We are searching data for your request:
Upon completion, a link will appear to access the found materials.
I am assuming that the basic reproduction number $R_0$ of a disease depends on the population size (or the number of susceptible individuals). When $R_0$ is reported it seems to be without such information. Is there a standard population size for reporting $R_0$?
From what I understand (as an ecologist/population modeller), $R_0$ in epidemiology is not dependent on host population size per se, at least not in its basic form. It is also not dependent on the number of susceptible individuals, since it is defined as the number of secondary infections in a fully susceptible population, see e.g. this section in Farrington et al (2001):
The basic reproduction number of an infectious agent in a given population is the average number of secondary infections which one typical infected individual would generate if the population were completely susceptible.
This is also consistent with the Wikipedia description ("… in an otherwise uninfected population"). $R_0$ is however dependent on the environment (dispersal routes, host-host interactions etc), which also why it is used to evaluate and compare the effect of control measures. Host-host interactions is also dependent on population density, so $R_0$ is indirectly influenced by host population density, and this is partially why estimated $R_0$ values for the same disease will differ between countries and populations.
Fundamentally and conceptually, $R_0$ is the same thing as the Net reproductive rate (often also labelled $R_0$) in demography and population modelling, and it was originally borrowed from demography to epidemiology. As such, the net reproductive rate is often used in for instance pest control to evaluate the potential impact of pest species on agricultural crops, in direct analog to the basic reproduction number in epidemiology (see Emiljanowicz et al, 2014 for a randomly picked, recent example). Most commonly, $R_0$ (both in population dynamics and epidemiology) is calculated from static demographic rates, but nothing prevents you from considering stochastic effects, density dependence or e.g. ways that demographic rates are functions of host density or population size. This could lead to formulations where $R_0$ is an explicit function of host population density, but that is not the standard use of $R_0$.
To me, it seems like you are asking for the effective reproduction number ($R(t)$), which can be defined by:
… the average number of secondary cases that result from an infectious case in a particular population (Box 1). R depends on the level of susceptibility in the population, in contrast to the basic reproduction number (R0), which is the average number of secondary cases arising from one infectious case in a totally susceptible population."
This section is from a WHO paper on meases (Chiew et al, 2013). So basically, $R(t)$ is the average number of secondary cases per primary case at time $t$ (including the effects of immunity and/or control measures). As you can see, $R(t)$ is a function of time, and it can e.g. be used to describe the temporal dynamics of SIR-type models.
Basic Reproduction Number
The basic reproductive ratio or basic reproduction number ( R0) is the average number of infected contacts per infected individual. At a population level, a value of R0 larger than one means that a virus will continue its propagation among susceptible hosts if no environmental changes or external influences intervene. An R0 value lower than one means that the virus is doomed to extinction at the epidemiological level under those specific circumstances. The basic models of infection dynamics were developed by R.M. Anderson, R.M. May, and M.A. Nowak, with inclusion of the following key parameters: rate k at which uninfected hosts enter the population of susceptible individuals (x), their normal death rate (u) (so that the equilibrium abundance of uninfected hosts is k/u), number of infected hosts (y), mortality due to infection (v) (so that 1/u + v is the average lifetime of an infected host), a rate constant (β) that characterizes parasite infectivity (so that βx is the rate of new infections and βxy is the rate at which infected hosts transmit the virus to uninfected hosts). These parameters are schematically indicated in Fig. 7.1 and they provide a theoretical value for R0 ( Anderson and May 1991 Nowak and May 2000 Nowak, 2006 Woolhouse, 2017 ).
Figure 7.1 . A schematic representation of the main parameters of viral dynamics that enter the equations that predict the rate of variation of uninfected and infected (internal horizontal lines in the human figure) individuals (shaded box on the left) and the R0 value (shaded box on the right). The meaning of parameters and literature references are given in the text.
R0 values are not a universal constant for viruses because, as discussed in Chapters 3 and 4 Chapter 3 Chapter 4 , virus variation may affect viral fitness and viral load in infected individuals, and the latter, in turn, may influence the amount of virus that surfaces in a host to permit transmission ( Delamater et al., 2019 ). Despite uncertainties, consistent R0 values have been estimated for different viral pathogens based on field observations. Values of R0 for human immunodeficiency virus type 1 (HIV-1) and severe acute respiratory syndrome (SARS) coronavirus range from two to five for PV the range is 5–7, and for Ebola virus is 1.5–2.5. For the measles virus (MV), which is one of the most contagious viruses described to date, the R0 reaches 12–18 ( Heffernan et al., 2005 Althaus, 2014 ). Most isolates of the SARS coronavirus that circulated months after the emergence of this human pathogen had modest R0 values, and this is consistent with SARS not having reached the pandemic proportions that were feared immediately following its emergence. In contrast, MV is highly transmissible, thus explaining frequent outbreaks as soon as a sizable population stops vaccinating its infants. This is an important problem, fueled by antivaccination campaigns without scientific basis. Since some of the parameters that enter the basic equations of viral dynamics depend on the nucleotide sequence of the viral genome, mutations may alter R0 values, allowing some virus variants to overtake those that were previously circulating in the population ( Fig. 7.2 ). Viral replication, fitness, load, transmissibility, and virulence are all interconnected factors that contribute to virus persistence in its broader sense of virus being perpetuated in nature. These parameters can affect both disease progression in an infected individual and transmissibility at the epidemiological level.
Figure 7.2 . Displacement of a virus variant by another in the field by virtue of the latter displaying a higher R0 value. The competing viruses are depicted as horizontal lines with a distinctive symbol. Differences in R0 recapitulate part of the determinants of epidemiological fitness ( Section 5.9 in Chapter 5 ). Concepts of competition among clones or populations within infected host organisms or cell cultures, treated in previous chapters, can be extended at the epidemiological level, with the appropriate choice of the key parameters. References are given in the text.
The difference between the number of infectious particles that participate in transmission and the total number of virus in an infected, donor organism provides a first picture of the indeterminacies involved in viral transmissions. The larger the population size and genetic heterogeneity of the virus in an infected individual, the higher will be the likelihood that independent transmission events have different outcomes. Individual susceptible hosts will receive subsets of related but nonidentical genomes. In a bright article that emphasized the molecular evidence and medical implications of quasispecies in viruses, J.J. Holland and colleagues wrote the following statement: “Therefore, the acute effects and subtle chronic effects of infections will differ not only because we all vary genetically, physiologically, and immunologically, but also because we all experience a different array of quasispecies challenges. These facts are easily overlooked by clinicians and scientists because disease syndromes are often grossly similar for each type of virus, and because it would appear to make no difference in a practical sense. However, for the person who develops Guillain-Barré syndrome following a common cold, or for the individual who remains healthy despite many years of HIV-1 infection, for example, it may make all the difference in the world” ( Holland et al., 1992 ). The ever increasing number of identified human genes whose allelic forms influence viral infections provides strong support for the predictions of Holland et al. Indeterminacies in the process of virus spread can be viewed as an extension of the diversification due to bottleneck events in the case of virus transmission, as visualized in Figs. 6.1 and 6.2 in Chapter 6 , when dealing with the limitations of the virus samples retrieved from an infected host as the starting material for experimental evolution approaches.
Human Coronavirus Infections—Severe Acute Respiratory Syndrome (SARS), Middle East Respiratory Syndrome (MERS), and SARS-CoV-2
David S. Hui , . Alimuddin Zumla , in Reference Module in Biomedical Sciences , 2020
Treatment is mostly symptomatic relief of pain and fever with rest. Due to its high infectivity with a basic reproduction number between 2 and 3, hospital admission is required for isolation and supportive therapy.
Systemic Corticosteroids: The use of rescue pulsed methyl prednisolone (MP) during clinical progression was associated with favorable clinical improvement in some patients with resolution of fever and radiographic lung opacities within 2 weeks in observational studies ( Sung et al., 2004 ). However, a retrospective analysis showed that the use of pulsed MP was associated with an increased risk of 30-day mortality (adjusted OR 26.0, 95% CI: 4.4–154.8) ( Tsang et al., 2003 ). Complications such as disseminated fungal disease and avascular osteonecrosis occurred following prolonged systemic corticosteroid therapy. Data based on a randomized controlled trial (RCT) suggest that pulsed MP given in the earlier phase prolonged viraemia in comparisons to the controlled group that received normal saline ( Lee et al., 2004 ).
