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John Cross, epidemic theory, and mathematically modeling the Norwich smallpox epidemic of 1819 [1]
['Connor D. Olson', 'Department Of Mathematics', 'Penn State University', 'University Park', 'Pa', 'United States Of America', 'Timothy C. Reluga', 'Huck Institute Of Life Sciences']
Date: 2024-12
In this paper, we reintroduce Dr. John Cross’ neglected and unusually complete historical data set describing a smallpox epidemic occurring in Norwich, England in 1819. We analyze this epidemic data in the context of early models of epidemic spread including the Farr–Evans–Brownlee Normal law, the Kermack–McKendrick square Hyperbolic Secant and SIR laws, along with the modern Volz–Miller random-network law. We show that Cross’ hypothesis of susceptible pool limitation is sufficient to explain the data under the SIR law, but requires parameter estimates differing from the modern understanding of smallpox epidemiology or large errors in Cross’ data collection. We hypothesize that these discrepancies are due to the mass-action hypothesis in SIR theory, rather than significant errors by Cross, and use Volz–Miller theory to support this. Our analysis demonstrates the difficulties arising in inference of attributes of the disease from death incidence data and how model hypotheses impact these inferences. Our study finds that, combined with Volz–Miller modeling theory, Cross’ death incidence data and population observations give smallpox attributes which largely cohere to those used in modern smallpox models.
Copyright: © 2024 Olson, Reluga. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
The paper proceeds as follows. First, we give a brief background on the state of numbers in infectious disease studies prior to 1819. We then introduce John Cross along with a history of smallpox epidemics before 1819 in the city of Norwich, England. After, we use Cross’ data and observations to describe the smallpox epidemic of 1819. We then interpret these data in terms of geometric models (Brownlee’s Normal law [ 10 ] and Kermack–McKendrick’s square hyperbolic secant law [ 4 ]), SIR and SI n R differential equation models, and finally, Volz–Miller random-network models. Along the way, we show how Cross’ inclusion of auxiliary demographic data in addition to epidemic dynamics data greatly improves our ability to make inferences. Finally, we compare the disease attributes we infer from these models to attributes of smallpox used by modern smallpox modelers.
Given how the fame of historical figures like John Snow has been at least partially a deliberate social construct [ 15 , 16 ] rather than a true attribution of credit, our motivation for producing this paper is rooted in the historical significance of the dataset we draw from, along with it’s pedagogical value. Cross [ 13 ] was potentially the first person to conjecture that the termination of the epidemic was due to a decrease in the number susceptible individuals. We contend that the data set he provides is a succinct historical example of epidemic termination by such an exhaustion. With the revelation of the inadequacy of the Bombay epidemic example [ 6 ], and human contact patterns of the 1820’s being potentially more consistent with the law of mass-action’s strong-mixing hypothesis, we campaign for the use of Cross’ work to fill the void as the canonical example of a simple immunizing SIR epidemic. The documentation of this smallpox epidemic and Cross’ accompanying analysis is a significant contribution of early epidemiological theory which deserves wider recognition.
Although Kermack and McKendrick’s plague example was misleading, other datasets pre-dating their work were consistent with their hypothesis. The analysis of Brownlee [ 10 ] is notable in this regard, though he himself subscribed to an alternate hypothesis that epidemic limitation followed the exhaustion of the causative agent. William Farr’s data sets on smallpox [ 11 ] and “cattle plague” [ 12 ] may be appealed to as well. But perhaps the most well-described epidemic to which SIR theory may be constructively compared is a 1819 smallpox epidemic in Norwich, England, excellently documented by John Cross [ 13 ], later referenced by Creighton [ 14 ] in his 1894 history of British epidemics, and now easily accessible online.
But as Bacaër [ 6 ] pointed out, the 1906 epidemic was less straight-forward. The data Kermack and McKendrick employed was a count of spill-over cases (human infections from an animal-borne disease) from a seasonal cycle of plague incidence in mice and rats that had been on-going since at least 1897, and would continue beyond 1911 [ 6 ]. Thus, seasonal forcing provides at least a partial explanation for this incidence data. The seasonality seems to have entirely escaped mention by Kermack and McKendrick [ 4 , 7 – 9 ]. So, despite Kermack and McKendrick’s assertions, the seasonality of plague incidence in mice and rats implies the 1906 epidemic wasn’t a novel epidemic in a population lacking immunity. Also, new cases were principally created by spill-over from mice and rats to humans via a flea vector rather than human-to-human transmission. Thus, in several aspects, the 1906 epidemic is contrary to fundamental assumptions of the SIR model Kermack and McKendrick present [ 4 ].
