The disease models used to guide policy for the COVID-19 pandemic must capture key complexities of transmission.
Read the Reflection, written 4 August 2021, below the following original Transmission.
The study of infectious disease dynamics can be divided into two areas. The first is pathology, which focuses on changes in the hosts due to the presence of the pathogen (e.g., dry cough, high fever, and acute respiratory syndrome). The second is transmission, or how the pathogen moves from an infected individual to an uninfected one in order to initiate a new infection. In sharp contrast to pathology, we almost never see transmission occurring — we can only infer that it has occurred when a healthy individual develops signs of infection. We then need to trace back their activities in the hopes of identifying an individual with symptoms that led to the transmission event.
This post will supply an overview of the problems that beset epidemiologists when we try to measure transmission, while familiarizing the reader with some key models used to measure transmission, and to prevent it.
Models that are used to describe the dynamics of infectious diseases fall into four broad categories:
• microparasites and macroparasites; [1-3]
• vector-borne diseases (VBDs);
• sexually transmitted diseases (STDs); and
• other infectious diseases (OIDs). 
An overview of the first three categories appears in the extended version of this post. For the purpose of understanding SARS-CoV-2, the virus that causes COVID-19, we shall jump to the fourth category — Other Infectious Diseases (OIDs).
OIDs are transmitted by free-living particles, expelled from the infectious hosts, that directly infect the susceptible hosts. The duration of time these infective stages last in the free-living stage is crucial — in the case of influenza and SARS-CoV-2, coughing and sneezing release a cloud of infective particles into the air that may only be infectious for a couple of minutes. In contrast, bacteria such as anthrax produce spores that may survive in the soil for decades. The parasitic worms that live in the guts of most vertebrates produce eggs that can survive for weeks to months. This creates a bifurcation in transmission mode; when infectious stages are short-lived, then transmission from one infected individual is a function of the density of susceptible hosts in their vicinity. You infect many more people when you sneeze in a crowded subway car than when you sneeze in a nearly empty one. This form of transmission, usually called density-dependent transmission, is assumed to be linear, with each infected individual infecting a fixed proportion of the susceptible individuals in their vicinity. In contrast, pathogens with long free-living stages, as well as vector-borne pathogens and STDs, tend to have saturating transmission functions: mosquitoes have to digest between blood meals, and even Casanova had to rest or dine occasionally. This form of transmission is modeled by frequency-dependent functions with the product of susceptible and infected hosts appearing in the numerator, and the total population in the denominator.
The shotgun splatter of direct transmission is not very specific. Occasionally, the pathogen infects the wrong host — this can occur when different species are forced to interact artificially when captured for food or the pet trade. These animals are often stressed, which leads them to release significant numbers of infective stages that can contaminate humans involved in the trade. Usually, the pathogen is unable to survive in the new hosts, as the cells it needs to infect in order to replicate are absent. However, when a pathogen does manage to infect novel cells, it can lead to the emergence of a new disease. This seems to be what is happening with COVID-19. Genetic evidence suggests its natural host is a bat species,  or possibly a pangolin. These species have very different physiology from humans (and most of our domestic livestock species); we rarely see any overt pathology in bats infected with these pathogens. This can change dramatically when the pathogen finds itself in the wrong host.
Bats have very different immune systems from other mammals, likely as a consequence of their ability to fly. [6,7] Humans and other non-volant mammals produce the B-cells of their immune system in their bone marrow. Because bats fly, they have hollow bones; the only place they have bone marrow is in their pelvises, so they produce B-cells at much lower rates. Similarly, active flight raises their body temperature to levels akin to fever in non-volant mammals, possibly constraining viral growth. Bats also do not store fat, as it compromises their aerodynamic ability. Instead, they can enter torpor to get through periods of limited food resources. These all act as constraints on viral pathogens that disappear when the pathogen finds itself in a novel host whose immune response may interact with that pathogen in ways that are detrimental to both the host and the pathogen. 
The dynamics of generalist pathogens provide important insights into the transmission dynamics of pathogens in structured human populations. Consider a pathogen that can infect multiple host species, each of which has a different body size, and thus different birth and death rates and population densities. The species with the smallest body size will have the highest birth and death rates and population density. The largest will have the opposite. If the pathogen follows simple dynamics, with within-species transmission far exceeding between-species transmission, then each host will interact independently with the pathogen and each will exhibit its own epidemic cycles: large and frequent outbreaks in hosts with low body mass, and slow, less dramatic cycles in larger hosts.  As we increase the relative rates of between-species transmission, these cycles will die out. Additionally, any tendency for epidemic outbreaks to occur is buffered by the pathogen’s constant jumps between host species, preventing any one species from becoming too abundant. If we increase between-species transmission to levels where it matches within-species transmission, then the small species can use the pathogen to drive the larger species extinct; small species are abundant and recover quickly from outbreaks, while rarer large species cannot recover from frequent epidemics. Ultimately, only the smallest species survive, and they revert to the epidemic behavior they exhibited when between-species transmission was rare.
