interaction between infectious and noninfectious diseases; and El Niño southern oscillations. The impact of some of these factors is discussed in Dye (2014); Eisenberg et al. (2007); Jones et al. (2008); Macpherson (2005); Morse (1995); and Patz, Graczyk, Geller, and Vittor (2000).

All these factors demonstrate the strong links, either directly or indirectly, between the environment and infectious diseases (Ayres, Harrison, Maynard, McClellan, & Nichols, 2010). Clearly, the word “environment” in this context has multiple meanings (see the discussion in Johnson et al., 1997). Here, “environment” refers to “the complex of physical, chemical, and biotic factors that act upon an organism or an ecological community and ultimately determine its form and survival” (Britannica, 2001). Of course, the strong interconnections between the environment and social, economic, and cultural factors are also important.

Although our understanding of the routes of transmission and the diverse abilities of pathogens to survive and grow in different environments has improved, there are still many unanswered questions. In particular, mathematical modeling is an important tool to help the scientific, medical, and public health community explore and possibly address these broad questions. Presented here, after a brief background, are some key examples demonstrating the potential of mathematical modeling in understanding the impact of the environment on infectious diseases.

# From Miasma Theory to Advanced Theoretical Frameworks

## Miasma Theory, Contagion Theory, Germ Theory, and Back to Environment

The link between the physical environment and diseases has been postulated since ancient times (Geller, 2001), as is revealed, for instance, by the etymology of some common names for infectious diseases: “malaria” comes from the Italian mala aria meaning bad air, as the disease was thought to be caused by foul air in marshy areas; “influenza,” a word from the Latin influentia, reflected the belief that the disease was caused by astrological and/or atmospheric influences; while the English word, “cold,” is self-explanatory. Other historical sources confirm the long-standing belief of the connection between environment and diseases: Empedocles (490–440 bc), according to Diodorus, managed to make the Sicilian city of Selinunte a healthy town by diverting two rivers so as to drain the local swamp land and improve the quality of the water supply (Stathakou, Stathakou, Damianaki, Toumbis-Ioannou, & Stavrianeas, 2009). This is thought to be one of the first known public health projects in history and implies attribution of a disease (malaria) to a source (swamp water), although it took more than 2000 years to identify the vector. Similarly, Hippocrates (460–370 bc) recommended to his medical followers that they should study the effects of the seasons, the hot and cold wind, and water on health (Hippocrates, trans. 1868).

Diseases were supposedly caused by evil exhalations from unhealthy environments that originated from decaying organic matter. This is the essence of “miasma theory” whose origins can be traced back to ancient times, although the term “miasma” (plural “miasmata” or “miasmatas”) probably did not appear for the first time until the 17th century (Collins English Dictionary, 1992). The theory was made explicit in 1717 by Giovanni Lancisi in his treatise De noxiis paludum effluviis (Of the poisonous effluvia of malaria; Porta, 2014). Miasma theory remained the dominant paradigm for centuries (Cipolla, 1992; Halliday, 2001; Howard-Jones, 1984), despite contemporaries’ awareness of its limitations. This is illustrated by a letter written in 1844 by an academic to the architecture journal, The Builder, describing miasmas as “unseen and subtle causes of disease, the existence of which we reason by analogy, and of which much has been said, although little is known” (Booth, 1844). In this letter, the author correctly advocates more salubrious buildings by ensuring adequate ventilation, effective drainage, reduced overcrowding, and so on (despite the scientific flaws in the theory exemplified by, nowadays humorous, statements such as “from inhaling the odour of beef the butcher’s wife obtains her obesity”; Booth, 1844, p. 350).

Undeniably, public health campaigns based on the flawed miasma theory effectively controlled many significant communicable pathogens (Cipolla, 1992; Eisenberg et al., 2007). Based on this fundamentally erroneous paradigm, during the 15th and 17th centuries, the states of northern and central Italy created an advanced system of public health and hygiene with the establishment of permanent health boards (magistracies) in the major cities (Cipolla, 1992). These boards would promote a variety of measures either during epidemics (administration of lazzarettos, creation of cemeteries reserved for the burial of plague victims etc.) or between epidemics (concerned with quality checks of food on sale, movement of beggars and prostitutes, quarantine of ships, sewers, etc.). Similarly, in 19th-century Britain, Sir Edwin Chadwick, one of the most important public health activists, social reformer, and confirmed believer in miasma theory, in his “Report on the Sanitary Conditions of the Labouring Population,” appropriately recommended measures such as the compulsory removal of all rubbish, improvement of drainage, ventilation, and street cleaning (Cipolla, 1992).

Even more remarkably, miasma theory was applied to the cholera outbreak in London in 1849 and was successfully supported by statistical models (Langmuir, 1961). More precisely, the great epidemiologist and statistician, William Farr, postulated that miasmas of cholera emanated from the River Thames and spread out over the various areas of the city. The spreading of miasmas was supposed to proportionally decrease with the elevation of the area above the Thames. Farr aggregated the cholera mortality for each district according to their elevation. This resulted in a clear correlation between cholera mortality and elevation above the River Thames with impressive agreement with observed data, as shown in Figure 1 (Farr, 1852; Langmuir, 1961).

