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Rates and Probabilities
There are broadly two ways of specifying how rapidly people move across a transition in the model
- Rate which is a number of people per unit time
- Probability which is the likelihood for an individual to transition per unit time
Although these appear similar, they behave differently when the integration step size is changed. Consider the example where 1200 people have transitioned in 1 year, in a population of 10000
- The rate is 1200 people/year
- The probability is 1200/10000 = 0.12 per year
Now, consider how these could be converted to per-month quantities. The rate can simply be divided i.e. 100 people/month = 1200 people/year. However, the probability of a transition is cumulative. That is, suppose was ask what the probability is of having at least one transition in the year, if we sample once every month. This would be
P(remain_for_a_year) = P(remain_for_dt)^(1/dt)
1-P(move_after_a_year) = (1-P(move_after_dt))^(1/dt)
1-P(move_after_dt) = (1-P(move_after_a_year))^dt
P(move_after_dt) = 1 - (1-P(move_after_a_year))^dt
where dt here is 1 month (the value would be 1/12 normalized by year). So here, the probability of moving per month is 0.0106. In a population of 1000, this corresponds to 106 people, so the difference is not so significant. But consider how this changes as a function of the probability of moving
| Flow | Prob | Prob per month | Flow per month | Probabilistic flow |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 500 | 0.0500 | 0.0043 | 42 | 43 |
| 1000 | 0.1000 | 0.0087 | 83 | 87 |
| 1500 | 0.1500 | 0.0135 | 125 | 135 |
| 2000 | 0.2000 | 0.0184 | 167 | 184 |
| 2500 | 0.2500 | 0.0237 | 208 | 237 |
| 3000 | 0.3000 | 0.0293 | 250 | 293 |
| 3500 | 0.3500 | 0.0353 | 292 | 353 |
| 4000 | 0.4000 | 0.0417 | 333 | 417 |
| 4500 | 0.4500 | 0.0486 | 375 | 486 |
| 5000 | 0.5000 | 0.0561 | 417 | 561 |
| 5500 | 0.5500 | 0.0644 | 458 | 644 |
| 6000 | 0.6000 | 0.0735 | 500 | 735 |
| 6500 | 0.6500 | 0.0838 | 542 | 838 |
| 7000 | 0.7000 | 0.0955 | 583 | 955 |
| 7500 | 0.7500 | 0.1091 | 625 | 1091 |
| 8000 | 0.8000 | 0.1255 | 667 | 1255 |
| 8500 | 0.8500 | 0.1462 | 708 | 1462 |
| 9000 | 0.9000 | 0.1746 | 750 | 1746 |
| 9500 | 0.9500 | 0.2209 | 792 | 2209 |
The reason this occurs is because the probability is computed for an individual, and thus the number of people leaving in any given month is proportionate to the number of people in the compartment. So the probabilistic flow here only applies to the first month, when there are 10000 people in the compartment. For example
| Month | People remaining | Number of people who transitioned |
|---|---|---|
| 0 | 10000 | 0 |
| 1 | 7791 | 2209 |
| 2 | 6070 | 1721 |
| 3 | 4729 | 1341 |
| 4 | 3684 | 1045 |
| 5 | 2871 | 814 |
| 6 | 2236 | 634 |
| 7 | 1742 | 494 |
| 8 | 1358 | 385 |
| 9 | 1058 | 300 |
| 10 | 824 | 234 |
| 11 | 642 | 182 |
| 12 | 500 | 142 |
Similarly, notice that a constant flow rate would correspond to an increasing probability of transitioning per person for each person remaining in the compartment. Both formulations are valid, but are suitable for different purposes. For example, consider the case where the compartments correspond to age (e.g. 20 years old and 21 years old) and the population's age is uniformly distributed. Then, we would expect the same number of people to transition from 20 to 21 years old each month, and in the last month, we know that anyone who has not already transitioned is guaranteed to transition. So in this case, the flow is uniformly distributed over the course of the year but the probability of transitioning is not. Similarly, suppose that we had a program for treatment that would move 500 people every month from the Infected compartment to the Recovered compartment. Again, this would correspond to a variable probability over the course of the year as the compartment empties, because the probability of a person being one of the 500 people each month increases as the compartment gets smaller. But if the disease had a fixed probability of resolving itself, then this would remain constant over the course of the year, so fewer people would have their diseases resolve as the compartment became smaller.