Mathematical epidemiology has its own example of mis-interpreted data. In their foundational papers (cited more than 10,000 times according to Google Scholar), Kermack and McKendrick [ 4 ] compared the results of the now well-known compartmental SIR model to data from a plague epidemic in Bombay (the colonial name for Mumbai) in 1906 [ 5 ]. This comparison “… implies that the rates did not vary during the period of the epidemic…” [ 4 ], and thus supported one of the central premises of the paper—that the course of an epidemic could be explained well by the partial but incomplete exhaustion of the supply of susceptible people, either by death or transformation through the acquisition of immunity.
While many scientists love in their hearts to feel themselves engaging in the perpetual accumulation of scientific knowledge, we must reluctantly admit that each piece of knowledge is a hard-won trophy of a mud-wrestling match, often fought as much with each other and ourselves as with data and facts. Not even the science of luminaries like Newton [ 1 ], Mendel [ 2 ], and Kepler [ 3 ] goes unblemished.
We also use Cross’ estimate of 1/6 from page 6 as the probability of mortality for infected individuals to simplify the fitting process. This value is directly inferred by Cross from his personal experience treating 200 cases, of which 46 terminated with death, along with the 530 smallpox deaths he recorded over the duration of the epidemic of which he says had infected “considerably above 3000 individuals.” With the exception of a single model, we utilize this probability to further reduce the number of model parameters we have to fit to the data.
We take care in explicating the many factors and information we have regarding the state of immunity to smallpox in Norwich because knowledge of the susceptible population is necessary to discern the transmission rate when the total population is held constant, which we assume for Norwich during the year of the epidemic. This estimate which we extrapolate from Cross’ data limits the total susceptible population to a realistic value given the history of smallpox and pox vaccination in Norwich at the time of the epidemic while also providing a fixed, informed value to then infer transmission from.
This susceptible population estimate explicitly depends on the assumption that Cross’ population sample is representative of Norwich in totality. As Cross has said, this epidemic was more prominent in the impoverished classes, so this sample potentially has fewer affluent families than is representative of Norwich, although Cross gives no particular class information about the city. This sample also suffers from the bias of all the families having members infected with smallpox. It is reasonable to believe that Cross’ data is representative of families which were effected, but is unclear how representative this selection of 112 families is of the city’s population. What is apparent is the estimate of 13,333 susceptible people forms an upper bound for this value.
How can we make sense of the many factors playing into the size of the susceptible population and come to a concrete number? Cross, while providing us all the context expounded above, also provides on page 7 data from his personal experiences with affected families during the height of the epidemic in 1819. From the beginning of March to mid August 1819, Cross attended to 200 cases of smallpox, from 112 families which totaled 603 people. Of the people uninfected in 1819, 15 had no history of infection or vaccination and thus resisted it, 91 had been vaccinated, and 297 had been previously infected. This gives a rough proportion 1 in 3 people in an effected family got infected. If we presume these families are a representative Norwich as a whole, then coupled with Cross’ population estimate of 40,000 people this gives an estimated 13,333 people susceptible to smallpox.
A majority of the first two chapters of Cross’ book are spent discussing the state of vaccination and general immunity in Norwich leading up to and during the epidemic of 1819. Cross stresses the difficulty he and the other medical men had in convincing people, especially in the impoverished classes, to immunize their children during the period between 1813 and 1819 when smallpox was non-existent in the city (pgs. 20–21). He continues with a discussion of the significant effort undertaken during the epidemic to vaccinate those yet uninfected, including a paid incentive for those who received a cowpox vaccination of which Cross’ provides data for on page 28. Because of the nature of smallpox, it is generally expected that someone who had been infected in a past outbreak, had been inoculated with smallpox, or had received the cowpox vaccine would be immune to further infection. Cross spends some pages discussing seeming exceptions to this general rule, and how the occasional failure of the cowpox vaccine complicated the vaccination effort (pg. 25). Norwich was also subject to a massive demographic shift during this period due to the rapid urbanization of rural populations facilitated by the industrial revolution (pgs. 12–13). These recent arrivals were less likely to be exposed to smallpox historically, while also being the mechanism by which Cross determines smallpox originally was introduced to the city in the summer of 1818.