Antiviral agents: Ribavirin alone had no significant in vitro activity against SARS-CoV and it caused significant hemolysis in many patients ( Sung et al., 2004 ). Lopinavir and ritonavir used as initial therapy was associated with lower overall death rate (2.3% vs. 15.6%) ( Chan et al., 2003 ). Additional beneficial effects included a reduction in use of steroids, less nosocomial infections, a decreasing viral load and rising peripheral lymphocyte count. A subgroup analyses showed that in those who received LPV/r as late rescue therapy after receiving pulsed MP treatment for worsening respiratory symptoms, the outcome was not better than the matched cohort ( Chan et al., 2003 Chu et al., 2004 ).
Convalescent plasma: Convalescent plasma obtained from patients and HCWs who recovered from SARS, contains high levels of neutralizing antibody. A study of 80 SARS patients, receiving early administration of convalescent plasma showed that discharge rate at day 22 was 58.3% for patients (n = 48) treated within 14 days of illness onset, compared to 15.6% in 32 patients treated beyond 14 days ( Cheng et al., 2005 ).
Interferons: Use of IFN-α 1 plus systemic corticosteroids in SARS patients found improved oxygen saturation, and more rapid resolution of radiographic lung opacities than systemic corticosteroids alone. However, this study was uncontrolled with a small sample size ( Loutfy et al., 2003 ).
We analyze empirical sequences of contacts between people. These data sets can be divided into physical proximity and electronic communication data. The former type could be interesting for studying information and disease spreading mediated by human contacts. The latter type is primarily of interest in the context of information spreading (bearing in mind that information spreading not necessarily follows the same dynamics as infectious diseases). In all data sets, nodes are human individuals. We list some basic statistics of the data sets in Table 1.
The epidemiologic concept of R naught (R0) is much in the news of late. This number, the basic reproduction number, is being used to calculate COVID-19 transmissibility and is a key part of the discussion on when to begin allowing cities and states to reopen.
What R Naught (R0) Means
R naught (R0), the basic reproduction number, is one of the most fundamental and often-used metrics for the study of the way a disease spreads. The symbol R represents the actual transmission rate of a disease and stands for reproduction. Naught, or zero, stands for the zeroth generation (patient zero). It refers to the first documented patient infected by a disease in an epidemic.
R0 is an indicator of the contagiousness or transmissibility of infectious and parasitic agents and represent the number of new infections estimated to stem from a single case in a population that has never seen the disease before. If the R0 is 2, then one person is expected to infect, on average, two new people (Anastassopoulou et al., 2020).
To provide some perspective, seasonal strains of flu have R0s between 0.9 and 2.1. The R0 value of the 1918 flu pandemic was estimated to be between 1.4 and 2.8, and for an extremely contagious disease such as the measles, R0 is thought to lie between 12 and 18 (Healthline, 2020).
R0 is one of the key values that can predict whether an infectious disease will spread into a population or die out. It is used to assess the severity of the outbreak, as well as the strength of the medical and/or behavioral interventions necessary for control (Breban et al., 2007).
Covid-19 R Naught
The R0 originally estimated for COVID-19 was between 2.2 and 2.7, but data collected from case reports across China reported a much higher R0. Results showed that the doubling time early in the epidemic in Wuhan was 2.3 to 3.3 days. From this data, researchers calculated a median R0 value of 5.7. This means that each person infected with the virus can transmit it to 5 to 6 people rather than only 2 to 3 as previously thought (Sanche et al, 2020).
How a Virus With a Reproduction Number (R0) of 2 Spreads
R0 describes how many cases of a disease an infected person will go on to cause—in this imagined scenario R0=2. Source: The Conversation, CC BY-ND.
History of R0
Mathematical demographer Alfred Lotka developed the Stable Population Theory during the early twentieth century to study the change and growth rate of certain populations. He proposed the reproduction number in the 1920s as a measure of the rate of reproduction in a given group of people and used it to count offspring.
In the 1950s, epidemiologist George MacDonald suggested using R0 to describe the transmission potential of malaria. He proposed that if R0 is less than 1, the disease will die out in a population, because on average an infectious person will transmit to fewer than 1 other susceptible person. On the other hand, if R0 is greater than 1, the disease will spread (Eisenberg, 2020). Since then the reproduction number has become widely used in the field of epidemiology.
How R0 is Used
R0 values indicate if a disease will spread or decline within a community and how far and how rapidly transmission will occur. It can also inform public health policy decisions used to mitigate spread.
The higher the R0, the more likely the disease will become an epidemic. There are three different possibilities that can be conveyed by R0 (Healthline, 2020):
- If R0 is less than 1, the disease will not spread and will eventually die out.
- If R0 is 1, the disease will remain stable in the community but will not cause an epidemic.
- If R0 is greater than 1, the disease will spread and may cause an epidemic.
How R0 is Calculated
R0 is determined using complex mathematical equations that look at data from the disease’s characteristics and transmissibility, human behavior, how often sick and susceptible people are expected to come into contact with each other, and where the affected community is located. Scientists may also add educated guesses.
One of the ways epidemiologists calculate R0 is by using contact tracing data obtained at the onset of the epidemic. Once an individual is diagnosed, that person’s contacts are traced and tested. R0 is then computed by averaging the number of secondary cases caused by diagnosed individuals (Breban et al, 2007).
However, counting the number of cases of infection during an epidemic can be extremely difficult, even when public health officials use active surveillance and contact tracing to attempt to locate all infected persons. Although measuring the true R0 value is possible during an outbreak of a newly emerging disease, there are rarely sufficient data collection systems in place to capture the early stages of an outbreak when R0 might be measured most accurately (EID, 2019).
As a result, R0 is nearly always estimated retrospectively from sero-epidemiologic data (which looks for the presence of antibodies in the blood) or by using theoretical mathematical models. The estimated values of R0 generated by mathematical models are dependent on numerous decisions made by the modeler (EID, 2019).
When mathematical models are used, R0 values are often estimated by using ordinary differential equations, but high-quality data are rarely available for all components of the model. The population structure of the model includes people who are exposed but not yet infectious, as well as assumptions about demographics such as births, deaths, and migration over time (EID, 2019).
The Effect of Vaccination
When examining the effect of vaccination, the more appropriate term to use is the effective reproduction number (R), which is similar to R0 but does not assume complete susceptibility of the population and therefore can be estimated with populations having immune members (EID, 2019).
Efforts aimed at reducing the number of susceptible persons within a population through vaccination would result in a reduction of the R value, rather than R0 value. In this scenario, vaccination could potentially end an epidemic if R can be reduced to a value <1. The effective reproduction number can also be specified at a particular time t, presented as R(t) or Rt, which can be used to trace changes in R as the number of susceptible members in a population is reduced. When the goal is to measure the effectiveness of vaccination campaigns or other public health interventions, R0 is not necessarily the best metric (EID, 2019).
The potential size of an outbreak or epidemic is often based on the magnitude of its R0 value, and R0 can be used to estimate the proportion of the population that must be vaccinated to eliminate an infection from that population—the higher the R0, the more people must be vaccinated (EID, 2019).
Vaccination campaigns reduce the proportion of a population at risk for infection and are highly effective in mitigating future outbreaks. This conclusion is sometimes used to suggest that an aim of vaccination campaigns is to remove susceptible members of the population in order to reduce the R0 for the event to less than 1. Although the removal of susceptible members from the population will affect infection transmission by reducing the number of contacts between infectious and susceptible persons, it will technically not reduce the R0 value because the definition of R0 assumes a completely susceptible population (EID, 2019).
Cumulative Incidence Models
Another more commonly used approach is to obtain R0 from cumulative incidence data which is “the probability of developing disease over a stated period of time.” Theorists construct models based on Ordinary Differential Equations (ODEs) which describe the dynamics of the expected population size in different disease stages without tracking individuals. These types of modeling assumptions are hypothetical and cannot be verified using population-level data (Breban et al., 2007).
ODE models are formulated in terms of disease transmissibility and progression rates in the population, which yield a threshold parameter for an epidemic. The epidemic threshold is a boundary where disease equilibrium becomes unstable (R0 is greater than 1) and an epidemic may begin (Breban et al., 2007).
Calculations of R0 that use cumulative incidence data often use three primary parameters:
- The duration of contagiousness after a person becomes infected (how long the virus can be transmitted by an infected person). The longer someone is contagious, the higher the R0 is.
- The likelihood of infection per contact between a susceptible person and an infectious person or vector.
- The contact rate (the rate at which an infected person meets susceptible people).
Sometimes other parameters are added, such as the availability of public health resources, the policy environment, various aspects of the built environment, and other factors that might influence transmission.
