Chapter VI: Epidemic Spread Mitigation in Population Networks
6.5 Simulation
˜
γk(d)(t) := 1 2
1
Ik(t)(Dk(t+ 1)−Dk(t))− X
n∈Ck
ln(1−γˆn(d)(t))
, for time t∈[0, Tsim).
Remark 26. The multiscale model can be reduced to either a full compartmental ODE or a full coupled HMM, models which are prevalent throughout much of the existing literature on epidemic modeling (see the beginning of the chapter for a review). In particular, the multiscale model can be reduced to a full CHMM model by using the Markov chain dynamics:
Xn(t+ 1) =x with probabilityPn(t)[Xn(t), x] for some x∈ X; the number of individuals per compartment at each time t can then be obtained by counting the number of individuals which belong in each phase, e.g., Sk(t) = P
n∈Ck1{Xn(t) = S}. If the CHMM module replaces the ODE module, a means of propagating individuals across each phase of COVID- 19, the multiscale model becomes fully stochastic, and further details about intra-community interactions may effectively remove the need to introduce the community structure. However, the key point is that when handling large-scale populations, individual-level modeling is not ideal due to computation time incurred from the large number of parameters that need to be estimated. Using coarser community partitions via the ODE module allows us to abstract away an appropriate number of details.
We choose the metric (6.23) in this way, as opposed to the standard L1 orL2 norms, so that any deviations away from the true matrix caused by a massive error in a single entry do not greatly impact the overall error.
For parameter estimation of both communities and individuals, we consider mean absolute- value differences:
1 K
K
X
k=1
|χˆ(t)−χ|, χ∈ {αk, γk(r), γk(d)}, 1 N
N
X
n=1
|χˆ(t)−χ|, χ∈ {βn(t), αn, γn(r), γn(d)}, (6.24) where χis a placeholder variable for the original parameters in θk orηn, andN ≤N is the number of tracked individuals in the total population.
Symptoms Observation Matrix: We consider the symptoms of 1) fever/headache/mi- graine, 2) difficulty breathing/blockage in lungs, 3) sore throat/scratchy throat/coughing, 4) stomach pain/indigestion/diarrhea,5) sneezing/runny nose/itchy nose, and 6) dead based on the real symptoms observed of a person infected with COVID-19. We assign the following concrete probability values, which were chosen based on the real-data statistics about the symptoms given by the CDC [29].
On(yn,j = 1|x) =
0.1 0.1 0.9 0 0
0.05 0.05 0.65 0.01 0 0.07 0.07 0.73 0.01 0 0 0 0.03 0.01 0
0 0.9 0.95 0 0
0 0 0 0 1
∈RB×|X |.
That is, the (j, x)th entry of the matrix above defines the probability of observing symptom j ∈ {1,· · · , B}from individualn (i.e.,yn,j = 1) given his/her current health status isx∈ X. Prediction and Forecasting: We substitute the estimated parameters
θ˜k(Tsim) := [{β˜kj(Tsim)}j∈{1,···,K},α˜k(Tsim),˜γk(r)(Tsim),γ˜k(d)(Tsim)]
obtained from Section 6.4 into the compartmental module (6.2) in order to forecast the evolution of counts up to some future time Tsim+ > Tsim. For the purposes of prediction, we keep the rates constant at the final value estimate at time Tsim. To determine the accuracy of the prediction, we consider the absolute-value difference between the true and predicted
counts X(t) over time t ∈ (Tsim, Tsim+ ], where X is a placeholder for one of the original compartments X ∈ {I, R, D}. Note that the susceptible and exposed individuals are not considered since they are difficult to obtain the true values of in the real world.
6.5.1 Performance of HMM Model: Parameter Estimation
We evaluate the performance of the parameter estimation scheme from Sections6.3.1and6.4 using two datasets, described below.
Dataset 1: We construct a dataset based off of [43], which is a collection of real-world contact-tracing time series data about the spread of COVID-19 in South Korea during 2020.