One complication is what happens when the compartment is replenished by inflowing people at each timestep. In that case, the effective flow rate could be greater than 1. Consider the case where 2209 people are added each month, so there is no net change in population each month. Then we would have
| Month | People remaining | Number of people who transitioned |
|---|---|---|
| 0 | 10000 | 0 |
| 1 | 7791+2209 new | 2209 |
| 2 | 7791+2209 new | 2209 |
| 3 | 7791+2209 new | 2209 |
| 4 | 7791+2209 new | 2209 |
| 5 | 7791+2209 new | 2209 |
| 6 | 7791+2209 new | 2209 |
| 7 | 7791+2209 new | 2209 |
| 8 | 7791+2209 new | 2209 |
| 9 | 7791+2209 new | 2209 |
| 10 | 7791+2209 new | 2209 |
| 11 | 7791+2209 new | 2209 |
| 12 | 7791+2209 new | 2209 |
However, this means that 26508 have left the compartment - although this is 265% of the initial compartment size, note that it is only 0.73 of all the people who were ever in this compartment at some point in the year. The flow rate of 9500/10000 naively giving a probability of 0.95 does not account for the fact that some people were able to both enter the compartment and leave the compartment within the same year. Similarly, the effective probability being 0.73 rather than 0.95 reflects the fact that people who entered the compartment late in the year are less likely to have left it that same year. Alternatively, if 0.95 of the population flowed out every month, then the net annual flow would be 25183 for the people who arrived during the year, plus another 9500 for the initial contents of the compartment. And we would have
| Month | People remaining | Number of people who transitioned |
|---|---|---|
| 0 | 10000 | 0 |
| 1 | 500+2209 new | 9500 |
| 2 | 135+2209 new | 2574 |
| 3 | 117+2209 new | 2227 |
| 4 | 116+2209 new | 2210 |
| 5 | 116+2209 new | 2209 |
| 6 | 116+2209 new | 2209 |
| 7 | 116+2209 new | 2209 |
| 8 | 116+2209 new | 2209 |
| 9 | 116+2209 new | 2209 |
| 10 | 116+2209 new | 2209 |
| 11 | 116+2209 new | 2209 |
| 12 | 116+2209 new | 2209 |
for a total outflow of 34183.
Essentially the answer would appear to be no. If we make the statement that '9500 left the compartment in 2010' then this does not provide any information about whether that number was uniformly distributed over the course of the year, or whether it scales with population size over the course of the year. In particular, based on the example above, converting the 9500 people to a probability requires knowing both the initial population size and the number of people entering the compartment and leaving the compartment due to other reasons. When the population size varies over the course of the year, it can be misleading to normalize the number of people transitioning over the course of the year by the initial number of people in the compartment. That is, if there were 10000 people present in the compartment in Jan 2010, it could be misleading to quote the flow rate as 9500/10000=0.95 because this does not account for people entering the compartment. It ought to be normalized by the number of people who were present in the compartment at any point during the year, but of course, this cannot be done prior to running the simulation. Rather, a flow rate that is provided as a net number of people per year corresponds to an unknown probability that is a candidate for calibration. On the other hand, if the probability is known, then the corresponding flow rate can be dynamically computed during the simulation.
Currently, in OptimaTB, if a fraction of people is provided, it is assumed to correspond to an annual probability i.e. if 'Fraction 0.95' is input, it is assumed to correspond to a probability of 0.95, which is converted to a timestep-based probability for integration. But if a number of people is provided, it is divided uniformly and distributed evenly across the course of the year. However, whether this target number of people can be reached depends on whether there are a sufficient number of people in the compartment e.g. if the birth rate is too low, there may simply be an insufficient number of people moved over.
An example of a parameter that may be provided as a fraction is the death rate or diagnosis rate - for an individual person, the probability that something happens to them. A parameter that may be provided as an absolute number is the number of notified cases.
(Check with Kedz) - It appears that parameters linked to transition tags, or dependent parameters used to compute a transition tag, are usually specified as a fraction, while parameters used solely as outputs are usually specified as numbers. This makes sense, because for every observed number of people there is a corresponding unknown underlying rate that governs the dynamics. The exception would be program-related transitions where a fixed number of people need to be moved - in this case, they are moved uniformly over the course of the year.