R0 can also depend on viral characteristics, how it spreads and how long it can survive in the air and on objects. It also depends on where the virus is found in the world. According to Paul Delamater, from the University of North Carolina at Chapel Hill, “There’s a host of social, cultural, and demographic characteristics of places that would make the R naught value differ from place to place.” For any given infectious agent, the scientific literature might present numerous R0 values (EID, 2019).
Difficulties Calculating R0
Despite its place at the forefront of mathematical epidemiology, the concept of R0 has many flaws and defining it can be difficult. Few epidemics are ever observed at the precise moment an infected individual enters a susceptible population, so calculating the value of R0 for a specific disease relies on secondary methods (Li et al., 2016).
In the hands of experts, R0 can be a valuable concept. However, the process of defining, calculating, interpreting, and applying R0 is far from straightforward. The simplicity of an R0 value masks the complicated nature of this metric. Although R0 is a biologic reality, the interpretation of R0 estimates derived from different models requires an understanding of the models’ structures, inputs, and interactions. “Because many researchers using R0 have not been trained in sophisticated mathematical techniques, R0 is easily subject to misrepresentation, misinterpretation, and misapplication” (EID, 2019).
Even if the infectiousness of a pathogen and the duration of contagiousness are constant, R0 will fluctuate if the rate of human-to-human or human-to-vector interaction varies. Any factor that can influence the contact rate—including population density, social organization, and seasonality—will ultimately affect R0 (EID, 2019). Since a pandemic occurs across many different populations, geographies, and climates, the R0 may vary considerably from country to country or even within a country.
Because R0 is a function of the contact rate, the value of R0 is a function of human social behavior and organization, as well as the innate biologic characteristics of a pathogen. More than 20 different R0 values were reported for measles in a variety of study areas and periods, and a review in 2017 identified feasible measles R0 values of 3.7 to 203.3. This wide range highlights the potential variability in the value of R0 for an infectious disease dependent on local societal behavior and environmental circumstances (EID, 2019).
There are many diseases that can persist with R0<1, while diseases with R0>1 can die out, reducing the usefulness of the concept as a threshold for an epidemic. For example, it is possible that a disease can persist in a population when already present but would not be strong enough to invade. Also, the threshold value that is usually calculated is rarely the average number of secondary infections, diluting the usefulness of this concept even further (Li et al., 2011).
Many of the parameters included in the models used to estimate R0 are merely educated guesses the true values are often unknown or difficult to impossible to measure directly. This limitation is compounded as models become more complex. So, although only one true R0 value exists for an infectious disease event occurring in a particular place at a particular time, models that have minor differences in structure and assumptions might produce different estimates of that value, even when using the same epidemiologic data (EID, 2019).
Public Health Measures That Decrease R0
When the R0 of a newly emerged disease indicates that an epidemic may occur, it is important to understand the processes that can limit transmission (R) of a disease in totally susceptible people in order to prevent epidemics from starting (or to limit their size). Once a country realizes that a new virus exists, measures must be taken to interrupt the chain of infection until treatments and vaccines can be developed.
Measures used successfully in previous epidemics, which have been shown to reduce the R0 of a disease are:
- Social distancing
- Tracking and tracing of exposed people and their contacts
- Providing healthcare workers with proper protective equipment
SARS-CoV-2 continues to spread. Although we still have limited information on the epidemiology of coronavirus disease, there have been multiple reports of superspreading events. During recent severe outbreaks of SARS, Middle East respiratory syndrome (MERS), and Ebola virus disease, superspreading events were associated with explosive growth early in an outbreak and sustained transmission in later stages (Frieden & Lee, 2020, June).
Superspreading events highlight a major limitation of the concept of R0. The basic reproductive number R0, when presented as a mean or median value, does not capture the heterogeneity of transmission among infected persons two pathogens with identical R0 estimates may have markedly different patterns of transmission. The goal of a public health response is to drive the reproductive number to a value <1, something that might not be possible in some situations without better prevention, recognition, and response to superspreading events. A meta-analysis estimated that the initial median R0 for COVID-19 is 2.79 (meaning that 1 infected person will on average infect 2.79 others), although current estimates may differ because of insufficient data (Frieden & Lee, 2020, June).
Countermeasures can substantially reduce the reproductive number on the Diamond Princess cruise ship, an initial estimated R0 of 14.8 (
4 times higher than the R0 in the epicenter of the outbreak in Wuhan, China) was reduced to an estimated effective reproductive number of 1.78 after on-board isolation and quarantine measures were implemented (Frieden & Lee, 2020, June).
In Wuhan, aggressive implementation of nonpharmaceutical interventions in the community, including a cordon sanitaire* of the city suspension of public transport, school, and most work and cancellation of all public events reduced the reproductive number from 3.86 to 0.32 over a 5-week period. However, these interventions might not be sustainable (Frieden & Lee, 2020, June).
*Cordon sanitaire: a quarantined geographic area, guarded to prevent the movement of people in or out of the area.
Although superspreading events appear to be difficult to predict and therefore difficult to prevent, understanding the pathogen, host, environmental, and behavioral drivers of superspreading events can inform strategies for prevention and control. This includes:
We discuss the context, content and importance of the paper ‘The population dynamics of microparasites and their invertebrate hosts’, by R. M. Anderson and R. M. May, published in the Philosophical Transactions of the Royal Society as a stand-alone issue in 1981. We do this from the broader perspective of the study of infectious disease dynamics, rather than the specific perspective of the dynamics of insect pathogens. We argue that their 1981 paper fits seamlessly in the systematic study of infectious disease dynamics that was initiated by the authors in 1978, combining effective use of simple mathematical models, firmly rooted in biology, with observable or empirically measurable ingredients and quantities, and promoting extensive capacity building. This systematic approach, taking ecology and biology rather than applied mathematics as the motivation for advance, proved essential for the maturation of the field, and culminated in their landmark textbook of 1991. This commentary was written to celebrate the 350th anniversary of the journal Philosophical Transactions of the Royal Society.
1. Introduction and historic context
The value of mathematical models for understanding the dynamics and control of infectious disease was recognized more than two centuries ago. In 1766, Daniel Bernoulli, working on a mathematical analysis of the benefits of smallpox inoculation, wrote ‘in a matter which so closely concerns the well-being of the human race, no decision shall be made without all knowledge which a little analysis and calculation can provide’ . Bernoulli's analysis, read to the French Academy of Sciences in 1760, addressed a topic hotly debated in society and government at the time—the value of ‘variolation/inoculation’. Using the first known model of infectious disease dynamics, he showed that despite the risks to individuals, inoculation with smallpox was beneficial for society as a whole as it increased average life expectancy by more than three years, even discounting the additional deaths the preventive measure would cause. Almost 250 years after Bernoulli's publication, infectious disease dynamics has grown into a field of science, with an established core of approaches and methods and a suite of case law and generic insights gathered by studying specific infectious agents, as well as general epidemiological phenomena in animals, plants and humans. Its history is rich in the sense that many authors have explored a broad range of questions throughout the three centuries.
Although substantial work exists from elsewhere, many important developments of lasting value towards building a body of methods and insights originated in the UK in the nineteenth and early twentieth centuries. In fact, essential groundwork was already started there much earlier at the start of the seventeenth century, when data on morbidity, mortality, cause of death and population size were starting to be routinely collected, at first sparked by outbreaks of plague, and later used, for example, in calculations of life expectancy. Notably John Gaunt in his use of the ‘bills of mortality’ was an important pioneer , as was Edmund Halley, who developed the first life table and published it in the Philosophical Transactionsof the Royal Society in 1693 . Bernoulli in 1760 used Halley's life table as the basis in his calculations of the expected change in average life expectancy under inoculation against smallpox. In the mid-nineteenth century, William Farr, Registrar General in the UK, was instrumental in gathering quality data and using these insightfully, for example, in his analysis of ‘the cattle plague’ (rinderpest) .
Our aim is not to provide a historic overview, but it is not without foundation to state that the UK played a dominant role in the field's genesis. It therefore makes sense to focus on the developments in the UK to sketch the ‘line of descent’ of the main proponents of our paper, Robert M. May and Roy M. Anderson, and the tradition out of which their contributions arose. A brief historical sketch up to 1975 can be found in the seminal book by Bailey , the first substantial textbook to provide an overview of the budding field, and published just before May and Anderson started to work on the topic.