Figure 1. Correlation of cholera mortality and elevation above the Thames River, London, 1849, as calculated by William Farr. Data extracted from Farr’s original report (Farr, 1852). See also Langmuir (1961). For visual purposes, the x-axis was square-root transformed.

The contagious nature of diseases (i.e., the ability to transmit from person to person via physical contact) has also been recognized for a long time. According to Thucydides’ detailed description of the Plague of Athens (430–428 bc) during the Peloponnesian War “there was the awful spectacle of men dying like sheep, through having caught the infection in nursing each other” (Thucydides, trans., 1933, p. 131). Thucydides also showed an understanding of cross-species transmission when he wrote: “All the birds and beasts that prey upon human bodies, either abstained from touching them (though there were many lying unburied), or died after tasting them. In proof of this, it was noticed that birds of this kind actually disappeared; they were not about the bodies, or indeed to be seen at all” (Thucydides, trans. 1933, p. 131). Thucydides, however does not propose any hypothesis to explain the causes of disease, stating: “All speculation as to its origin and its causes, if causes can be found adequate to produce so great a disturbance, I leave to other writers, whether lay or professional; for myself, I shall simply set down its nature, and explain the symptoms by which perhaps it may be recognised by the student, if it should ever break out again” (Thucydides, trans., 1933, p. 129).

In contrast, the Roman writer Marcus Terentius Varro (116–127 bc), in his book De Re Rustica (On Agriculture), clearly expresses the idea that diseases were caused by contact with tiny, invisible living organisms, according to whom “[p]recautions must also be taken in the neighborhood of swamps, both for the reasons given, and because there are bred certain minute creatures [animaculae in Latin] which cannot be seen by the eyes, which float in the air and enter the body through the mouth and nose and there cause serious diseases”(Loeb, Goold, & Hooper, 1934). A concept also referred to as “seeds of diseases,” which had supporters since ancient Greek times, was fully elaborated by the physician and mathematician Girolamo Fracastoro (1478–1553; Nutton, 1983), who distinguished three forms of contagion: direct contact (typical of syphilis and gonorrhea); transmission by fomites; and infection at distance mediated by air (typical of tuberculosis and smallpox; Karamanou, Panayiotakopoulos, Tsoucalas, Kousoulis, & Androutsos, 2012). Fracastoro argued that infections are caused by transferable seed-like beings, although these seeds are conceived not as living microorganisms but as chemical substances liable to evaporation and atmospheric diffusion (Karamanou et al., 2012). Thus, the concept of disease transmission and contagion was well understood before microorganisms were actually identified.

The existence of microscopic organisms was revealed by Anton van Leeuwenhoek (1632–1723) using simple microscopes; he observed bacteria and protozoa, although he made no connections between these organisms and disease. Even when later scientists observed microorganisms in the blood of diseased people, the belief was that these microorganisms spontaneously generated because of the disease, rather than the other way round (“Germ Theory,” n.d.). The “spontaneous generation” theory, according to which life arises spontaneously from nonliving matter, remained the theoretical paradigm until the experiments of Francesco Redi (1626–1697) in 1668; Redi disproved the theory for higher organisms such as flies. And finally Louis Pasteur (1822–1895), in 1859, disproved the theory for microorganisms such as bacteria. Following this research, Pasteur proposed that microorganisms are the cause of diseases (“germ theory”). At the same time, Robert Koch (1843–1910) formulated the criteria and procedures (Koch’s Postulates) necessary to establish that a particular microbe and no other was the cause of a particular disease (Karamanou et al., 2012).

The laboratory-based research of Robert Koch and Louis Pasteur, which provided the scientific proof for the germ theory, was also complemented by the epidemiological work (i.e., the study of epidemic disease, including its spread, causes, and methods of control) of John Snow during the 1854 London cholera epidemic. Snow postulated that cholera was caused by contagious agents, which multiplied in an ill individual; these agents were transmitted by a direct fecal-oral route and through the contamination of the water supply, see, for example, Fisman (2007). Snow’s explanation, which Farr eventually accepted and supported, was correct. The comma-shaped bacillus Vibrio cholera was isolated in 1854 by Filippo Pacini (1812–1883), although his contribution was only recognized posthumously. It is unlikely that John Snow and Robert Koch (Koch also identified the agent Vibrio cholera 30 years later) were aware of Pacini’s paper (Howard-Jones, 1984).

The extensive work on cholera was pivotal in shifting from miasma theory, which is intrinsically associated with environmental factors (e.g., proximity to marshy areas, presence of smell), toward the germ theory paradigm, which correctly focused on microscopic organisms as pathogenic agents rather than the environment (still, in the 1874 international sanitary conference, representatives of 21 governments voted unanimously that “ambient air is the principal vehicle of the generative agent of cholera” (Howard-Jones, 1984, p. 380).