This exercise suggests that understanding rates of between-species transmission is an additional, vital component of disease dynamics. To that end, Who Acquires Infection From Whom (WAIFW) matrices provide a framework to examine how the pathogen moves between different groups of hosts and allows us to identify which section of the population acts as a reservoir to maintain the infection and which are subject to spillover events. The matrices were originally developed to study the transmission of pathogens such as measles between different age classes in human populations. [10, 11] They were crucial in understanding how HIV moved between different sections of the population during the AIDS epidemic,  and they will be central to understanding the efficacy of the social distancing now being put in place for the COVID-19 epidemic.
These matrices are informing the social distancing guidelines now being put in place for the COVID-19 epidemic. Essentially, we are trying to massively reduce the strength of interactions across all age classes and completely remove the interactions between young and old in order to protect people who seem to be more susceptible once infected. Quantifying the structure of these transmission matrices is crucial for understanding the size of the epidemic and how to control its spread.
Santa Fe Institute
- Anderson, R.M. and R.M. May, Population biology of infectious diseases: Part I. Nature, 1979. 280: p. 361-367.
- May, R.M. and R.M. Anderson, Population biology of infectious diseases: Part II. Nature, 1979. 280: p. 455-461.
- Anderson, R.M. and R. May, Infectious Diseases of Humans. 1992: Oxford University Press.
- Lockhart, A.B., P.H. Thrall, and J. Antonovics, Sexually transmitted diseases in animals: ecological and evolutionary implications. Biological Reviews, 1996. 71(3): p. 415-471.
- Andersen, K.G., et al., The proximal origin of SARS-CoV-2. Nature Medicine, 2020.
- Dobson, A.P., What links bats to emerging infectious diseases? Science, 2005. 310: p. 628-629.
- Brook, C.E. and A.P. Dobson, Bats as ‘special’reservoirs for emerging zoonotic pathogens. Trends in Microbiology, 2015. 23(3): p. 172-180.
- Brook, C.E., et al., Accelerated viral dynamics in bat cell lines, with implications for zoonotic emergence. eLife, 2020. 9: p. e48401.
- Dobson, A., Population dynamics of pathogens with multiple host species. American Naturalist, 2004. 164(5): p. S64-S78.
- Bolker, B. and B.T. Grenfell, Space, Persistence And Dynamics Of Measles Epidemics. Philos Trans Roy Soc B, 1995. 348: p. 309-320.
- Grenfell, B.T. and R.M. Anderson, The estimation of age-related rates of infection from case notifications and serological data. Journal of Hygiene (Cambridge), 1985. 95: p. 419-436.
- Anderson, R.M., et al., The influence of different sexual-contact patterns between age classes on the predicted demographic impact of AIDS in developing countries. Annals of the New York Academy of Sciences, 1989. 569: p. 240-274.
Read more posts in the Transmission series, dedicated to sharing SFI insights on the coronavirus pandemic.
Listen to SFI President David Krakauer discuss this Transmission in episode 27 of our Complexity Podcast.
August 4, 2021
Understanding the Complexities of Transmission Is Key to Controlling Viral Pandemics
COVID-19 has fulfilled the traditional Chinese definition of chaos: an event that creates both crisis and opportunity. For those interested in epidemiology and the dynamics of infectious agents, boundless opportunity has appeared: over fifty thousand COVID-19 papers have appeared since a December 30, 2019, report on ProMED described a cluster of patients around the Wuhan market in China with an unusual respiratory disease.1 Since then, more than four million confirmed cases have been fatal, and more than two hundred million people have been infected. Perhaps ten times as many as this have actually been infected, and we will probably never know how many additional fatalities have occurred indirectly because of COVID-19. The spread of the virus has provided deep and vital insights into multiple complex systems, particularly the structure of all currently extant human societies and the transportation networks that facilitate connections between them. Papers referring to SEIR (Susceptible–Exposed–Infected–Removed, where Removed signifies recovered, resistant, or deceased) models have been published by epidemiologists, but also people with little background in public health, from physicists through mathematicians to almost everyone who can spell SEIR; this has created considerable insight and significant confusion. In my earlier Transmission, developed in the first months of COVID-19, I raised questions about what we already knew as epidemiologists that might be helpful. Here, I’d like to focus on other important considerations that are central to understanding more subtle complexities of the dynamics and evolution of COVID-19.