The great majority of work on infectious disease dynamics before the start of the twentieth century was driven by the desire to understand specific infectious diseases and specific public health problems. This tradition culminated in the elaborate work of Ronald Ross, who received the second Nobel Prize in Medicine for his work on the transmission of malaria. Ross wrote that the epidemiology of infectious diseases must be considered mathematically, and that the mathematical method of treatment is really nothing but the application of careful reasoning to the problems at issue . Ross introduced the fundamental insight that not all mosquitoes had to be eliminated to stop the malaria parasite from spreading, but that depression of the number of mosquitoes per human host in a population to a value below a critical level was sufficient. He expressed that critical level using parameters that could be measured in the field, and based on a model describing the proposed mechanisms of the parasite transmission process. Ross's ideas about applying mathematical reasoning to infectious disease dynamics originated from his ambition to understand malaria transmission and control. However, he was the first to develop, in an appendix (called the ‘theory of happenings’) to his 1911 book and in subsequent papers with Hilda Hudson, a general theory (which he called ‘a priori pathometry’) of infectious disease dynamics not specifically tailored to a particular pathogen or public health problem [7,8]. This marks the start of infectious disease dynamics as a scientific field, with its own research philosophy and set of tools, and it is this tradition that Anderson and May would come to systematically explore and expand. Ross's ideas on thresholds and critical community size play a large role in the way of thinking that Anderson and May adopted.
The work by Ross sparked the interest of more theoretically inclined researchers, resulting in decades of progress on mathematical tools and analysis, not specific to particular diseases or public health problems. A series of influential papers by McKendrick and Kermack in the 1930s generalized Ross's initial ideas of critical thresholds for malaria to critical size of a community of susceptible individuals necessary for an infectious disease to become established in a population . Mathematicians and statisticians started to dominate the field, with most contributors from the UK and the USA. The British mathematicians Maurice Bartlett and Norman Bailey had enormous influence in the 1950s, 1960s and 1970s, both contributing to stochastic models for infectious diseases [5,10]. Roy Anderson was a postdoctoral fellow with Bartlett at the Department of Biomathematics in Oxford in the early 1970s, possibly the first department of its kind. Bartlett established the concept of a critical community size for outbreaks to occur . This plays a large role in the analysis and discussion of results in the subject of our review, the paper ‘The population dynamics of microparasites and their invertebrate hosts’, by R. M. Anderson and R. M. May, published in the Philosophical Transactions of the Royal Society as a stand-alone issue in 1981 .
A concept related to that of the critical community size that has played a pivotal role in mathematical epidemiology is the basic reproduction number, , called the ‘basic reproductive rate’ in the Anderson & May paper and other papers from that period, and denoted there by R. It is defined as the average number of new cases of an infection caused by a single infected individual in a population consisting only of susceptible individuals, mirroring similar concepts in demography [13,14]. The idea of arose from the work by Ross on the critical threshold for malaria, but was implicit there. It was developed into a well-defined epidemiological quantity mainly by the zoologist George Macdonald (working at the London School of Hygiene and Tropical Medicine), again in the context of malaria . When an infectious agent entering a population of susceptible individuals will cause an epidemic when the infectious agent cannot spread in that population. The concept of the basic reproduction number is both simple and powerful and has become one of the most used and useful ideas in understanding infection dynamics. Under certain assumptions on how transmission opportunities in a population (i.e. the number of contacts where infection could be transmitted) change with increasing population size, one can interpret the condition as being equivalent to the condition that the size of the susceptible population into which the infectious agent is introduced is larger than the critical community size. These properties are exploited by Anderson and May throughout their paper to explain the results of their models. The concept of is closely linked to quantities such as ‘net fertility’ or ‘net reproductive rate’ in demography (introduced mainly through the work of Alfred Lotka), and ‘absolute fitness’ or ‘reproductive fitness’ in population genetics (introduced mainly through the work of Ronald Fisher and Sewall Wright), although these concepts did not evolve from each other in a linear manner . They all describe the average contributions of members of a given generation to the next generation, in terms of new infections caused, the birth of daughters, or genotypes produced.
The advances that Anderson and May brought to the growing field were twofold. First, their main aim was to explain epidemiological patterns that could be observed without concentrating on any one infectious agent in particular. This differs from the generic approach taken in the decades before, because the patterns they sought to understand were taken from empirical observations and data and the models they developed were firmly rooted in biological assumptions about mechanisms that could be behind the observed patterns. Possibly, this deviation in approach can be understood from their primary interest that did not come from mathematics or medicine, but from zoology and ecology.
The second advance was that, in contrast to the situation so far where researchers worked mainly in isolation, Anderson and May collaborated in larger groups with biologists and mathematicians, thereby establishing and educating the first real generation of dedicated epidemiological modellers. Many of their earlier collaborators and students are influential to the present day, having gained professorships and contributed further generations of students and colleagues. One can certainly speak of a dynasty of researchers in infectious disease dynamics, started by Anderson and May and now scattered over several continents. From its beginnings in understanding public health problems, Anderson and May brought the maturing field back from the once essential direction of mathematical abstraction, and firmly steered it into being a field of biology rather than applied mathematics. The driving forces became the understanding of empirically observed biological patterns, and the need for evidence-based public health decisions, for which a mathematical approach became essential.
To appreciate their approach and philosophy, it is historically important to emphasize the context in which their collaboration evolved and their careers developed, as well as the role May played in the ‘golden age’ of theoretical ecology. Both May and Anderson were part of a very active group of ecologists, entomologists, zoologists and theoreticians that arose under the guidance of Richard Southwood at the Silwood Park campus of Imperial College London. The activities and influence of this group has recently been documented , allowing us to place the Anderson & May paper in perspective.
By the time they met in the summer of 1973, May had just been appointed to a chair at Princeton University and had completed his seminal book Stability and Complexity in Model Ecosystems . This and subsequent work opened up an entirely new way of looking at phenomena in the natural world in terms of nonlinear dynamic behaviour that could be generated from simple models . Silwood Park was very attractive to May, and he established close and diverse collaborations there. There were field and empirical ecologists, entomologists and zoologists, with interesting data and problems, and who were very welcoming to a theoretician who wanted to firmly root theory in real ecology [16,19]. Anderson, a zoologist working on helminth parasites of fish, had started using mathematical models linked to empirical data and observations. Epidemiological data, particularly from parasite and invertebrate systems, were available and this possibly made the combination of epidemiology and ecology an attractive one for both May and Anderson. It seems only natural that they would strike up a close collaboration, which would turn out to be the most fruitful one of their long (and continuing) careers. Their paper in Philosophical Transactions  is the fifth in, what in hindsight can be called, a series of careful, systematic, almost tutorial-like papers that established the authors' names and influence early in their prolific collaboration, clearly outlining a structured philosophy in research into infectious disease dynamics, and culminating in the comprehensive textbook on infectious disease modelling in 1991 .
The first papers in this series were two connected publications in the Journal of Animal Ecology [20,21] dealing with macroparasites (broadly speaking worm parasites or helminths) of vertebrate hosts, together comprising 50 printed pages, and addressing the possibilities for regulation of the host population dynamics by the parasite. This seems a natural starting point for the collaboration, given Anderson's origins and interests in studying helminths, and May's interest in the stability of ecological systems. These were followed a year later by a new pair of connected publications, this time in Nature, reviewing the state of the art of the dynamics of both macroparasites and microparasites (broadly speaking, viruses, bacteria, protozoa) of vertebrate hosts [22,23].
In all of these papers, their approach originated from an ecological interest, rather than an epidemiological interest. The point is that both for ecology (where parasites and pathogens were largely ignored at the time and the emphasis was on understanding predator–prey interactions), and for (mathematical) epidemiology (where progress in understanding was guided by either interesting mathematical problems generated by unspecified infectious agents, or by assumptions that were relevant to the human context), their way of thinking brought entirely new perspectives. Assumptions such as a constant host population size were no longer tenable when looking at infections in populations other than human. Relaxing this assumption was not only a natural step when coming at this topic from ecology rather than medicine, it also opened the way to asking new questions about infectious disease dynamics and studying a much richer set of observed phenomena. Once taken, these steps also turned out to be essential for understanding infectious disease dynamics in human populations.
The step to extend the initial sets of papers to address the regulation of invertebrates by microparasites is a natural one, as May had already worked on parasitoids of insects with Michael Hassell at Silwood Park , where there was a clear interest in the topic. An illustration of this interest is that the CABI Institute of Biological Control moved to Silwood Park in 1981 . Also from the point of view of empirical data with interesting unexplained phenomena and patterns, the invertebrates and their infectious agents must have appealed as a promising topic. For example, the observation that forest insect pests, such as the larch budmoth, show long-period cycles (8–9 years) of population explosion, had already fascinated ecologists for a number of years, and there were several competing hypotheses . Anderson and May added a hypothesis of their own, leading to extensive and heated debate among ecologists in the years following its publication. Another issue that was gaining prominence at the time was the potential use of infectious agents as a means of biological control of invertebrate pest species. These ecological and applied issues can only be meaningfully investigated with models that allow for a varying population size. In the next section, we give a brief overview of the paper.