Yet, microbiological arguments alone are not sufficient (nor necessary as William Farr’s miasmatic model of cholera demonstrated (Langmuir, 1961) to explain the geographic distribution and seasonality of many infectious diseases (Altizer et al., 2006; Fisman, 2007). Recognition of the ecology (i.e., study of the relationships between organisms and their environment) of host and pathogen represents an important conceptual advancement, reconciling the microbiology paradigm (i.e., infections are caused by living organisms) with the observation that the patterns of many diseases are often associated with environmental factors. This theory is supported by increasing evidence that the environment can affect pathogen abundance, survival, or virulence, as well as host susceptibility to infection (Fisman, 2007), leading to the general consensus that environmental factors, including climate change, are one of the main drivers of disease emergence.

Accordingly, many conceptual frameworks to identify the impact of environmental factors on infectious disease have been proposed (Eisenberg et al., 2007). The objective of understanding and mapping the impacts of the environment on infectious diseases, however, is far from accomplished. Mathematical and statistical approaches are powerful tools to investigate this impact, but they still face many challenges (Lo Iacono et al., 2017). Table 1 lists some of these challenges

Table 1. Some challenges for mathematical modelers to estimate the impacts of environmental change on infectious diseases (these and other more technical, though important, challenges are discussed in the systematic review by Lo Iacono et al. (2017).

 Disentangling multiple transmission pathways and identifying the biophysical mechanisms of how the environment affects disease and seasonality. Reducing uncertainty and bias in reporting, which can mask the impact of environment. Developing quantitative frameworks to study the potential effect on infectious diseases due to anthropogenic climate change, evaluating intervention, adaptation, and co-benefits Identifying and quantifying the different sources of the temporal lag (e.g., the time required for potential growth of pathogen population in the environment, exposure dynamics, incubation period, delays in reporting) from the start of the pathway to infection to disease detection. Studying the evolution of pathogen in response to climate and other environmental changes. Investigating the effects of time-varying factors (e.g., seasonal exposure to a risk factor) on transmission patterns. Dealing with different spatiotemporal scales (e.g., daily stochastic fluctuations, seasonal variation, or longer term El Niño oscillations. Developing models for a truly One Health (Rabinowitz et al., 2013; Zinsstag et al., 2012) approach, that is,, an integrated, holistic approach to understand intersections between disease dynamics, environmental drivers, economic systems, and veterinary and public health responses. Overcoming difficulties in acquiring suitable datasets to base the model on or test hypothesized associations between disease rates and weather.

# A Brief Overview of Mathematical Modeling and Infectious Diseases

Models, by definition, are simplified frameworks describing natural phenomena. In particular, they have been used successfully to understand the dynamics of infectious diseases. The scope of mathematical modeling is broad: models can provide precise predictions, assessments of risk, tests of conjectures and hypotheses, and estimation of unknown parameters for statistical inference (i.e., drawing conclusions based on data), or they can simply offer heuristic insights (Hethcote, 2000). Process-based models are theoretical representations of the biophysical mechanisms under investigation (Dubitzky, Wolkenhauer, Cho, & Yokota, 2013). These models usually arise from first principles and/or are based on the empirical functions describing the fundamental processes (e.g., a phenomenological curve describing how the growth rates of certain pathogens depend on temperature or relative humidity). Process-based models are particularly suited to incorporate specific responses to altered environmental conditions; and they offer significant advantages in predicting the effects of global change as compared to purely statistical or rule-based models based on previously collected data (Cuddington et al., 2013).

An important class of process-based models is the group of compartmental models. In these types of models, the population of interest is usually partitioned into relevant epidemiological categories, such as susceptible, exposed, infected, and recovered individuals, see Box 1 and Figure 2 (Anderson & May, 1981, 1991; Hethcote, 2000; Keeling & Rohani, 2008). Key tasks of these compartmental models are to model transitions from one category to another and to calculate changes (such as population size and age composition) for each category (Box 1 and Figure 2). There are different methods to achieve these tasks. Some examples are illustrated in Table 2.

Figure 2. Schematic of compartmental models with Environment compartment.

Table 2. Some examples of different, general, methods also used within the scope of compartmental models (other than solving differential equations).

 Agent-Based Models Agent-Based Models can be used to mimic the relevant processes with a computationally aided set of autonomous, interacting agents, for example, individuals belonging to a particular epidemiological category (Macal & North, 2010). Network Analysis Network Analysis focuses on the connections between individuals to study how infectious disease propagates (Danon et al., 2011; Keeling, 1999). Branching Processes Branching Processes (Jacob, 2010) have been applied, in particular to estimate the probability of extinction of one of these epidemiological categories. Hawkes Processes In Hawkes Processes, the transition from the susceptible to the infected category is governed by Poisson processes, with a memory of past events when the rate of infection is changing because of past infection or the depletion of the susceptible category (see the section “How Can We Model the Impact of Animal Reservoir on Uuman Infections? ” and Lo Iacono et al. (2016).