Other sources also provide population estimates for Norwich around 1820. Creighton [ 14 ] cites 50,000 as the population at the time. Edwards [ 33 ], in a survey of mortality bills up through 1830, reports a population census in Norwich of about 50, 000 in 1821, and based on interpolation through adjacent census periods, would suggest an estimate of 49, 000 in 1819. Although larger, they aren’t significantly different from Cross’ estimate, so for the remained of the paper we will utilize Cross’ 40, 000 person estimate.
In fitting a model, one of the important questions is how many people were present in the city of Norwich in 1819, and how many of them were susceptible to smallpox infection. On page 13, Cross tangentially suggests the population of the city N ≈ 40, 000 souls at the time of the epidemic. This is consistent with his other comments, such as on page 33 where he says 10, 000 = N/4, and on page 6 where he suggests 3, 000 = N/13 (implying N ≈ 39, 000).
There is the potential that Cross’ has made errors in recording deaths, either mis-attributing them to smallpox, or not recording deaths which were caused by smallpox. The former is unlikely as smallpox infection is visually apparent during the eruptive phase, and the doctors at the time will have been familiar with diagnosing smallpox. The latter is a possibility, but we do not directly address it in our subsequent models. We leave it for future work whether a nuanced approach to handling missing data gives models which predict the data well while also cohering with smallpox biology.
Along with this time-series data, Cross provides us with data on the age ranges of those that perished due to the disease which we recreate in Table 3 . It is worth noting that of the 530 recorded deaths to smallpox, 477 were born after the 1813 smallpox outbreak which had the last mentioned cases of smallpox in the city. These numbers are consistent with the biology of smallpox, as those who have survived infection are imparted with immunity to further infection. Therefore, we expect that many of the susceptible individuals in the population will have been born after the last epidemic, which is certainly reflected in the death data. The models we study later in this paper do not consider age structure, but utilizing this data which Cross provides would be a logical next step for making more realistic models.
In 1819, in the city of Norwich, England, a smallpox epidemic occurred which spanned the entire year. Over the duration of this epidemic, the deaths of 530 individuals were attributed to smallpox infection. Cross traces the origin of this outbreak to a girl who was exposed in a market town during a journey from York to Norwich and broke out in smallpox upon arrival in Norwich. This index case occurred at the latter end of June, 1818. Cross states that “the earliest smallpox cases seen by any medical man could be traced to this origin.” He traces the infection from this origin through the handful of cases which occurred at the end of 1818, resulting in only 2 deaths, but he does not provide us with extensive data on the outbreak until January 1819. Because we are without data for this 6 month time period, it makes using this initial date difficult for fitting the model. The primary time series data which we utilize for model fitting is presented in Table 1 , which is a recreation of the page 5 table of [ 13 ]. For our purposes, we re-express the important data in Table 2 .
First, in reference to possible issues of ethics, all individuals from whom the following data has been collected are dead, and the data is fully anonymized.
Cross’ clear-eyed documentation of the epidemic is rather remarkable, given the state of science when he was writing. Most others chronicling epidemics, such as Boylston’s account of smallpox in Boston in 1724 [ 32 ] and those reviewed by Creighton [ 14 ], fail almost universally to synthesize their observations into useful tables and numbers the way Cross does. Cross’ clarity on the demographics of his city’s population is particularly notable, as such denominators can be vital in model fitting (as we will see momentarily), yet were seldom attended to by those focused on the practice of medicine.
He identifies that this outbreak likely ended because it exhausted the susceptible population. This clearly fore-shadows our modern understanding, and precedes a similar conjecture by Hamer [ 31 ] by nearly a century.
It therefore remains to be accounted for, why the disease spread thus rapidly, expiring, as it were, of starvation, in a few months, whilst commonly it might have gone on for two or three years, before the same number of individuals, distributed amongst a population of 40,000 inhabitants, would all have taken it (13).
