Monkeypox scenario modelling

– June 17th, 2022 –

I do not pretend I have a working model of the 2022 monkeypox outbreak. The aim is to produce believable scenarios of what could happen in the near future. All modelling below is based on the data up to June 17th, 2022. As new data emerge, the picture is likely to change!


This page is a companion to my The Conversation piece on possible scenarios for the Monkeypox (hMPXV) epidemic. The purpose of this page is to add more details to the mathematical models I used to explore the scenarios.

As discussed on my COVID-19 modelling page, the aim of the Monkeypox model is not to predict exactly the type, size and timing of the hMPXV epidemic in Summer 2022, or its longer-term course, but rather to provide an illustration of potential scenarios. As such, it is kept very simple. I have done some fitting to the existing data, as discussed below, but it is not meant to be a definite picture of how hMPXV spreads in the UK or globally.

Model

The model used to produce the graphs is a basic SEIR model. The population of N individuals is split into four subpopulations, susceptible S, latent E, infected I, and recovered R. No death has so far been recorded for the 2022 epidemic.

The basic model is described in terms of four differential equations:

    \[\begin{array}{l*{10}{c}c} \frac{dS}{dt}=d (N- S)-\beta(t) S\frac{I}{N} \\[10pt] \frac{dE}{dt}=\beta(t) S \frac{I}{N} - (a  + d) E  \\[10pt] \frac{dI}{dt}=a E - (g + d) I \\[10pt] \frac{dR}{dt}=g I - d R\]

where \beta is the rate of spread, 1/a is the length of the latent period, 1/g is the length of the infectious period. In addition, d is the rate at which immune individuals become susceptible again and is considered in Scenario 3 (endemic). The rate of spread, \beta is time-independent in Scenarios 1 (basic), 3 (endemic) and 4 (repeated epidemics), but is assumed to decrease exponentially for Scenario 2 (large outbreak),

    \[\beta(t)=\beta_0+\left(\beta_1-\beta_0\right) \exp{(-w t)}.\]

Data

The model has been used to describe the UK hMPXV epidemic, with the cut-off point of June 17th. I did not want to do it for global data, as there are obvious problems with reporting across different countries. Also, the epidemic was initiated in different countries at different times which resulted in “start-and-stop” periods such a simple model cannot cope with.

I have used two sources of data. The UKHSA released a report on 10th June that published a very detailed data set obtained by interviewing infected individuals and tracing the contacts. The full data are plotted in black points in the graphs below. However, the last 14 days or so are unreliable due to under-reporting, as highlighted in the UKHSA report. The data points until that point – the ones on which I based the model – are marked in red. A comparison with the second data set clearly shows the “back-filling” due to late reporting.

It is worth noting that the UKHSA data set represents dates of infection, not of reporting.

The second data set comes from Our World in Data and is marked in blue. This data set contains cases by date of confirmation. Due to a long latent period, these data points are shifted compared to the UKHSA data. The shift is of course variable, probably longer at the beginning and shorter later in the epidemic. However, it is not unreasonable to assume that about 10 days lapse between the infection (UKHSA) and confirmation (OWD). I used an 11 days shift and obtained a pretty good agreement between the two data sets, except early where the lag between infection and confirmation – and hence between the UKHSA and OWD data – is larger.

Model fitting and results

The model was fitted by eye to the UKHSA data early in the epidemic and to the OWD data late. The incubation period is assumed to be 10 days (although more recent estimates are at a lower value, possibly 8.5 days). The infectious period is set at 10 days. The population size N is varied between Scenarios, as is the birth/death rate, d. Finally, \beta also varies between Scenarios as changing N and d requires a readjustment of the rate of spread to agree with the data in the period up to June 17th, 2022.

Scenario 1 (small population at risk)

The basic fit (Scenario 1) is shown below (as of June 17th, 2022), for N=1000 and \beta=0.53~day^{-1}. The corresponding R_0 is about 6; clearly too much for the hMPXV as spreading in Africa.

Scenario 1: red and black points: UKHSA data; blue points: OWD data shifted by 11 days; solid line: model under Scenario 1. The size and duration of the outbreak are for illustration purposes only and do not constitute a detailed prediction of what might happen in the future.