Most compartmental models, however, are built over a set of differential equations describing the rate of change of the population in each category (Box 1 and Figure 2) (Anderson & May, 1981, 1991; Hethcote, 2000; Keeling & Rohani, 2008). By solving the system of differential equations, it is possible to estimate the number of individuals in each compartment at any one time. Although a model for smallpox was already formulated and solved by Daniel Bernoulli in 1760 (Heesterbeek & Roberts, 2015; Hethcote, 2000; Murray, 2002), the roots of these compartmental models can be traced back to the beginning of the 20th century due to the work of Kermack and McKendrick (1927), Ross, Macdonald, and several other mathematicians and scientists (Smith et al., 2012). This class of models has been widely extended to other situations, including stochastic effects (Keeling & Ross, 2008), spatial variability, and other forms of heterogeneity in the network of contacts (Keeling, 1999), multiple species (Dobson, 2004), and evolutionary dynamics (Day & Gandon, 2007; Day & Proulx, 2004; Lo Iacono, van den Bosch, & Gilligan, 2013; Lo Iacono, van den Bosch, & Paveley, 2012).

# Incorporating the Impact of Environment into Compartmental Models: Two Examples

## Incorporating the Impacts of Environment in Models for Vector-borne Diseases

Climate and ecosystem change and globalization are impacting the ecology of arthropod species, and thus the occurrence of vector-borne diseases. Climate change has been shown to modify the geographic distribution of arthropod species: temperature and other weather variables directly affect their survival, reproduction, and biting rates, while changes in the availability and size of water bodies (due to new irrigation patterns, dam constructions, and changes in rainfall patterns) affect the accessibility to breeding sites and the rate at which the vectors lay their eggs. Temperature also affects the period between the infection of and transmission from vectors (i.e., pathogen extrinsic incubation periods), which usually require pathogen replication at ambient temperatures.

The impact of environment is often introduced directly in the epidemiological (and/or ecological) parameters in the compartmental models (e.g., in the infection rate β‎ and recovery rate γ‎ in Box 1) An increasing number of eco-epidemiological models for vector-borne diseases explicitly include the population dynamics of the vectors in the models. This can be done by using stage-structured models (see, e.g., Tuljapurkar & Caswell, 1997), where the population of a vector (e.g., mosquitoes) is partitioned into its different life stages—the egg, larva, pupa, and adult stages. These approaches typically model transitions from one stage to another, and calculate changes in the population size for each stage (e.g., Otero, Schweigmann, & Solari, 2008; Otero, Solari, & Schweigmann, 2006).

Important input parameters explicitly depend on weather variables. For example, the oviposition rate depends on the availability and size of the water bodies, which in turn depend on the precipitation and hydrology of the region (Asare, Tompkins, Amekudzi, Ermert, & Redl, 2016; Shaman, Spiegelman, Cane, & Stieglitz, 2006; Soti et al., 2012, 2013); the mortality and developmental rates, the biting rate, and the activity of the vectors depend on temperature and on other climatic variables; as well as the susceptibility of the vector to the pathogen and the extrinsic incubation period of the pathogen (the literature is vast; see, e.g., Rogers & Randolph, 2006, and the references therein). Explicit relationships are often available in the literature, and the time series of temperature could be readily used as model inputs, resulting in compartmental models with time-varying parameters (Box 2).

## Incorporating the Impact of Environment in Models for Water-borne Diseases

Other approaches directly model the dynamics of the pathogen population in the environment. Capasso and Paveri-Fontana (1979), for example, proposed a simple deterministic mathematical model for Vibrio cholerae, consisting of a system of two ordinary differential equations describing the change of the human-infected population in a town community and of bacterial abundance in the ocean. This naturally leads to an extension of the compartmental models by coupling the set of differential equations, similar to those shown in Box 1 and Figure 2, with an additional differential equation describing the concentration of the pathogen in the environment.

The set of equations can be explicitly regulated by environmental variables. For example, the growth and survival of V. cholera can be modeled based on inputs of weather and/or climatic drivers such as temperature, salinity, sunlight, and pH (Lipp, Huq, & Colwell, 2002). Precipitation is also expected to alter the contact rates between the contaminated water and the hydrological connections of local human populations (Righetto et al., 2013). Infections occur when a susceptible individual comes in contact with this additional category. Infected people can excrete pathogens, and this feeds back into the environmental compartment. Human-to-human transmission has also been incorporated in these models by allowing for infection transmission when a contact between a susceptible and infected person occurs.

This class of models, largely applied to Vibrio cholerae (Gatto et al., 2012; Righetto et al., 2012, 2013; Torres Codeco, 2001), is particularly suitable for the study of the impact of the environment on water-borne disease. Nevertheless, the environment can have an indirect impact on the epidemiology of these diseases, for example, by changing the population size (e.g., due to increased urbanization) and the rates of contact among people, or by altering the patterns of contact at the people–environment interface, leading to models that include human behavior such as population mobility (Mari et al., 2012).

Despite significant progress, the impact of the environment on infectious disease is still open to many questions. Here we discuss in more detail these broad issues and, in some cases, propose potential approaches to address these questions.