Cross claims that three quarters of the susceptible population came down with the disease during the epidemic (pg. 13). Making an observation much before its time, Cross writes
The neglect of vaccination, it will be presently shown, was one of the reasons of the extensive prevalence of this pestilence (12).
In his book, Cross provides a brief history of the prevalence of smallpox in the city of Norwich. This history suggests smallpox epidemics were sporadic occurrences in Norwich rather than annual events like the plague epidemics in Bombay. Cross begins with an epidemic from 1805, about which he says that beforehand smallpox had almost been extinct in Norwich for some time. From this we can infer that the city of Norwich had not been exposed to a major outbreak of smallpox for a relatively long period of time. The cause of such a drought is left unspecified. Shortly after this epidemic, Cross describes another outbreak which lasted from 1807–1809 and brought with it 203 recorded deaths due to smallpox. Cross sites the duration and destruction of this epidemic to be due to the portion of the population which had gone unvaccinated. The last outbreak to occur before the focus of Cross’ text occurred in 1813, and lasted from February to September, leading to the death of 65 individuals. After this epidemic, until the outbreak in 1819, Cross does not believe that more than a single person, who did not spread this infection, was infected with smallpox within the city. Those 6 years of absence primed the city for the epidemic on which we will now focus.
The author of the chronicle we use was a British surgeon named John Green Cross (1790–1850) [ 27 ]. Cross was born in 1790 to a farming family in Suffolk, England. Early on in his life, John was apprenticed to a surgeon-apothecary in Stowmarket, the daughter of whom he married in 1815. Upon conclusion of his apprenticeship, Cross traveled to London to study at St. George’s Hospital. After spending some time studying in Dublin and Paris, Cross settled in Norwich in March of 1815. He served as witness of the epidemic we study here, of which he published his account “A History of the Variolous Epidemic which occurred in Norwich in the year 1819” [ 13 ] in 1820. The book was favorably reviewed by the North American Review [ 28 ] and the Edinburgh Medical and Surgical Journal [ 29 ] at the time of publication. In 1823, Cross became assistant-surgeon to the Norfolk and Norwich Hospital, and became surgeon in 1826. Cross was awarded the Jacksonian prize at the College of Surgeons of England in 1833 for his work on “The Formation, Constituents, and Extraction of the Urinary Calculus.” Over his life, he mentored forty apprentices, one of which went on to be the first professor of surgery at Cambridge. John Cross died on June 9, 1850, and was buried in Norwich Cathedral [ 27 ]. After his death, his book was cited by Simon [ 30 ] and Creighton [ 14 ], but scarcely since then, with the notable exception of a popular history by Glynn and Glynn [ 20 ].
By 1819, there was, at least among a few people, an understanding of (1) the value of demographic statistics, (2) the necessity of numbers and calculation in understanding certain healthcare trade-offs, and (3) the importance of collecting data to support statistics and calculation. But, despite numerous efforts [ 14 ], the syntheses of these ideas was still rather limited, and we have not yet found evidence that anybody had developed a complete self-contained statistical picture of an epidemic wave prior to 1819.
The second line was more medical. The introduction of variolation into England in 1721 by Lady Mary Wortley Montagu, and it’s subsequent promotion by Hans Sloane [ 19 ], forced physicians to consider more nuanced perspectives on preventative medicine. It was universally accepted that smallpox infection was potentially lethal [ 20 ]. However, variolation could also be lethal [ 20 ]. Thus, English physicians faced a difficult choice between a treatment that might kill an otherwise-healthy patient, or inaction that left the patient vulnerable to a disease that also might kill them. Some physicians believed that there was still a balance to be struck. For example, Hans Sloane is said to have explained to Princess Caroline that he could not recommend variolation on its own, but that the dangers of smallpox infection might be much worse than those of variolation [ 19 ]. But without the language of statistics and systematic data to support them, the debate had difficulty rising above randomized experiences of individual physicians. Fortunately, such a language was almost at hand. In 1722, James Jurin attempted to use data and calculation to demonstrate that in the environment of the day, the risks of variolation were greatly out-weighted by the risks of smallpox itself [ 21 ]. It was almost immediately pointed out by Isaac Massey (in the primitive language of the time) that the calculations could not be taken at face-value because access to variolation was positively correlated to economic status and the associated benefits that could improve outcomes post-infection [ 22 ]. Further data and calculations with regard to the issue were communicated over the next century by many others [ 17 , 23 ] including statesman Benjamin Franklin [ 22 ] and mathematician Daniel Bernoulli [ 24 , 25 ]. The debate receded with Jenner’s introduction of safer cowpox-based vaccination in the late 1700’s [ 26 ], but quantification had been firmly established as part of the smallpox conversation by that point.