The study of infectious disease dynamics can be divided into two areas: (1) pathology, which focuses on the changes in the physiology and abundance of the pathogen’s hosts due to its presence, and (2) transmission, which infers how the pathogen moves from an infected individual to an uninfected one. We almost never see transmission occurring; we can only infer retrospectively that it has occurred when a healthy individual develops signs of infection. We then need to trace back their activities in the hopes of identifying an individual, with or without symptoms, that led to the transmission of the virus to the known infected individual and potentially other individuals not yet showing symptoms. Understanding transmission matters.
The traditional epidemiological framework is designed mainly to consider transmission rates between people of unspecified age. The emergence of HIV extended this framework to consider rates of interaction between people with different categories of sexual activity, as well as activities such as needle-sharing between intravenous drug users and potential transmission by blood transfusion in the early stage of the epidemic. Now, the models used to guide policy for the COVID-19 pandemic must capture key details of transmission; increasingly, we realize that these are driven by rates of interactions between people of different ages who operate in different sectors of the economy and interact at different social locations and at different intensities. Determining these rates of mixing and interaction will be a major challenge for the next generation of COVID-19 models. These epidemiological models will then need to be combined with economic models for a multisector economy; hybrid economic–SEIR models can then examine how to reduce activities in different sectors of the economy in order to reduce transmission at a minimum cost to economic productivity. Initial exploration of such models creates almost an entirely new set of interdisciplinary studies in econo-epidemiological complexity.2
WHO ACQUIRES INFECTION FROM WHOM?
WAIFW matrices (Who Acquires Infection From Whom) provide a framework to examine how a pathogen moves between different groups of hosts and allow us to identify which sections of the population act as a reservoir to maintain the infection and which are subject to spillover events. The matrices were originally developed to study the transmission of pathogens such as measles between different age classes in human populations.3, 4 They were crucial in understanding how HIV moved between different sections of the population during the AIDS epidemic.5 They will be central to understanding the efficacy of the social distancing introduced for the COVID-19 epidemic. Social distancing is, essentially, an effort to significantly reduce the strength of interactions across all age classes and completely remove the interactions between young and old age classes in order to protect people who seem to be more susceptible once infected. Quantifying the structure of these transmission matrices is crucial for understanding the size of the epidemic and how to control its spread.
Social mixing and disease transmission are strongly determined by contact rates between people working in different economic sectors. During the COVID-19 pandemic, a major challenge has been how to restructure WAIFW matrices to reflect those interactions. Age is less important than economic welfare in determining the structure of these matrices. For example, people working in service industries—places like bars and restaurants—will be in contact with many more people than those who can readily work from home. Some agricultural workers will be relatively isolated, while others will be working intensively with others, often in crowded conditions. Children and students add additional levels of contact across these economic networks, both with each other and their teachers at school or college, and with their parents and grandparents at home. The dynamics of measles, mumps, and chicken pox are driven by the mixing activities of children until they are either vaccinated or infected and recover as immune hosts. COVID-19 (as well as other emergent pathogens) has very different dynamics, as everybody is initially susceptible and transmission can occur wherever people gather and interact. If we can quantify the background level of contact between people in different economic and educational sectors, we should be able to examine both the economic and epidemiological impact of partially closing down or constraining different sectors of the economy.
In the simplest epidemiological models for a respiratory pathogen like COVID-19, we can divide the population into two parts: those who are willing to follow recommended public health guidelines and those who can not (or will not). We can also assume that rates of mixing within each of the two sections of the population are higher than between the two parts. Mask-wearing reduces transmission, so this will reduce infection numbers in the mask-wearing proportion of the population. Vaccination also reduces transmission and significantly reduces pathology and resultant mortality. In both cases, if there is any mortality associated with infection, the group that is mask-wearing and/or vaccinated will increase in relative abundance to those not undertaking either of these two actions. Over a four-year electoral cycle in a population that is initially evenly split between the two behaviors, this would eventually give mask-wearers and the vaccinated a significant numerical electoral advantage over those refusing to participate in these interventions. Why would their political leaders encourage them to avoid these activities? More subtly, if there is mixing between vaccinated and nonvaccinated hosts, it is likely that exposure of vaccinated hosts to infected, nonvaccinated hosts will boost the immunological protection of the vaccinated hosts, while concomitantly reducing the life expectancy of those not vaccinated. This is an unwitting and paradoxical form of altruism, with one group sacrificing their personal health to boost the immunity of others.