## Environment as Biodiversity

Biodiversity or biological diversity refers to “the variability among living organisms from all sources including, among other things, terrestrial, marine and other aquatic ecosystems and the ecological complexes of which they are a part; this includes diversity within species, between species and of ecosystems” (Convention on Biological Diversity, Article 2).

## Does an Increase in Biodiversity Reduce the Burden of Disease?

This concept is known as the “dilution effect.” According to this hypothesis, preserving intact ecosystems and their endemic biodiversity would generally protect against infectious, zoonotic (i.e., diseases transmitted from animal to humans) diseases (Civitello et al., 2015; Johnson, Ostfeld, & Keesing, 2015; Keesing et al., 2010; Lacroix et al., 2014; LoGiudice, 2003). Biodiversity is usually measured by the number of different species present in a particular location, which in turn is regulated by other factors such as climate and land use (Walther et al., 2002). Different mechanisms have been proposed to explain how biodiversity impacts disease transmission. For directly transmitted zoonotic disease, such as Hantavirus pulmonary syndrome, rodent hosts living in areas with high species richness are more likely to come in contact with heterospecific mammals (different biological species) and less likely to come in contact with rodent hosts of the same species. If the heterospecific mammals are not susceptible to the infection, they break the chain of transmission, reducing the burden of diseases among rodents and thus reducing the risk of spillover to humans (Clay, Lehmer, St. Jeor, & Dearing, 2009; Keesing et al., 2010). For vector-borne diseases, the dilution effect can be explained in terms of “wasted bites,” that is, when the vector preferentially feeds on a species that does not develop infection. Accordingly, in some communities in Africa, people keep cattle or sheep near their houses, assuming that this will distract mosquitoes carrying malaria away from people. Similarly, some midges show apparent preference for cattle over sheep, so in South Africa deploying cattle to protect sheep from bluetongue has been proposed as a way to control the disease (Nevill, 1978). Mammal biodiversity at the global scale, however, is associated with an increased risk of emerging zoonosis (Jones et al., 2008). Accurate measurements of the dilution effect are difficult, as longitudinal studies of infection prevalence on different hosts are required. Not surprisingly, this hypothesis is still a matter of debate, with only partial evidence supporting it (Bouchard-Côté & Jordan, 2012; Ostfeld, 2013; Randolph & Dobson, 2012, 2013; Salkeld, Padgett, & Jones, 2013).

Process-based models can provide useful insight into a sometimes polarized debate. A theoretical work on African Horse Sickness, a fatal viral disease transmitted and amplified by Culicoides biting midges, demonstrated that the influence of noncompetent hosts is complicated by two potential, but contrasting, effects: a dilution effect, whereby vectors exhibit a feeding preference for a noncompetent, nonequid host; and an amplification effect, whereby increased vertebrate–host densities result in increased vector abundance (Lo Iacono, Robin, Newton, Gubbins, & Wood, 2013). Lo Iacono et al. explored how the Basic Reproductive Number (Box 1) in the presence/absence of the noncompetent host changed under different scenarios, depending on how the abundance of Culicoides midges is affected by the density of the host and the different values of feeding preference toward a particular host. The authors showed that the dilution effect is not universal, but rather depends on the particular scenario and parameter values. The approach can also provide quantitative estimates for this potential effect: for example, how many noncompetent hosts are needed in a farm to ensure that the disease is mitigated.

## How Can We Model the Impact of Animal Reservoir on Human Infections?

Zoonotic diseases are at the origin of the majority of human pathogens (see Jones et al., 2008, and references therein). Perhaps HIV-1 is the most spectacular case of a human pandemic that emerged from an endemic infection of chimpanzees in Central Africa. Measles, smallpox, and diphtheria are examples of established human diseases that probably have zoonotic origins (Slingenbergh, Gilbert, de Balogh, & Wint, 2004). The Severe Acute Respiratory Syndrome (SARS) and swine influenza pandemics demonstrate that, although the transmissions of pathogens from animals to humans (“spillovers”) are rare events, they can transmit worldwide rapidly and have devastating health impacts.

The lack of a satisfactory answer to simple questions, such as “why do certain zoonotic diseases remain confined in the region where they originated while others result in a pandemic?” shows the challenges that the scientific, medical, and public health communities face. Cross-species transmission is driven not only by the physiology of the hosts and the biology of the pathogens, but also by the complex interactions of many environmental factors. These comprise weather and climatic factors (e.g., precipitation affecting vegetation and in turn the abundance of rodents), ecological factors (e.g., the presence of hosts with differing degrees of susceptibility and periodicity in their abundance), epidemiological and genetic factors (e.g., a broad set of pathogen life histories and periodicity of infection prevalence), and anthropogenic activities (e.g., land-use and behavioral changes affecting direct and indirect interactions with reservoir hosts). Despite some theoretical progress (Blumberg & Lloyd-Smith, 2013; Cauchemez et al., 2013; Kubiak, Arinaminpathy, & McLean, 2010; Kucharski et al., 2014; Lo Iacono et al., 2016; Reluga & Shim, 2014), disentangling the many complex aspects of transmission at the animal–human interface is still a compelling task for future studies (Allen et al., 2012; Lloyd-Smith et al., 2009).