Cross’ work lies in a transitional phase in the history of the quantitative understanding of infectious diseases. Over the century and a half preceding 1819, quantification in public health and medicine had been developing along two complementary lines. The first was a drive in government to develop natural laws of the state—“political arithmetic”. This began with John Graunt’s “Natural and Political Observations … upon the Bills of Mortality”, which appeared in 1662 [ 17 ]. The subsequent progress in demography and the related development of statistical mathematics for data analysis eventually infiltrated the public-health movement in the first half of the 1800’s (i.e. [ 18 ]).
Results
We now explore the interpretation of Cross’ smallpox data in Table 1, in light of classical modeling. Aside from Farr’s auto-regressive modeling [12], the earliest models were a Gaussian in the fashion of Evans and Brownlee [10], and a square hyperbolic secant following Kermack and McKendrick [4], which were early developments in mathematical epidemiological modeling. Then we discuss an SIR-type model, which reveals the shortcomings of using compartmental models to fit this data, and we present a model in the vein of the Volz and Miller [34–36] SIR extension. This final model rectifies the inferred attributes of the Norwich epidemic we obtain from Cross’ data with attributes of smallpox used in the wider literature. There are many more complex smallpox models one can fit to this data set, but as this paper is focused on providing a convenient historical example to ground standard SIR theory while also revealing its shortcomings and methods of remedy, we will focus solely on these simpler models.
To reiterate what we discussed above in more detail, unless otherwise stated, we use the susceptible population estimate of 13,333 which we derived from Cross’ estimate of the Norwich population as 40,000 and his direct observations of families afflicted with the disease. Although this isn’t an observation directly from Cross but a product of our inferences from his observations, we will occasionally refer to this value as “Cross’ S 0 ” in the proceeding discussion. Because Cross only provides us with time series data for deaths, we need to fix the death rate, or the models will generally have more parameters than we can specify with the given data. Again we lean on Cross and his estimate of 1/6 of infected individuals dying to handle this issue. These two informed assumptions are sufficient for us to fit reasonable models to Cross’ data.
SIR theory Kermack and McKendrick’s SIR theory is a mechanistic epidemic theory that categorizes people based on their disease-state, and describes the expected dynamics of the counts in each category, based on simple principles of population mixing, person-to-person transmission, and elementary disease progression. It is a synthesis of preceding research by Ross, Hamer, Lotka, and Hudson. The simplest form of SIR theory is the compartmental SIR model, which is a system of three ordinary differential equations tracking the number of susceptible (S), infected (I), and removed (R) individuals under the hypotheses that the population is sufficiently strongly mixed for the law of mass-action to apply, that individuals can transmit the disease immediately after infection, and that the duration of infectiousness is exponentially distributed. To align the SIR model with Cross’ data, we will split the removed compartment into recovered R and dead D. The rates of change of numbers in each compartment are governed by the system (4a) (4b) (4c) (4d) where β is the transmission rate, 1/γ is the expected duration of infectiousness, and m is the probability of mortality. For those unfamiliar with Newtonian derivative notation is the time derivative. These are solved from the initial condition (S 0 , I 0 , 0, 0). A key quantity in SIR theory and our general understanding of epidemics is the basic reproductive number , defined as the expected number of infections generated in a naïve population from a single infected individual. In these SIR models, , but in practical application when S 0 and N are significantly distinct as Cross claims is the state of Norwich, more is reflective of the reality of smallpox early in the epidemic. Given Cross’ own estimates of an initial susceptible population S 0 = 13, 333 and a probability of mortality m = 1/6, the simple compartmental SIR model has 3 free parameters: , γ, and I 0 . Estimating these parameters is roughly equivalent to estimating the initial growth rate, the duration of the epidemic, and the start-time. The estimates that minimize the root mean-square error are shown in the first column of Table 5. The fit is slightly better than those obtained by the geometric models discussed above. However, the parameter estimates themselves are not consistent with those described in other sources like Costantino et al. [37] (see Table 7). The basic reproductive number is too small and the duration of infectiousness is much too short. PPT PowerPoint slide
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TIFF original image Download: Table 5. Errors and best-fit parameters for the three SIR variants we present. Units on β are people-1 × days-1, and of 1/γ is days.