SUBPOPULATIONS OF HOSTS
COVID-19 dynamics have been subtly different in rural and urban areas, mainly because of differences in population density, but also because facilities to identify and treat people are much diminished in rural areas, particularly in sub-Saharan Africa, India, and South America. This generates weakly coupled sets of epidemics in areas and countries where subpopulations are coupled together at different levels of interaction. The dynamics of generalist pathogens provide important insights into the transmission dynamics of pathogens in structured human populations.6 Consider a pathogen that can infect multiple host species, each of which has a different body size and thus different birth and death rates and population densities. The species with the smallest body size will have the highest birth and death rates and population density, while the one with the largest physical size will have low birth and death rates. If the pathogen follows simple SEIR dynamics and we initially assume that between-species transmission is much lower than within-species transmission (<10-3), then each host will interact independently with the pathogen and each will exhibit its own epidemic cycles: large and frequent outbreaks in the hosts with low body mass, slow and less dramatic cycles in the case of larger hosts.7 As we increase the relative rates of between-species transmission (10-3–10-1), these cycles will die out and any tendency for epidemic outbreaks to occur will be buffered by the pathogen constantly jumping between host species and preventing any one species from becoming too abundant. If we increase between-species transmission to levels where it matches within-species transmission, then the small species can unwittingly use the pathogen to drive the larger species extinct; small species are abundant and quickly recover from outbreaks, while large species become rare and cannot recover from frequent epidemics. Ultimately only the smallest species persist, and they revert to the epidemic behavior they exhibited when between-species transmission was rare.
Similar dynamics are likely to be important in three areas of COVID-19 dynamics where heterogeneity creates weakly coupled structures within the host population.
Geoffrey West and Chris Kempes have suggested that city size is crucial in determining the dynamics of COVID-19 and other directly transmitted viral pathogens (this volume/these essays). I strongly suspect that the transmission dynamics of cities of different sizes are coupled together in similar ways as different species of hosts are coupled together. This would imply that small to intermediate levels of mixing by people commuting between different cities, towns, and villages would strongly buffer any innate tendency for cities to exhibit epidemic cycles at different frequencies. Instead, we would see a more homogeneous, low-level epidemic, with much longer persistence and no dominant epidemic frequency. Interestingly, this also suggests that when we observe differences in the epidemic dynamics of a pathogen in different-sized cities, this provides an alternative and indirect test of the efficacy of lockdown at each location. Similar logic applies to the dynamics of a pathogen in countries isolated from each other by transportation restrictions.
The rapid development of vaccines against COVID-19 has been a major scientific success. Logistical constraints, religious beliefs, and blatant misinformation have led to much lower rates of vaccination uptake than are needed to attain herd immunity. It now also seems that vaccinated hosts can acquire infection and transmit the virus, but for much shorter periods of time than those who have not been vaccinated. Their mortality rates are also significantly reduced. This will again generate heterogeneity in the epidemiological structure of the host population, buffering the tendency for epidemics in a fashion similar to that described for host species of different physical size, or cities of different size.
The physical size of the virus may also be important in determining key aspects of the dynamics of the epidemic. There is a curious and poorly explored relationship among the physical size of pathogens, the efficacy of vaccines, and the duration of immunity: measles seems to lie at a perfect “sweet spot” on a spectrum producing lifetime immunity from either a highly efficacious vaccine or natural infection. Larger viruses, bacteria, protozoa, and parasitic worms produce immunity of shorter durations, with vaccines of diminishing efficacy (Dobson et al, in prep). Ironically, viruses that are physically smaller than measles, such as influenza, produce prolonged immunity, but mutate so rapidly that new strains appear at rates that are increasingly less recognizable to the human immune system. Curiously, there is a 99% correlation between physical size and genome size for all parasitic organisms, suggesting that efficacy of immunity is mainly a function of the physical size of the pathogen and some scaling law underlies the relationship among virus size, vaccine efficacy, and duration of immunity. COVID-19 is a very large virus when compared to measles,8 and the immunity of the commonly circulating coronaviruses is relatively transient (around a year). Although vaccinated and previously infected hosts do seem to recover more readily from challenge infections, it seems likely that vaccination against COVID-19 will need to be an annual event. This will require significant capacity building and infrastructure investment across the entire biomedical enterprise.