This issue is even more complicated when, as well as animal-to-human transmissions, human-to-human infections also occur, and the environment is likely to have a differential impact on these two routes of transmission. To address these tasks, Lo Iacono et al. (2016) proposed modeling these infections using a stochastic framework known as Hawkes Processes. According to this model, each infection is governed by a Poisson process; the rate of infections, however is not constant, but rather depends on the time-varying number of susceptibles and infected at each previous time step. New infections can generate other infections (“self-exciting process”) but may also cause the depletion of the susceptibles, with a damping effect on the progression of the infection (“self-correcting process”). The key environmental and socioeconomic processes can be incorporated into the time-dependent rate of infections in a simple manner (Box 3). The model can then be used to understand the potential different impacts of environmental drivers on the spillover from animal-to-human and human-to-human transmissions.

# Potential Impact of the Environment on the Evolution of Pathogens

There is now evidence that environmental factors in general and climate change in particular affect the geographical distributions and abundance of many species, including pathogens and vectors of diseases, as well as the timing of growth and reproduction (Altizer, Ostfeld, Johnson, Kutz, & Harvell, 2013; Hoffmann & Sgrò, 2011; Moritz & Agudo, 2013).There are different adaptive responses to environmental change (Hoffmann & Sgrò, 2011; Hoffmann & Willi, 2008; Santini, 2015). One response is to exploit phenotypic plasticity, that is, the capacity of a genotype (the entire set of genes of a living organism) to exhibit variable phenotypes (its physical characteristics) in different environments, enabling an individual to adapt to changed environments without the need for novel genetic mutations. Another is migration, directly or facilitated by vectors, to more suitable habitats. Another response is evolving new attributes, although direct evidence of genetically based adaptation to climate change over time remains limited (Moritz & Agudo, 2013).

Examples of evolutionary adaptation are mainly available in agro-ecosystems, owing to the large selection pressures originated by growing use of genetic technologies (e.g., the use of disease-resistant crops), the widespread use of chemical controls promoting herbicide and insecticide resistance, and intense changes in land use (Lo Iacono et al., 2012; Santini, 2015; Thrall et al., 2011). There are, however, other examples relevant to human health. For instance, in Bangladesh the dominant pathogenic strain for Vibrio cholera was the Classical strain until the 1970s; then the mutant El Tor strain invaded the resident population and replaced the Classical strain, which can be explained by changes in monsoon rainfall patterns (Koelle, Pascual, & Yunus, 2005). Another example is Lyme disease; according to Kurtenbach et al. (2006), the strains of the Lyme bacterium, Borrelia burgdorferi, are subjected to different selection pressures, depending on the synchrony and asynchrony of the seasonal activity of the larval and nymphal ticks. More precisely, “asynchrony of infecting nymphs and uninfected larvae favours pathogen persistence strategies, whereas synchrony of these tick stages combined with a short annual period of activity should favour short-lived ‘boom-and-bust’ strategies and the capacity for co-feeding transmission” (Kurtenbach et al., 2006, p. 666).

The patterns of seasonality are also expected to be an important driver of pathogen evolution (Altizer et al., 2006; Donnelly, Best, White, & Boots, 2013), as periods of high transmission are followed by “population bottlenecks,” reducing strain diversity and causing rapid genetic shifts (Patz et al., 2003). Another important example arises from the interaction between environmental and demographic stochasticity. More precisely, epidemics are characterized by stochastic fluctuations, with their own frequencies. These “natural frequencies” depend on the parameters of the system, for example, the infectious and incubation period, the birth rate of the host, and so on (see, e.g., Keeling & Rohani, 2008). When an external periodic perturbation, such as seasonality in temperature or in the application of control measures, has a frequency close to the natural frequencies of the epidemics, the result will be enhanced fluctuations in the epidemics (resonance), promoting the emergence or suppression of particular strains of the pathogen (Lo Iacono, van den Bosch, et al., 2013).

Mathematical modeling of multistrain, multi-host pathogens remains, in general, a major challenge (Dobson, 2004; Kurtenbach et al., 2006). Not surprisingly, extensions of this class of models in response to environmental and climate change are even more challenging.

# Impact of the Environment on the Stability of Infectious Diseases and Their Potential Vectors

Let us consider vector-borne diseases again. Human mobility, wild and domestic animal trade, and climate change are increasing the opportunity for the vector, and also the pathogens, to reach and potentially become established in different parts of the globe. Several examples now support this position: the incursion of the mosquito Aedes albopictus (known vector of chikungunya virus, yellow fever virus, dengue virus, zika virus, and dirofilariasis) into Europe through the international trade in used tires and lucky bamboo (Medlock et al., 2012); outbreaks of mosquito-borne chikungunya fever in northeastern Italy in 2007 (Beltrame et al., 2007; Rezza et al., 2007); autochthonous cases of dengue fever in 2010 in France (La Ruche et al., 2010), in Croatia (Gjenero-Margan et al., 2011) and Madeira in 2012 (European Centre for Disease Prevention and Control, 2013; Tomasello & Schlagenhauf, 2013); and the unexpected, epizootics of Culicoides-borne viral diseases such as bluetongue and Schmallenberg virus (Koenraadt et al., 2014) in northern Europe probably due to climate change. These and other examples raise simple but only partially answered questions, some of which are listed in Box 4.

