Models, spatial dependence and mortality
06/03/2021
Theory
Premises
- All models are wrong, but some are useful
- As all models are wrong -> any single model must also be wrong
- If a model is wrong, how do we know if it is useful?
- A model is useful if it predicts something we are interested in
- Predictions must be compared with something
- We can compare predictions with reality or …
- We can compare predictions of one model with the predictions of competing models
- Using a single model to make predictions implies that the model may be right!
- But we already have decided that all models are wrong!
- So …. we should never rely on a single model
Predictions
- Making predictions is hard .. especially about the future.
- Quantitative predictions involving single numbers are tested against empirical data from the future
- This is hard … and all models are wrong!
- So all quantitative predictions will also be wrong
- Quantitative predictions can be useful if they are accurate.
- In order to ensure accuracy we need to lower precision.
- An imprecise model can still be accurate, but useless for decision making.
- An inaccurate, highly precise model which is also wrong is harmful for decision making
- As the difficulty of making predictions increases with time from the present useful quantiative predictions can only be made about the short term.
Model comparison
- If models are not compared to reality then we compare them to other models
- Model comparison can involve qualitative model behaviour
- Comparing multiple models behaviour leads to multiple working hypotheses
- These hypotheses can be tested through matching patterns (but not quantities) with reality
- So such models predict hypothetical futures
- Given the initial premise that all models are wrong, none of these hypothetical futures will actually match reality
- However .. investigating the extent to which model predictions approximate reality can be useful
- A useful model is easier to understand than reality
Heuristics
- “What if” modelling
- Use “Occam’s electric razor”
- Make models as simple as possible, but not more so
- Communicate models to model users (hypothesis formation)
- Use multiple model structures and multiple parameterisations
- Compare models with models, do not (yet) compare models to the real world
- Match patterns and model behaviour, not quantitative predictions
SIR
\(\frac {dS}{dt} = - \beta SI\)
\(\frac {dI}{dt} = \beta SI- \gamma I\)
\(\frac {dR}{dt} = \gamma I\)
\(R_0 \approx \frac{\beta}\gamma\)
- A complex “jumble” of parameters representing transmission \(\beta\)
- Another complex "jumble of parameters representing time infectious \(\gamma\)
- Epidemic does not take off if there is herd immunity at the outset.
\(HIT = 1 - \frac 1 R_0\) where \(R_0 \approx \frac \beta \gamma\)
- However when a model is run overshoots of the HIT occur.
- Reality is yet more complex. (Gomes et al. 2020) and (Aguas et al. 2020) derive a mathematical extension of the HIT equation based on consideration of the coefficient of variability of transmissions.
\(HIT= 1-\frac 1 {R_0}^(1+{CV}^2)\)
S = Susceptible fraction at risk
I = Infected and infectious fraction
R = Removed fraction (either dead or immune)
Scaling
Outbreak (transmission between people within a setting)
Epidemic (transmission between settings)
Pandemic (transmission between countries)
Endemic (endogneous) - spontaneous cases with no pattern of transmission
The SIR could be argued to be applicable to susceptible, infected and removed settings or even countries under a “metapopulation” framework ..as in (Hanski 1998)
Simple network models
Panmixia
To reproduce the classic SIR results the model can be setup with long distance links between the nodes. This ensures that the epidemic can spread throughout the whole susceptible population. The notable result of complete panmixia is the speed at which the epidemic (outbreak) rises and falls. If the population is completely susceptible at the outset the epidemic ends when the whole population has become infected and subsequently removed from the pool of susceptible through either recovery with immunity or death. Such an epidemic typically lasts weeks rather than months.
Moving front
If long distance connections are removed the panmixia assumption is not met. The consequence of this is to add more structure into the spread of the epidemic. This structure may be spatial or more conceptual in nature, as the social connections may not be defined by strict geographical proximity. The outbreak now takes more time to move through the population, even with the same intrinsic transmission between linked individuals. Contact tracking and tracing would identify this form of spread and could potentially mitigate and/or supress it in the early stages.
Effective lockdown (rapid supression)
The model can reproduce the expected result of successful repression if links are removed early on the progression of the epidemic. This is simulated by a break point in which the network is rebuilt with many fewer linkages. The outbreak declines following this measure being adopted. However, this model assumes a successful lockdown. If lockdown measures are ineffective in completely suppressing the virus the subsequent results will simply follow one of the patterns shown by any other model with population of susceptible individuals “seeded” with infection.
Initial blocking immunity
If a substantial proportion of the population are initially immune the epidemic can only spread if transmission between linked individuals is very high. If this is not the case then herd immunity supresses all possibility of spread early on. With high levels of “blocking” immunity a highly transmissible virus eventually finds pathways to spread to the susceptible population, but the epidemic trajectory can be slower. In many respects this hard to distinguish from the pattern produced by an effective lockdown.
Fast waning immunity
If immunity is assumed to be very short lived the model can reproduce multiple waves of infection that occur after the first outbreak. To reproduce this the assumption has to be made that almost all the population that were initially infected lose their immunity in a relatively short space of time. These waves do not occur if only a minority of those infected are restored to the susceptible population, as herd immunity prevents them from re-occurring.
Take home messages
- All models display some form of spatial structure, at least initially
- Panmixia quickly erodes spatial structure
- Endemic (endogneous) processes display esoteric and unpredictable spatial structure that does not correspond with any model involving transmission.
Mortality and epidemic spread
Premises
- All mathematical and simulation models of epidemics share a key assumption
- All deaths are caused by exposure to the pathogen. Death would not occur during the duration of the epidemic without exposure.
- Therefore the spatio-temporal pattern of mortality during an epidemic should differ from the expected pattern in the absence of the epidemic
- Evidence for a change in the temporal pattern is provided by the sharp, unseasonal, spike in late March to May.
- What is the evidence for a change in spatial pattern?
Spread between countries
https://www.ons.gov.uk/visualisations/dvc895/mapchangeovertime/index.html