https://doi.org/10.1371/journal.pone.0312744.t005 What is the source of the discrepancy between the estimated parameter values and those used by modern studies? Of the information we have derived from Cross’ account of the epidemic, it is possible that either Cross’ directly observed probability of mortality m = 1/6 or the initial susceptible population S 0 we inferred is responsible. To investigate, we fit two additional SIR models, the first determining the best-fit initially susceptible population S 0 while keeping m as originally specified, and the second the best-fit probability of mortality m while keeping S 0 as we inferred. The details of each model are contained in Table 5, and the graphics are in Fig 3. PPT PowerPoint slide
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TIFF original image Download: Fig 3. (A) This plot compares the SIR model with Cross’ S 0 versus the best-fit S 0 and m along side Cross’ data. (B) The residuals of the two best-fit models are here indistinguishable, and the performance is much improved during the height of the epidemic. (C) Here the incidence, or rate of infection, of the three models are presented. We observe the best-fit m has a much larger incidence due to the larger number of people infected during the epidemic. A structural difference is both best-fit models have an earlier peak incidence, while the peak incidence of the Cross’ model aligns more with the function models.
https://doi.org/10.1371/journal.pone.0312744.g003 Looking at the errors, we see the model solely utilizing Cross’ estimates is an improvement over the two integrated function models, but its performance is much closer to those models than the two subsequent SIR models, which are a two-fold improvement. One attribute which is inferable from the parameters in Table 5 is the infectious period, which is captured in the quantity . For the first SIR model, this is an infectious period of 3.23 days, while the two best-fit models have infectious periods of 20.71 days, much more inline with the expected duration of a smallpox infection. The models also predict the number of total infections which occurred during the epidemic, of which the predictions are respectively 3,172 people, 3,198 people, and 11,142 people. Cross’ estimate that at least 3,000 people were infected conforms nicely with first two models, but his use of “considerably more” in describing how many infections did occur does not rule out the third model’s prediction explicitly. The first two totals of infection being around 3,000 is unsurprising since the total number of deaths Cross recorded was 530, and a fixed death rate of 1 in 6 necessitates that the model should predict around 3,180 total infections which both models align with. Cross’ speculates that the epidemic ends due to the exhaustion of the susceptible population, which the two best-fit models conform to with around 84% of the population getting infected under both models, while the original SIR model predicts only around 24% of the susceptible population suffering infection. Studying Fig 3, especially plots A and B, along with the Error and in Table 5, suggest that the two best-fit SIR models are essentially the same regarding the data, even though they have entirely different parameters due to their different S 0 and m, and Plot C reveals this distinction. What is occurring here is S 0 and m are confounding in the form of the quantity mS 0 , which is approximately 2,222 for the first model using the estimates of S 0 and m from Cross’ text, while for the two best-fit models mS 0 ≈ 638. For such an SIR model and an a > 0, the parameters (aS 0 , aI 0 , β/a, m/a) give the same number of deaths, but not the same number of infections. Thus, with only death data, we can only determine the best-fit mS 0 and the family models conforming to that. This analysis informs that there is no point in determining the best-fit model treating both S 0 and m as free parameters, as they are tied together and only the quantity mS 0 is identifiable within our data. The nearly two-fold improvement in performance of these best-fit SIR models naïvely suggests that the inferences we made from Cross’ text incorrectly estimate mS 0 despite these coming from his direct observations. One could speculate as to which quantity of S 0 and m is the primary culprit, where the two models we fit presume either one or the other is a correct quantity, but this speculation is spurious. Yet, the results of these best-fit SIR models need not undermine Cross’ work or our inferences from it.
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