In the system dynamics model from Shiflet and Shiflet (2014), one person is infected at start. Infected people can infect susceptible people. The population of infected will always increase by (number of infected * number of susceptible * InfectionRate * change in time dt). The infected people may recover. The amount of people that will recover in an iteration is always equal to (number of infected * RecoveryRate * change in time dt). Figure 1 illustrates the system dynamics process while Figure 2 shows the SIR process as a flowchart.
|Figure 1. System Dynamics process (source: Shiflet and Shiflet, 2014)|
|Figure 2. System Dynamics flowchart|
The Agent-based Model
|Figure 3. Agent-based Modeling: agent decision process|
|Figure 4. Display of the ABM. Green = susceptible. Red = infected. Blue = recovered.|
Figure 5 shows the changing process of the cells while Figure 6 shows the display of the CA model.
|Figure 5. Cellular Automata cell changing process|
|Figure 6. Display of the CA model. Green = susceptible. Red = infected. Blue = recovered.|
The Discrete Event Simulation Model
In a Discrete Event Simulation model (aka. queuing model), there are three abstract types of objects: 1) servers, 2) customers, 3) queues, which is quite different from the CA and ABMs.
|Figure 7. Discrete Event Simulation process.|
Results from the Implementations
|Figure 8. Results for the different models. Clockwise from top left: SD model, ABM, DES and CA|
First of all, the SD model has the smallest number of iterations before no one is infected. The number of iterations shown on the graph are the average of the ten runs, since the runs range from smaller to larger numbers (except for the SD model, which only has one run). The SD model only took 17451 iterations to stop, while the ABM took 19145 iterations (on average), the DES model took 18645 iterations (on average). The CA model took the longest time on average for no more individuals to be infected, it took 25680 iterations (on average).
The results of the SD, ABM and DES models while appearing to be very similar to each other. In the sense, that the number of infected people increase fast at first and reaches a peak number of over 1500 at more than 2000 iterations (2272 for SD, 2403 for ABM, 2538 for DES). On the other hand, in the CA model, the number of infected people increases much slower due to the diffusion mechanism of the CA model and never reaches an amount as high as in the former models.
An important characteristic of the SD model is that there is no randomness in the model, so no matter how many times you run this model, you will get the same result. In the other three models, getting infected or recover always depend on a probability function, so there is difference in every run.
Furthermore, people in the SD model and the DES model are homogeneous, and everyone has the same probability to becoming infected or recovering from an infection, although these rates change over time, they do not vary among the different people in the population. On the other hand, in the ABM and the CA model, people (represented by moving agents or static cells) are heterogenous in the sense that they have different locations. Only susceptible people around an infected individual can be infected. It is interesting that when people can move around, like in the ABM, the result is similar to the SD model, though the ABM takes a little more time to recover (19145 iterations in ABM vs. 17451 iterations in SD). When people are static and the number of people on the same space is limited (one cell in one space in this case), like in the CA model, the infection process becomes slower and it takes longer for everyone to recover.
To test how the models are sensitive to a specific parameter we now present what happens if we increase the infection rate in each model from 0.002 to 0.02 and show the results shown in Figure 9. As to be expected as the infection rate increased, the number of susceptible people decrease at a much faster rate. However, the SD, the ABM, and the DES models are still similar to each other, while the infection in the CA model is slower. The average number of iterations for these models are: 15807 (SD), 15252 (ABM), 16937 (CA), 16677 (DES). By increasing the infection rate the total number of iterations of each model has decreased, with the CA model still taking the longest time to converge. The peak of infected people in each model are on average: 2363 people at 255 iterations (SD), 2310 people at 363 iterations (ABM), 2035 people at 1019 iterations (CA), 2340 people at 286 iterations (DES). The CA model takes a longer time and reaches a lower peak.
|Figure 9. Results for the different models with infection rate = 0.02. Clockwise from top left: SD model, ABM, DES and CA.|
These models are only simple examples of how a SIR model can be implemented in different modeling techniques, but in reality, if we were to model disease propagation in more detail we would need to consider many other things such as people could be both moving through space (i.e. traveling to work) and static (i.e. staying at home), and the capacity of each cell is always limited to some amount.
Gilbert, N. and Troitzsch, K.G. (2005), Simulation for the Social Scientist (2nd Edition), Open University Press, Milton Keynes, UK.
Shiflet, A.B. and Shiflet, G.W. (2014), Introduction to Computational Science: Modeling and Simulation for the Sciences (2nd Edition), Princeton University Press, Princeton, NJ.
More information about the models and to download them please visit Yang Zhou’s website.Continue reading »