Mathematically, the response of dynamic systems like infectious diseases to external changes can be addressed by using stability analysis (May, 2001). Stability analysis measures the resistance to change of the system under investigation. For nonseasonal compartmental models, stability analysis proceeds by (1) identifying the equilibrium solution, that is, when the populations in all the compartments (e.g., susceptible, infected, recovered) are no longer changing with time; and (2) checking if the solution is stable, that is, the system returns to the equilibrium solution after being perturbed. Examples of equilibrium solutions can be the situation when the diseases fade out or an endemic situation with fixed coexisting populations of infected and susceptible individuals.

A crucial aspect for many biological systems, however, is that these exhibit periodic patterns. Periodicity might arise from the seasonality in the environmental drivers or from the internal dynamics of the system such as predator–prey cycles. In the context of seasonal vector-borne diseases, the system might result, for instance, in sustained oscillations in the populations of vectors and/or infected individuals. As in the nonseasonal case, a key question is whether or not these periodic solutions are stable. This can be done by crude numerical simulations or, more elegantly, by employing Floquet Theory, a well-established mathematical tool that provides stability analysis for periodic systems (Grimshaw, 1993; Klausmeier, 2008). The biophysical interpretation of stability analysis is extremely important. For instance, through international trade, batches of mosquitoes imported into a naïve region could either fade out or grow, and after a transient phase, eventually the new species will invade and establish in the region. Stability analysis is able to quantify the range of the key environmental factors ensuring the extinction, establishment, or even chaotic behavior, of the mosquito population. The same analysis can be repeated, focusing on the pathogen dynamics rather than on the vectors.

# Quantifying the Time Lag between Environmental Stimuli and the Occurrence of Disease

The effects of the environment on the occurrence of disease are not instantaneous. This issue is further complicated as multiple causal pathways can be involved. For instance, in the case of Lyme disease, a tick-borne disease, environmental factors such as temperature, day length, and relative humidity continuously affect the physiology of the ticks, namely, the developmental, diapause (i.e., a period of suspended development in an insect), and mortality rates (Belozerov, 2009; Dobson, Finnie, & Randolph, 2011; Dobson & Randolph, 2011; Gilbert, Aungier, & Tomkins, 2014; Knülle & Rudolph, 1982; Ludwig, Ginsberg, Hickling, & Ogden, 2016; Ogden et al., 2005; Oliver, 1989; Ostfeld & Brunner, 2015; Randolph, Green, Hoodless, & Peacey, 2002). These individual, not-instantaneous responses will have a detectable effect on the abundance and questing activity of the ticks only after a certain time lag. Environmental factors will change the behavior of people too (e.g., their choice of outdoor activity and type of clothing), affecting the chance of being bitten by an infected tick. Assuming that the tick bite results in an infection, this infection will not be detected until symptom onset, introducing a further time lag. Finally, the infection will be (hopefully) reported to the health authorities after a certain time lag, depending on the country’s health system (Dhôte, Basse-Guerineau, Beaumesnil, Christoforov, & Assous, 2000; Marinović, Swaan, van Steenbergen, & Kretzschmar, 2015). Furthermore, the typical incubation periods and disease notification depend on the particular clinical form of the infection (i.e., the incubation periods range from about two weeks for patients showing erythema migrans alone to 96 weeks for patients presenting with the neurological complications termed acrodermatitis chronica atrophicans (Dhôte et al., 2000).

Detection of diseases can also be affected by the environment. For example, it has been reported that the incidence of Lassa fever in Sierra Leone is higher during the dry season (Shaffer et al., 2014; Webb et al., 1986). Recent analyses, however, indicate that this could be an artifact of data collection methods, for example, because of impassable roads reducing the ability to seek medical care during the rainy season (Bausch, Moses, Goba, & Grant, 2013; Grant et al., 2016).

Thus, the time lag from some changes in the environmental factors triggering the causal pathways of infection (which can be challenging to define) to disease detection is the cumulative effect of many time lags arising from different sources. These are typically represented by random variables drawn from adequate distributions, for example, a log-normal distribution (Sartwell, 1995). This is more complicated if there are multiple routes of transmissions (e.g., human-to-human as well as animal-to-human or a water-reservoir-to-human), as this introduces further time lags from different origins. Estimating the time lag between an environmental variable and infection is a common task encountered in the literature. In most cases, the assessment is based on assessing potential correlations between the time series of disease incidence and the time series of environmental variables at 1,2 … n weeks before the date of the reported case. The methods used, however, rarely distinguish among the different sources of time lags, nor do they incorporate the mechanisms of these delays.