2. The population dynamics of microparasites and their invertebrate hosts
The style and content of the paper  fit naturally in the approach established by Anderson and May in their previous sets of matching papers ([20,21] and [22,23]) but this was their longest and most structured publication so far. It is 74 pages in print, including technical appendices and references, and the editors of Philosophical Transactions are to be commended on accepting the paper despite its non-standard length. Robert May in his cover letter (undated, but of 18 April 1980 according to ) expressed the hope that it ‘might be published as a free-standing issue’, but there was of course no guarantee. The alternative would have been to split the paper, as they had done previously. However, in contrast to the two earlier submissions, there is not a natural point in the paper to make the cut.
As an aside, it is interesting to note that although this paper is a very recent one in this anniversary collection of the Philosophical Transactions, a lot has changed in the 35 years since. For example, a substantial part of the paper would now be published as electronic supplementary material. In producing the paper and its results, calculations were done on a hand-held calculator programmable with a magnetic strip reader, and graphs were drawn by hand and turned into figures with Letraset (R. M. Anderson 2014, personal communication). For the journal, finding suitable reviewers was a problem. Those suggested by the authors were both from the Silwood Park group, and perhaps not seen as sufficiently independent. In any case, according to the journal archive they were not approached. Other names suggested by the editor, among them Bartlett and Kendall, were unavailable. The single (!) reviewer who eventually assessed the manuscript has passed away, and can be revealed to be John Maynard Smith. The referee report (dated 20 June 1980) reads (in full): ‘Professor Harper asked me if I would look at the enclosed manuscript. I think it is entirely suitable for publication in the ‘Transactions’. It is an important contribution to knowledge. It is clearly written, and as brief as it could be in the light of the field covered’. Although substantial discussion about assumptions, choices and conclusions would have been warranted, it is possibly only with today's knowledge that any profound criticism can be formulated. Maynard Smith was by no means an expert in infectious disease modelling, but when the paper was submitted there was relatively little modelling in infectious diseases that went beyond simple, low-dimensional models of homogeneous populations.
The paper is didactically well thought out, carefully written and comprehensive. It presents a study and systematic exploration of the influence of infectious agents on populations of invertebrate hosts that are varying in size, carefully exploiting the possibilities of simple models. It provides a set of tools and an approach that set the stage for a vast amount of subsequent work. It is a tutorial, explaining biological assumptions, observed phenomena to be studied, terminology and mathematics in detail. Its way of thinking goes far beyond speaking only to invertebrate species and their pathogens but is exemplified by the urge to explain and understand observed phenomena and patterns in biological terms, using models as tools. Careful use of low-dimensional models, staying close to data and biology, using analytical results for robustness, clever use of figures to highlight changes in the stability of steady states in parameter space, translating these results into biological terms and (in hindsight) intuition for increased understanding are all hallmarks of its philosophy.
The authors’ approach was influential, not necessarily because they were always right, but because biologists could relate to the careful reasoning. The models were explained so that they could be understood by biologists, they had links to data, and despite their simplicity generated relevant and often surprising insight. Also, concepts and new ways of looking at things were introduced and/or brought to prominence. The authors may not have always been the first, but many ideas and techniques only gained traction once they showed how they could be used effectively.
(a) The fundamental philosophy of the approach
The first 10 pages of the paper are spent carefully setting the scene for modelling the interaction between parasites and their invertebrate hosts. The model discussed would now be referred to as SIS with constant population size, in other words a logistic differential equation. The model is justified by comparison with data from an experimental epidemic that follows the familiar sigmoid curve over time. The authors recast the model in terms of prevalence of infection (proportion infected), y(t′), for a rescaled time variable, and obtain
if R > 1 the infection persists within the host population, and the prevalence approaches the value y = 1 − 1/R over time and
if R < 1 the infection cannot persist, and the prevalence approaches zero.
R is defined to be the expected number of secondary infections (βH) produced within the infectious period, 1/(α + b + γ), of one newly introduced host. That is, R is the basic reproductive rate of the parasite … , precisely analogous to the conventional ecologists' and demographers' ‘expected number of offspring’, R0…. (p. 460)
This is one of the earliest references in the work of Anderson and May to what is now the accepted concept, the basic reproduction number (or ratio), , discussed in our introduction. At the time, its use was increasing in the analysis of simple epidemiological models, for example through the work of Klaus Dietz . Somewhat surprisingly, the concept does not feature at all in Anderson and May's influential set of Nature papers [22,23].
The authors define the threshold host density, HT, as that value of H for which R = 1. Clearly, H > HT implies R > 1 and the infection persists, so HT may be interpreted as the critical community size. Although the authors are never explicit about whether H is the number of hosts in the population or a population density, they refer to a closed population and are careful to be consistent with units. The logistic model is useful for illustrating the concepts of R and HT, but has the curious property that deaths due to infection are compensated by increased birth rates, to keep the population constant. For the rest of their paper, the authors ‘break new ground’ by treating H as a dynamic variable. The idea is introduced in their discussion of ‘Model A’, the most basic of the seven models the authors discuss: Models A–G.
If host population size is not constant, one can meaningfully address the question of regulation. In the present context, this means the question whether the infectious agent is able to markedly influence the population dynamics of the host. For example, for host populations that would grow exponentially in the absence of the infectious agent, it may be that they continue to grow but at a slower rate, that they stop growing and reach an equilibrium level, or that they exhibit more complicated behaviour such as oscillations in size.
(b) Basic dynamics of host–parasite associations: Model A
In their ground-breaking paper from 1978 , the authors demonstrated that parasites could regulate a wild animal population. Starting with Model A, they investigate a similar concept in their 1981 paper—could a pathogen regulate an insect population? Assuming that the host population would grow exponentially at rate r in the absence of infection, with infection increasing the host mortality rate by α as before, they show that if α > r then the pathogen would regulate the host population size at a level
In illustrating the dependence of H* and y* on model parameters (Fig. 6 in ), the authors note that if a pathogen were to be selected for biological control, rather than seeking the most pathogenic (highest value of α), an intermediate value would result in the lowest population density. This optimum value is not given explicitly. In their discussion of Fig. 6, they remark that a large β is required for persistence when host populations turn over rapidly. This statement refers to the definition of HT, which is inversely proportional to β. Reference to Fig. 6 shows that parasite abundance, H*y*, is also inversely proportional to β when other parameters are fixed.
In this section, the authors' main concern is whether ‘natural populations of invertebrates typically have microparasitic infections capable of regulating them’ (p. 466). They acknowledge that few field studies have provided estimates of α and r, and although they present results from laboratory studies showing sufficiently large α (table 1 in ), they concede that these results may not apply to natural populations. So, although the model shows regulation to be feasible, and the data are encouraging, the cautious conclusion is that infections may ‘contribute, wholly or in part, to the regulation of their invertebrate host populations’ (p. 467).
(c) Elaborations on the model: Models B–F
The next five sections of the paper are devoted to generalizations of Model A, denoted by Models B–F. The added complications introduced in these sections do not, in the main, greatly change the conclusions from Model A. Models B and C include parasite-induced reduction of host reproduction, and vertical transmission of the pathogen, respectively. Of more interest is Model D, which includes a latent period of infection. Certain combinations of parameter values are shown to lead to stable periodic solutions instead of equilibrium values. However, in the absence of any available data to suggest this occurs for natural host–pathogen systems, the discussion is treated as of mathematical interest and confined to an appendix. For Model E, host stress due to overcrowding is linked with increased pathogenicity. This is modelled by replacing α in Model A with . The authors conclude that ‘when pathogenicity is related to host abundance, the parasite will always be capable of regulating population growth, and its main problem is to transmit itself fast enough to counter-balance the rapid death of infected hosts’ (p. 474).
In Model F, the authors introduce a second density-dependent constraint independent of the pathogen, by replacing the host mortality rate b with b0 + sH. Hence, in the absence of the pathogen the host population dynamics are governed by a logistic equation, and H tends to the carrying capacity K. Analysis of the model then leads the authors to conclude that the disease can be maintained if the ‘basic reproductive rate’ R is greater than one, where
They conclude that this requires β > s and HT < K, where
It is argued that if H > K then the logistic dynamics will reduce H below K, but H < HT implies R < 1. What is overlooked in this analysis is that H is a time-dependent state variable, not a constant. If we define the basic reproduction number to be
Fig. 13 of the paper  shows a (β, K) section through parameter space, and the locus of defining the boundary between parasite extinction and persistence. A new concept is defined: d = 1−H*/K measures the degree to which the host population size is depressed below the disease-free level by the infection. It is shown that maximum d, and hence optimal sustainable control of the invertebrate population, is achieved with an intermediate value of pathogenicity α.