The solid line represents the reported cases,

    \[dX/dt=a E,\]

assuming the reporting immediately follows the transition from the latent to the infectious class, i.e. the appearance of symptoms:

The agreement first with the UKHSA data (red points) and then with the OWD data (blue points) is quite remarkable. Now, I do not pretend that I have got a reliable model for the hMPXV spread in the UK. In particular, I do not claim that the epidemic will be over by mid-July and will infect about 1000 people.

Scenario 2 (large population at risk)

In this scenario I assumed that 40% of the population is susceptible; this corresponds to all males under 50 years old: 50% of the population are males and 80% of those are under 50 years old. The first assumption reflects the fact that currently, the hMPVX appears to spread predominantly in males. The second corresponds to the apparent residual cross-immunity from smallpox vaccines.

Scenario 2: red and black points: UKHSA data; blue points: OWD data shifted by 11 days; solid line: model under Scenario 2. The size and duration of the outbreak are for illustration purposes only and do not constitute a detailed prediction of what might happen in the future.

The value of R_0 from Scenario 1 needs changing for later in the epidemic. In order to properly describe the initial stages, a relatively high value of \beta needs to be chosen. But, this is not consistent with what we know about the hMPXV and would result in a massive outbreak.

For Scenario 2, the rate of spread, \beta is assumed to decline exponentially (w=0.027~day^{-1}) to 30% of the original value. The resulting R_0=1.8 is much closer to what we know about hMPXV. Other parameters: N=24,000,000, \beta_0=0.198, \beta_1=0.66:

Clearly, this is a very pessimistic scenario, but then no NPIs or vaccination campaigns are present. The total number of infected individuals is nearing 20,000,000 and the epidemic lasts for over a year.

Scenario 3 (endemic)

In this Scenario, hMPXV becomes endemic. This could happen by several routes:

The first option is a loss of immunity leading to repeated infections, as for COVID-19. There is not much evidence of this happening with hMPXV and evidence to the contrary from smallpox which appears to produce a lasting, “sterilising”, immunity.

Secondly, newborn individuals are susceptible and so there will always be an influx into the population of those who can become infected. The question is of course whether the virus can survive in the population in large enough numbers to be able to persist until the population rebuilds.

The third option – and explored here – is based on a slow replacement of individuals from the original population by new, susceptible, ones coming from outside. In this process, immune individuals leave the population (“die”) and are replaced by new, susceptible, ones (“birth”). It is similar to the first option, but it can be much faster.

Here, we are back to Scenario 1 parameters, but \beta=0.6~day^{-1} (to compensate for the removals) and d=1/60~day^{-1}; all the other parameters are the same. This corresponds to a single new individual replacing an old one every 2 months.

Scenario 3: red and black points: UKHSA data; blue points: OWD data shifted by 11 days; solid line: model under Scenario 3. The size and duration of the outbreak are for illustration purposes only and do not constitute a detailed prediction of what might happen in the future.

Scenario 4 (new incursion)

This is essentially the same as Scenario 1, but at a later point in time, a new (small) population is made available to the virus. Thus, at t=T, the number of susceptible individuals increases by N=1000.

Scenario 4: red and black points: UKHSA data; blue points: OWD data shifted by 11 days; solid line: model under Scenario 4. The size and duration of the outbreak are for illustration purposes only and do not constitute a detailed prediction of what might happen in the future. In particular, the location and size of the second peak are for illustration purposes only.

For the record, I am aware of the “atto-foxes” problem. Yes, the model can predict a restart of the epidemic but the variable I will be so low by the time t=T that in practice it will mean no infected individuals are left. In the original story, about multiple rabies waves in England, the numbers of infected foxes required for this to happen were of the order of 10^{-18}, i.e. “atto-foxes”.

So, I also inject a single infected individual at the same time point, t=T, to make sure the epidemic starts again properly.

Are these Scenarios compatible with the current data?

Remarkably, all four models fit the data up to June 17th pretty well, only with discrepancies later in the epidemic.

Comparison of short-term dynamics for different Scenarios. Red and black points: UKHSA data; blue points: OWD data shifted by 11 days; solid line: model.
The model is based on the data until June 17th, 2022

As stressed above, this is really a toy model which only pretends we know what will happen next. I suspect a version of Scenario 1 or possibly Scenario 3 is likely to happen; Scenario 4 is also very likely in a longer-term. But of course, the future will be much more dependent on the application of NPIs and ring vaccination than such simple transmission models can handle.

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