# Environment and Human Behavior

The importance of human behavior in the progression of infectious diseases is unquestionable. Accordingly, a growing body of research is incorporating human and other animal behavioral changes into infectious disease models (Verelst, Willem, & Beutels, 2016). Many studies have focused on reactive behavioral change, such as vaccination prompted by the emergence of risk of infection. Further progress will come by validating these theoretical models with data (Verelst et al., 2016). This concern is expected to be valid for models of human behavior in response to environmental factors. However, some attempts in behavioral models have been validated with data. For example, as part of an exploration of the risk of African Horse Sickness transmission in Great Britain, Lo Iacono et al. (2013) modeled the likelihood of the location of horses in Great Britain based on simple assumptions of human behavior, data on land use, and qualitative data collection based on questionnaires.

# Conclusion

Mathematical modeling is essential to understand the complex interaction between the environment (in the broad sense, including animals, humans, climate, etc.) and infectious diseases, and possibly predict its impact. Many significant questions remain, however; only a limited number of these issues are discussed and illustrated here. Apart from the expected problems associated with the development of new mathematical and computational techniques, these issues pose other challenges, which can be summarized as (1) undefined scope; (2i) limited data; and (3) partial attainment in developing a “One Health” approach.

## Undefined Scope

The term “environment” is a collective word comprising many biological and physical factors. Behavioral and socioeconomic aspects also have direct and/or indirect links with the environment. For instance, economic and political reasons alone might promote the construction of a new dam or changes in irrigation patterns; this decision could result in critical changes in the environment that are likely to affect water-borne and vector-borne infectious diseases. Entangled multifactors are not the only source of ambiguity. Even when focusing on a single well-defined measurable driver, multiple effects are possible. For example, temperature directly changes the life history parameters of mosquitoes and associated diseases, but it can also affect human behavior. The first effect can be treated as a “dose-response relationship,” and its incorporation in mathematical models is relatively straightforward. In contrast, the inclusion of human behavior in mathematical approaches is still in its infancy and is seldom included in these models.

## Limited Data

Within the United Kingdom, the hiatus caused by the bubonic plague in 1665 contributed to the routine collection of mortality data by cause of death, and Farr in the 19th century improved the provision of standardized disease surveillance data that is essential for disease epidemiology modeling and intervention. Data are still subjected to many limitations, however. “Disease reporting bias” is a well-known problem (Gibbons et al., 2014), and approaches to deal with surveillance artifacts such as bank holidays, vacations, travel, and weekends have been used (Nichols, Richardson, Sheppard, Lane, & Sarran, 2012). Reliable data on pathogen incidence are scarce, especially in developing countries, although this is expected to change with the advancement of genome sequences (Grad & Lipsitch, 2014) and improved surveillance efforts (Nichols, Andersson, Lindgren, Devaux, & Semenza, 2014). Weather data are relatively easy to collect; even so, they are subjected to important limitations. In developing countries, long-term, high-resolution weather data are often absent. In developed countries, the data might be available, but fine spatiotemporal linkage of these data with infection cases is rare (Djennad et al., 2017). For many diseases, the microclimate at the spatiotemporal domain where infection occurred (e.g., indoor temperatures and relative humidity) is important information. However, this type of information is seldom accessible. A similar problem is encountered when estimations of wildlife abundance and distribution of animal host are required.

## Partial Attainment in Developing a Truly One Health Approach

The One Health perspective is a holistic approach that recognizes the interconnections between human health, animal health, and the environment, including socioeconomic factors. Despite a growing support for these integrated cross-disciplinary approaches (Restif et al., 2012), many One Health initiatives are still subjected to the barriers that segregate its contributing disciplines (Manlove et al., 2016). This is somehow expected, owing to researchers from different disciplines with different perspectives on approaches. They have been described as wanting to “share frameworks about as much as they like to share toothbrushes” (Gorman, Kincannon, & Mehalik, 2001).

Of course, the challenges in cross-disciplinary studies are not only a matter of scientists’ parochial attitudes. Developing a solid One Health approach, and even more for a Planetary Health approach, which extends the philosophy of One Health by focusing not only on the present generation but on past and future civilizations (Whitmee et al., 2015), requires clear conceptualization of the broad problem (i.e., to define the scope) and access to data from many sources—two key problems already discussed (sections “Undefined Scope” and “Limited Data”). It can be logistically and financially daunting. For instance, a three-year scientific grant is unlikely to be long enough for longitudinal model-guided fieldwork. The drawbacks of this partial integration inevitably will cascade into the selection of the modeling approaches. Despite these challenges, integrative approaches combining process-based, empirical, and participatory modeling have the potential to improve models by guiding the design of the models possibly anchored with data (Grant et al., 2016; Scoones et al., 2017).

# Acknowledgments

This work was supported by the National Institute for Health Research Health Protection Research Unit (NIHR HPRU) in Environmental Change and Health at the London School of Hygiene and Tropical Medicine in partnership with Public Health England (PHE), and in collaboration with the University of Exeter, University College London, and the Met Office; and the UK Medical Research Council (MRC) and UK Natural Environment Research Council (NERC) for the MEDMI Project. The views expressed are those of the authors and not necessarily those of the NHS, the NIHR, the Department of Health, or Public Health England.

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