(d) Free-living infective stages: Model G
In presenting Model G the authors return to the initial format of Model A, and then modify it to include a free-living stage. This is achieved by adding an equation for the ‘population of free-living infective stages’, W, assuming that an infected individual produces these stages at a rate λ, and a susceptible individual becomes infected at rate vW instead of the mass-action rate βY. In the exposition, the units of W are not specified. However, the requirement that during the infection process the free-living stages are removed at a rate vH implicitly determines the units of λ and hence W. This observation calls into question the authors' claim based on data in their Table 5 that λ is ‘always vastly greater than α + b + γ’ (p. 481), but the assumption simplifies the exposition without materially affecting the results. The authors provide a comprehensive analysis as an appendix. The major result from this section is presented in their Fig. 16. Four (α, λ) sections through parameter space each show four regimes of dynamical behaviour: pathogen extinction, pathogen persistence in a growing population, the pathogen regulating the host population to a stable equilibrium and host–pathogen limit cycles. Conditions are given for the system to be in each of these regions, and the authors conclude that ‘highly pathogenic microparasites producing very large numbers of long-lived infective stages are likely to lead to non-seasonal cyclic changes in the abundance of their invertebrate hosts and in the prevalence of infection’ (pp. 482–483). This model is then discussed in greater detail, and diagrams are presented showing how the periods of limit cycles, when they occur, vary with the model parameters. The model is applied to the dynamics of the larch budmoth, Zeiraphera diniana, and its infection with a granulosis virus. The agreement between model and data is said to be encouraging, and the authors conclude that the model is ‘sufficient to account at least for most long-term population cycles in forest insects’ (p. 490). A similar analysis, with a more extensive review of parameters, was presented by Anderson & May in 1980 .
Population cycles of forest insects are a favourite example in teaching dynamical systems (e.g. ). The reason for the outbreaks is invariably presented as a result of hysteresis generated by the fast timescale of the insect population and the slow timescale of the trees . Pathogens are not usually implicated. Bowers et al.  extended Model G by replacing exponential growth in the absence of pathogen with logistic population dynamics. Their analysis concluded that host–pathogen interactions by themselves could not generate the observed patterns, although they may contribute to their generation. The argument was taken up again by Berryman . His view was that the observed cycles were due to interactions with insect parasitoids, rather than with a virus or with the forest foliage.
(e) Dynamics in a fluctuating environment
In their next section, the authors discuss the persistence of microparasites in fluctuating host populations. Here we move away from host–pathogen interactions giving rise to cyclic behaviour, but focus on the characteristics of the pathogen that enable it to survive fluctuations in host population density. The discussion is centred around the mechanisms that may have evolved to enable microparasites to survive when the host population size fluctuates below the critical value for pathogen maintenance, HT. In the first part of this discussion, host population fluctuations are modelled as seasonal, but the authors then consider cycles generated by interactions between the host and slowly regenerating food supplies. This brings us back to the spruce budworm example. Clearly, the authors favour long-lived infective stages being an effective strategy for pathogen survival. The idea of threshold population size was an important one to emerge from the paper, and is highlighted in their Table 10 in their conclusions. This was not the last word on the subject, and the discussion has continued for more than 20 years .
(f) Biological control and evolutionary trends
Biological control of an insect pest was not a new idea in 1981 substantial modelling effort had been devoted to the problem from the early 1970s, for example by Whitten and collaborators . Prout  provides an overview of the literature at the time, where the efforts were focused on genetic control via sterile males the paper is an elaborate study of the release of sterile males in a density regulated population, using a ‘philosophy’ similar to that of the Anderson & May paper of using simple models and detailed analysis, but focusing on models with discrete generations. This literature is not cited by Anderson & May in 1981, but particularly Prout  would have deserved a mention. As far as we are aware, there was relatively little effort before their paper to model biological control using infectious agents.
In the section on biological control in the paper, it is shown that the host population would be driven to extinction if free-living infective stages of the pathogen were introduced at a rate exceeding , where is the equilibrium population size of infected hosts (with no biological control). The authors find it plausible that both these quantities could be estimated, but as we have seen the value of λ should be subject to a rescaling. The authors do issue caveats, including a note that no allowance has been made for spatial heterogeneity. While they have movement of the host population in mind, this does raise the question of appropriate scaling and units in all of the models. The authors also note that pathogens exert a selection pressure on their hosts, hence the control measure is aimed at a moving target. This leads into the final section of the paper.
Section 15 on evolutionary trends is perhaps the least convincing part of the paper. It is postulated that the ‘production of transmission stages typically entails some sort of sexual process, where genetic exchange occurs’ (p. 499). This does not sit well with the exposition of microparasite transmission, and no mention of mutation is made. The authors confess that the models in this section are very preliminary, but come to the conclusion that a parasite's optimum strategy is one of intermediate pathogenicity: a concept now widely accepted. Although they contradict themselves in saying that pathogens have a shorter generation time than their hosts, having already determined that to be a counter-productive strategy, they conclude that for invertebrates host and pathogen coevolve. This preliminary analysis paved the way for a much more detailed exposition that appeared the following year .
3. Impact and present-day developments
The volume and diversity of the literature on infectious disease dynamics, and the extent of its methodology and insights are impossible to sketch here. Instead, we very briefly highlight recent advances related to three dimensions of Anderson & May's paper. The most direct is the topic of infectious diseases of invertebrate hosts. The second is to take the applied view and look at biological control. The third is a view of the integration of ecological and epidemiological questions.
While a substantial part of the study of infectious agents in invertebrate hosts views the hosts as pest species, understanding the role of infectious agents in regulating invertebrates is rapidly becoming a conservation issue. This relates to the potential loss of ecosystem services provided by invertebrates, most notably those species essential as pollinators . The role of pathogens in bee colony collapse and bee decline is an area of intense analysis, including the use of population models [37–40]. It is also an area that has interesting parallels to the situation following Anderson & May's paper, given their new hypothesis and analysis trying to explain the long-term cycles of outbreaks of forest pests such as the larch budmoth. Many different hypotheses and contributing factors have been advanced, of which the role of infectious disease agents is one. A recent point of view from the inherent dynamics of complex systems, when a factor of influence changes slowly leading to population collapse , could apply to either area. With regard to recurrent outbreaks of insect pests, similar analyses have recently been presented, for example for the tea tortrix Adoxophyes honmai .
The dimension of biological control is still relevant today, and the issue has broadened beyond the scope envisaged in the Anderson & May paper. On top of their role as pest species, invertebrates have received growing attention in recent decades for their role as vectors for infectious diseases of animals and humans. In the paper, the idea was to study a species of infectious agent A, which could directly regulate the population of an invertebrate pest species, either preventing or reducing the size of outbreaks of the pest (i.e. host) species. One criterion could be to prevent the reproduction number of the invertebrate species from exceeding one. There is currently a different but related interest in using regulation by an infectious agent A to reduce an invertebrate population to such a level that the reproduction number of another infectious agent B, for which the invertebrate is a vector, is reduced below one. The difference is that in the former case one is only interested in depressing the host population, whereas in the latter case it may be that the infectious agent reduces the competence of individuals of the vector species in transmitting pathogen B. Competence could be reduced, for example, by reducing the lifespan of the invertebrate when infected by pathogen A, thereby reducing the infectious period for transmitting B, or by interfering with replication of pathogen B within a host that is also infected with pathogen A. Attempts to use the bacterial species Wolbachia pipientis to control Aedes aegypti mosquitoes, the main vector transmitting dengue virus between humans, are important recent examples  (for a wealth of information and a brilliant cartoon of the main idea see http://www.eliminatedengue.com/en/program). The models used in the Anderson & May paper would need significant modification to deal with these issues and the subtleties involved. Not least, the human host population for the invertebrate species should be included, as this is involved in gauging the effect of biological control by pathogen A on transmissibility of pathogen B.