EVOLUTION OF NEW STRAINS
The emergence of new strains of the virus was inevitable; viruses have a high mutation rate and constantly produce novel varieties, the vast majority of which are dysfunctional and fail to replicate. Occasionally, a functional novel variant will appear by mutation and it will quickly replace currently existing strains if it is either more transmissible or if the hosts it infects transmit the virus for longer periods of time. When levels of immunity in the host population are low, only novel strains with higher transmission success can persist and replace older strains. 9 As the appearance of these strains is essentially random, the rate at which novel strains appear will vary directly with the number of people infected. If you halve the number of people infected, then it will take twice as long for new strains to appear, while if you reduce infections by 90%, it will take ten times as long. Complications arise as levels of immunity to prior infection increase in the population; this creates selection pressure on the virus to circumvent the host’s immunity to prior infection. Potentially, this can occur at maximum rates when intermediate numbers of people are immune, and the virus is circulating in a large proportion of the remaining unvaccinated population.
Long-term research on an emerging bacterial pathogen of house finches in the US, which has very similar dynamics to COVID-19, suggests that selection for asymmetrical immunity (where one strain induces host immunity against itself and all other strains, while weaker strains only induce host immunity against themselves) is a powerful driver of virulence in emerging pathogens, with later strains generating a stronger immunological response that prevents reinfection of recovered hosts infected by earlier strains.10 Mathematical models on the evolution of virulence in this system suggest that the ability of the pathogen to transmit before symptoms of virulence are apparent can lead to selection for significantly higher levels of virulence. 11 All these evolutionary considerations strongly underline the importance of getting as many people vaccinated as quickly as possible. Furthermore, the size considerations mentioned above suggest this will need to be done globally on an annual basis. This will require an increase of at least an order of magnitude in our global ability to produce and distribute vaccines.
THE BOTTOM LINE
Contact patterns between infected and susceptible people are what ultimately determine the dynamics of an epidemic. These contact patterns develop significant asymmetries within and across different sectors of the economy. They are further exasperated by people’s resistance to wearing face masks, or politically based refusal to believe in either the pathogen or the benefits of vaccination. I remain shocked at how politically polarized discussion of these issues has become, particularly as it seems obvious that the last thing you want to do as a politician is reduce the health and life expectancy of your voters in ways that significantly reduce the probability that they will survive to reach the polling booth.
Welcome to the Covidocene! Annual vaccination against this virus will be the new normal. Unfortunately, COVID-19 will not be the last novel pathogen to invade humans—all the available evidence suggests we should see at least one or two more emergent viruses in the next decade.12 This can be averted, or halted at an earlier stage of spread, but it will require considerable international cooperation.
Read more thoughts on the COVID-19 pandemic from complex-systems researchers in The Complex Alternative, published by SFI Press.
1 ProMED-mail, “Undiagnosed Pneumonia - China (Hubei): Request for Information,” December 30, 2019, https://promedmail.org/promed-post/?id=6864153
2 Boucekkine, Dobson, Loch-Temzenides, Ricci, Gozzi, Pascual; manuscript in preparation
3 B. Bolker and B.T. Grenfell, 1995. “Space, Persistence, and Dynamics of Measles Epidemics,” Philosophical Transactions of the Royal Society B 348: 309–320, doi: 10.1098/rstb.1995.0070
4 B.T. Grenfell and R.M. Anderson, 1985, “The Estimation of Age-Related Rates of Infection from Case Notifications and Serological Data,” Journal of Hygiene 95: 419–436, doi: 10.1017/s0022172400062859
5 R.M. Anderson, T.W. Ng, et al., 1989, “The Influence of Different Sexual-Contact Patterns between Age Classes on the Predicted Demographic Impact of AIDS in Developing Countries,” Annals of the New York Academy of Sciences 569: 240–274, doi: 10.1111/j.1749-6632.1989.tb27374.x
6 A. Dobson, 2004, “Population Dynamics of Pathogens with Multiple Host Species,” American Naturalist 164(5): S64–S78, doi: 10.1086/424681
8 E.C. Holmes, 2009, The Evolution and Emergence of RNA Viruses, Oxford, UK: Oxford University Press.
9 T. Day, S. Gandon, et al., 2020, “On the Evolutionary Epidemiology of SARS-CoV-2,” Current Biology 30(15): R849–R857, doi: 10.1016/j.cub.2020.06.031
10 A.E. Fleming-Davies, P.D. Williams, et al., 2018, “Incomplete Host Immunity Favors the Evolution of Virulence in an Emergent Pathogen,” Science 359(6379): 1030–1033, doi: 10.1126/science.aao2140
11 E.E. Osnas and A.P. Dobson, 2010, “Evolution of Virulence when Transmission Occurs before Disease,” Biology Letters 6(4): 505–508, doi: 10.1098/rsbl.2009.1019
12 A.P. Dobson, S.L. Pimm, et al., 2020, “Ecology and Economics for Pandemic Prevention,” Science 369(6502): 379–381, doi: 10.1126/science.abc3189