Before the 1980s, infectious disease epidemiology had long been focused on understanding the interaction of a single infectious agent in a population of a single host species and has thrived because of this focus. Almost exclusively, and understandably, these species were either humans or farm animals, with much less attention paid to plants and wildlife. During the 1980s, ecological aspects were studied, for example, to understand the dynamics of infections in wildlife and consequences for wildlife conservation however, a one-on-one interaction prevailed. In recent decades, ecologists have taken a more structured approach to infectious disease agents, studying these agents in multihost settings, and more recently in ecosystems where host species and non-host species of specific infectious agents interact ecologically. Interactions between ecology and epidemiology, particularly in food webs and ecosystems, give rise to many interesting phenomena, and empirically studied systems are abundant . Theoretical studies have concentrated on infectious agents in systems consisting of one predator and one prey species. From the work initiated by May on ecosystems and stability before he (also) became interested in epidemiology, it has emerged that organization and weak interactions in food webs and ecosystems can have decisive effects on stability, even in complex systems . This raises the question whether the incredibly abundant organisms that are infectious agents may, even if they have co-evolved in ecosystems to interactions that are weak with most host species, have important roles in ecosystem structure and stability that have hitherto been unexplored. To study this, one needs a more structured approach to studying infectious agents in ecosystems, a topic that is likely to attract substantial attention from ecologists and mathematical modellers in coming decades . Such studies are also needed if we are to understand the emergence of infectious agents in new hosts (such as humans) from co-evolved ecosystem settings, especially in response to anthropogenic changes in these ecosystems. Some 40 years after May first addressed the stability of multispecies communities, and some 30 years after he and Anderson started to transform infectious disease epidemiology, the two fields can now be merged and studied in a way that would satisfy them both.
The study of infectious disease dynamics has grown into a proper active and important field of research. Although many researchers have contributed to its genesis, the many ways in which Anderson and May contributed have been essential. They were very active in research, with a keen eye for areas that needed attention, frequently sparking interest by a wider community—reflecting the view by May of himself as an ‘R-selected researcher’, quickly exploring new territory before moving on. They interacted with biologists, and later the medical and public health communities. They organized influential meetings (e.g. [47,48]), training a generation of influential epidemiological modellers (and indirectly an ever growing second generation). Of course, more people in the field combined several or all of these qualities, but never this extensively or effectively. And most of that first generation would admit to being influenced and inspired by the five seminal publications of which this 1981 paper was one.
Hans Heesterbeek is professor of Theoretical Epidemiology at Utrecht University. His current research interest is on the dynamics of infectious agents in food webs and ecosystems, as well as on the history of infectious disease epidemiology. He is a co-author of two textbooks on mathematical modelling of infectious diseases and is currently writing a book on the history of the study of epidemics. He is a founder and editor-in-chief (with Neil Ferguson) of the journal Epidemics, and editor of Proceedings of the Royal Society B. He has known May and Anderson since 1993, when he was a postdoc with Robert May at the department of Zoology of the University of Oxford at the time that Roy Anderson became head of department. In the same year, he met his current co-author Mick Roberts, with whom he has very productively collaborated ever since.
Mick Roberts is Professor in Mathematical Biology at Massey University, Auckland, New Zealand. His research interest is to understand the epidemiology of infectious diseases and optimize strategies for their control, using modern methods of mathematical analysis and developing new methods as necessary. He was previously a scientist at the Wallaceville Animal Research Centre near Wellington, where he was the leader of programmes in modelling infectious diseases and parasitology. He has held visiting fellowships at Oxford, Cambridge and Utrecht Universities. Professor Roberts was elected a Fellow of the Institute of Mathematics and its Application (FIMA) in 1992, and a Fellow of the Royal Society of New Zealand (FRSNZ) in 2008. He received the New Zealand Mathematical Society Research Award in 2006. He is the author of over 120 peer-reviewed publications, including more than 90 journal articles.
5. A model for West Nile virus
Some diseases, for example, West Nile virus, dengue fever, malaria, Zika virus are transmitted through a vector (for these diseases the vectors are various species of mosquitoes), rather than directly from person to person. Female mosquitoes bite to obtain a blood meal that is essential for reproduction, so only female mosquitoes need be considered. West Nile virus can kill birds and humans, but infected mosquitoes remain infectious for life and do not die of the virus. Birds can transmit the virus back to mosquitoes, whereas humans appear to be dead end hosts, and so may be excluded from a simple model. Since the life-cycle of mosquitoes is much shorter than that of birds, mosquito demography should be included in a model, but bird demography can be ignored. For a simple mosquito-bird (vector-host) model, take S and I compartments in each population, giving a system of four ODEs as formulated by Wonham and Lewis (2008). The equations for this system with nonnegative initial conditions are
where the variables and parameters are defined as
- • SB, SM: number of susceptible birds, mosquitoes
- • IB, IM: number of infectious birds, mosquitoes
- • αB, αM: probability per bite of virus transmission to bird, mosquito
- • β: biting rate of mosquitoes on birds
- • δB: bird death rate from virus
- • bM, dM: mosquito birth, natural death rate
The disease is transferred by an infectious mosquito biting a susceptible bird, or by a susceptible mosquito biting an infectious bird, This cross infection between mosquitoes and birds is illustrated in the flow diagram in Fig.ਃ . The transmission is assumed to be frequency dependent see Wonham and Lewis (2008) for more discussion on the model and transmission assumptions.
Flowchart for the West Nile virus model by Wonham and Lewis (2008).
Assuming that bM = dM so the bird population is constant, there is a DFE with all birds and mosquitoes susceptible, with populations denoted by SB = SB0, SM = SM0 and IB = IM = 0. Using the next generation matrix at the DFE
The first ratio under the square root represents the number of bird infections caused by one infectious mosquito, and the second represents the number of mosquito infections caused by one infected bird. The square root represents a geometric mean. In the literature the square root is often omitted, giving the same threshold for stability at 1, but taking the average number of secondary infected humans resulting from a single infected human see, for example, Roberts and Heesterbeek (2003).
The relative importance of each parameter for control can be estimated by computing elasticity indices. For this model, ϒ α B ℛ 0 = ϒ α M ℛ 0 = 1 2 , ϒ β ℛ 0 = 1 , ϒ d M ℛ 0 = ϒ δ B ℛ 0 = − 1 2 . Thus reducing the biting rate of mosquitoes has the largest proportional effect on reducing ℛ0.
Targeting the mosquito to bird transmission, the target reproduction number has S = < ( 1,2 ) >, thus
If the transmission from mosquitoes to birds can be reduced by a fraction at least 1 − 1 / ℛ 0 2 , then the vector-host disease will die out.
This simple model has in particular neglected the period during which mosquitoes are in the larval stage, and also the exposed period of infected mosquitoes during which the viral load becomes sufficiently high for bites to be able to transmit the disease. Both these periods are significant fractions of the mosquito life-span. Including the two extra compartments of larval and exposed mosquitoes, gives a 6-dimensional ODE system, as formulated in Wonham and Lewis (2008). The resulting ℛ0 is not changed by the larval stage, but the exposed class reduces ℛ0 by a factor that is the square root of the probability of an exposed mosquito becoming infectious. Numerical model simulations in Wonham and Lewis (2008) show that the inclusion of these classes delays the disease outbreak.
Jiang, Qiu, Wu, and Zhu (2009) consider a bird-mosquito West Nile virus model (proposed by Bowman, Gumel, van den Driessche, Wu, and Zhu (2005)) and find that backward bifurcation is possible for some parameter values, thus the initial numbers of mosquitoes and birds are important in determining whether or not the disease dies out, even if ℛ0 < 1. More complicated vector-host models require appropriate parameter values to estimate elasticity indices and so to guide control planning see, for example, Manore etਊl. (2014) for a comparison of dengue and chikungunya dynamics, and Cai, Li, Tuncer, Martcheva, and Lashari (2017) for a malaria model that may also exhibit backward bifurcation. Temperature plays an important role in the reproduction of many vectors that transmit vector-borne diseases. Thus climate change and variations in temperature may have an important influence on ℛ0 and disease persistence. A detailed study of this effect for Lyme Disease, which is transmitted by ticks, using data from northeastern North America is given by Ogden etਊl. (2014), and concludes that climate warming may have partly driven the emergence of Lyme disease in this region. More recently, Wang and Zhao (2017) develop a periodic time-delayed model of Lyme disease, compute ℛ0 from data, and show that ℛ0 can be driven below 1 if the recruitment rate of tick larvae is reduced.
In this dissertation, we focus on the development and analysis of time-delayed mathematical models to represent real world applications in biology and epidemiology, especially then analyze the models using various theorems and methods in the literature, such as, the comparison principle and the method of fluctuations, to study qualitative features of the models including existence and uniqueness of solutions, boundedness, steady states, persistence, local, and global stability, with respect to the adult/basic reproduction number Ra/R₀, which is a key threshold parameter. Firstly, we discuss ecological models in Chapters 2-4. In Chapter 2, we derive a single species-fish model with three stages: juveniles, small adults and large adults with two harvesting strategies depending on the size and maturity. We study the population extinction and persistence with respect to Ra and find that the over-harvesting of large matured fish after a certain age can lead to population extinction under certain circumstances. Numerically, we investigate the influence of harvesting functions and discuss the optimal harvesting rates. In Chapter 3, we develop a model for the growth of sea lice with three stages such that the development age for non-infectious larvae to develop into infectious larvae relates to the size of adult population size. As a beginning, we describe the nonlinear dynamics by a system of partial differential equations, then, we transformed it into a system of delay differential equations with constant delay by using the method of characteristics and an appropriate change of variables. We address the system threshold dynamics for the established model with respect to the adult reproduction number, including the global stability of the trivial steady state, persistence, and global attractivity of a coexistence unique positive steady state. As a case study, we provide some numerical simulation results using Lepeophtheirus salmonis growth parameters. To explore the biological control of sea lice using one of their predators, "cleaner fish", we propose a model with predator-prey interaction at the adult level of sea lice in Chapter 4. Mathematically, we address threshold dynamics with respect to the adult reproduction number for sea lice Rs and the net reproductive number of cleaner fish Rf, including the global stability of the trivial steady state with Rs < 1, global attractivity of the predator-free equilibrium point when Rs > 1 and Rf < 1, persistence and coexistence of a unique positive steady state when Rs > 1 and Rf > 1. Furthermore, we discuss the local stability of the positive equilibrium point and investigate the Hopf bifurcation. Numerically, we compare between two cleaner fish species, goldsinny and ballan wrasse, as a case study. For epidemiological models, in Chapter 5, we propose an SEIRD model for Ebola disease transmission that incorporates both the transmission of infection between the living humans and from the infected corpses to the living individuals, with a constant latent period. Through mathematical analysis, we prove the globally stability of the disease-free and a unique endemic equilibria with respect to R₀. Moreover, we find that the long latent period or low transmission rate from infectious corpses may reduce the spread of Ebola. In Chapters 6, we consider the influence of seasonal fluctuations on disease transmission and develop a periodic infectious disease model where asymptomatic carriers are potential sources for disease transmission. We consider a general nonlinear incidence rate function with the asymptomatic carriage and latent periods. We implement a case study regarding the meningococcal meningitis disease transmission in Dori, Burkina Faso. Our numerical simulation indicates an irregular pattern of epidemics varying in size and duration, which is consistent with the reported data in Burkina Faso from 1940 to 2014. In summery, in population growth models, we find that the basic reproduction ratio depends on maturation time, indicating that this key parameter can play an important role in population extinction and persistence. In disease transmission model, we understand that latent period can play a positive role in eliminating or slowing a disease spread.
Exponential and Logistic Growth
A common discussion in epidemics involves exponential growth. This type of growth is very rapid, and increases over time. Exponential growth is very powerful. Take an infection that doubles once a week. There may only be one infection to start, but you very quickly end up with a large number of infections. However, growth cannot remain exponential forever. And even in the simplistic models discussed above, we see that there is an inflection point, where the curve moves from being exponential to being less than linear. The real shape of this curve is known as a logistic curve.
Our findings show that the basic reproductive number (R0) is associated with population density, even when percent of individuals that use private transportation and median income were accounted for. In these settings, greater population density may potentially facilitate interactions between susceptible and infectious individuals in densely-populated networks, which sustain continued transmission and spread of COVID-19. Moreover, we see that population density continues to have an important impact on disease transmission regardless of transportation accessibility and median income, suggesting that the opportunity for effective contacts are mostly driven by crowding in denser areas, increasing the contact rates necessary for disease spread. However, we did not see that density-dependence is differential across transportation accessibility, as demonstrated by the non-significant interaction of population density and transportation. Our findings are consistent with previous research that have demonstrated a strong relationship between population density and other infectious diseases [4, 6, 7, 9, 35]. In the current SARS-CoV-2 pandemic, recent literature has been conflicting, where some research also suggests a density-dependence of COVID-19 transmission [17, 36] and other measures of the severity of the outbreak [19, 37], while other research suggests that there are other factors that can better explain the pandemic [18, 38]. However, to our knowledge, our results are one of the first to show that population density is an important driver of COVID-19 transmission, even in areas where residents rely more on private modes of transportation. Moreover, even though transmission is less in lower density areas (i.e. rural areas), rural settings may eventually disproportionately be more vulnerable to COVID-19 morbidity and mortality. Individuals in rural areas are generally older, have more underlying conditions, have less access to care, and have fewer ICU beds, ventilators, and facilities needed for severe COVID-19 treatment [39–42]. Further research is needed on the overall burden of COVID-19 across the spectrum of population density.
Geographic estimates of R0 of SARS-CoV-2 need to take into account the specific area’s population density, since the R0 estimate is dependent on both the pathogenicity of the virus as well as environmental influences. In countries where cases are only on the starting to climb, such as countries in Latin America and Southern Africa [1, 43], or there is a resurgence of cases, such as India, Iraq, and Israel , area-specific density can assist in predictions of R0, which is important because epidemiological forecasts and predictive models are sensitive to small changes in R0 inputs. Accurate estimation of R0 consequently leads to more precise approximations of the epidemic size, so that governments can appropriately allocate resources and coordinate mitigation strategies. Moreover, as cities and states reopen in the United States, and if there is a second-wave of infections, areas with higher density accessibility will likely have greater SARS-CoV-2 resurgence.
Our study has a number of limitations. While we demonstrate that population density is associated with R0, the estimation of R0 can be biased depending on the data and assumptions adopted. However, our main aim in this analysis was to evaluate the association between population density and R0, and not to accurately estimate R0. Thus, any biases in estimation of R0 due to underlying assumptions would likely be non-differential across counties, and would still yield similar results. In addition, we estimated R0 based on the number of reported cases therefore, the incidence of COVID-19 across US counties may be underestimated at varying rates due to differential testing. Testing data at the county-level currently do not exist, and we were unable to adjust for the number of tests performed. Confounding of true epidemic growth by increase in testing could also be a potential constraint to the robustness of the analysis. To mitigate this limitation, we included a random intercept term to adjust for state-level effects, and thus differential testing across states were accounted by our model. Differential testing by local governments within states are less likely to strongly impact our findings, as most funding and budgets for COVID-19 is distributed at the state-level [45, 46]. We also conducted a sensitivity analysis using death data which demonstrates the robustness of our findings. Furthermore, we utilized a number of assumptions based on previous findings to calibrate the exponential growth period, which ensured that the virus had taken hold and allowed a sufficient number of days and case counts to estimate exponential growth. There are potential for biases in our method for example, there is the possibility that some NPIs were introduced in the initial outbreak stage of COVID-19 in some counties however, if this was the case, then case counts and subsequently R0 would even higher than we calculated, and thus our associations of density and R0 was underestimated. However, we implemented numerous ways to limit the biases. The exponential growth period was restricted to approximately 14 days at the start of the epidemic, where we would expect limited increases in testing and thus would not affect R0 substantially. Moreover, we plotted the calibrated exponential growth curves of all the counties included in our analysis, which gave us reasonable curves that approximated exponential growth for case and death data. Another limitation is that we had to only include counties that had sufficient case data in order to accurately estimate R0 however, if we included all counties, the true association between population density and R0 would likely be greater than what we report in our analysis given our findings that the counties excluded in the analysis had a significantly lower density and expected very low R0 due to lack of cases. Another limitation is that our model also assumes homogenous mixing, which may can be an oversimplification of the heterogeneity in contact patterns within populations [4, 47]. However, previous research has shown that population structure only changes R0 estimates slightly , and assumptions of well-mixed populations are valid in small-to-medium spatial scales . Moreover, our method loses spatial granularity in assessing R0 in counties, especially in counties with spatially heterogenous clustering. The aim of our study, however, was to provide a generalizable estimate of the association between population density and R0, in order to appropriately estimate potential for disease transmission, rather than a microspatial estimate that may not be generalizable to other settings. Finally, an important confounder that we were unable to adjust for is the number of importations of SARS-CoV-2 in these counties, as more urbanized areas are more likely to have links with countries and other states where the virus could have originated from. Even so, we still see that once an area is seeded with COVID-19, the growth rate is greater in denser areas during the time period prior to implementation of NPIs.