The communities corresponding to the SEIRD compartmental module from Section 6.2 are constructed based off of Korea’s provinces, so there is onlyM = 1 type of community for this dataset. Individuals within each province are assumed to interact according to a strongly- connected graph, while the edge weights between each pair of provinces are designed based off passenger traffic data collected in 2018 from KOSIS [84]. The total number of individuals considered in this dataset is approximately 5000, and each individual v ∈ V is modeled as a CHMM with nominal parameter valuesηv(t) calculated based on their “age”, “symptom- onset-date”, “confirmed-date”, and “released-date” fields from the original contact-tracing dataset.
Dataset 2: We construct a more artificial dataset in the following way. We choose a specific number of communities K, each of some size Nk, k ∈ {1,· · · , K} which is randomly chosen from some range. From each community Ck, a subset of members belong to more than one community, and another subsetCk ⊆ Ck of members individual is chosen to be tracked. Some initial distribution of health statuses across the total population is chosen, and the “true”
behavior of disease spread throughout the network is emulated by propagating the CHMM of every individual (tracked or not). This generates a sample path of compartment counts over time per community, as well as health status sequences and observed symptoms over time for each tracked individual in the population; this sample path of values is precisely our dataset. In one of our experiments, we consider different parameters (e.g., community structures, proportion of population which is tracked) in order to demonstrate their effects on the forecasting performance of our multiscale model.
We use a network withK = 10 communities andN = 116 individuals in the population. The communities are roughly equally-partitioned: [15,21,19,26,18,18,20,24,14,20] and there
Code Hyperparam Definition
K Num communities
community size range [min, max] number of members per community multi communities prop Prop. of total population in multiple communities range simult communities [min, max] num communities one person can be in
tracked prop Prop. of people in each community tracked
Table 6.1: Possible simulation parameters to adjust in the code (Dataset 2).
Home Work School
α 0.8×4/14 0.8×4/14 0.8×4/14 γ(r) 0.979/14 0.979/7 0.979/7 γ(d) 0.021/7 0.021/14 0.021/14
Table 6.2: True parameter values for each type of community in Dataset 2.
are a total of 55 tracked individuals. The results can be observed in Figure6.8and Figure6.9.
6.5.2 Impact of Network Topology on Virus Propagation
In this simulation, we apply the full multiscale model described in Sections 6.2 to 6.4 to Dataset 2. We especially determine the effect of the population network structure on the speed and breadth of the virus spread. The number of communities is chosen at random and interconnected according to various topologies:
1. Dense Graph: the number of edges in each community is between 60% and 80% among all possible edges in the community. The number of edges between any two commu- nities is between 1% and 2% among all possible edges. More specifically, the three communities have 564, 1044, and 1274 many edges respectively;
2. Sparse Graph: the number of edges in each community is between 1% and 2% among all possible edges in the community. The number of edges between any two communities is between 1% and 2% among all possible edges;
3. Tree with Multiple Branches: each community is a tree, and the number of branches for each node is between 3 and 6. This corresponds to a structure which is somewhere
Figure 6.8: The estimated parameters ˆβn(1)(t) and ˆαn(t) over time, with constant red line denoting the true value.
Figure 6.9: The estimated parameters ˆγn(r)(t) and ˆγn(d)(t) over time, with constant red line denoting the true value.
in between the densities of the Dense graph and the Sparse graph.
Using the parameters estimated via the HMM model in Section 6.5.2, we now consider the propagation of disease at the large scale. After a similar experiment is performed for Dataset 2 and converting the transition probabilities to transition rates, we use the values given in Table 6.2.
We visualize the trajectories of each compartment in Figure 6.10. Even for this smaller network of 200 nodes, the Sparse Graph took approximately an hour to run on a 2.2-GHz Intel Core i7 Macbook Air. This result provides further experimental motivation for why the CHMM module should be used strictly for small-scale parameter estimation; the ODE module is better suited for community-wide propagation.
We verify the hypothesis that reducing the frequency of contacts between people via inter- ventions methods such as quarantine is helpful in preventing the spread of the virus. For a fixed simulation time of Tsim = 2000, the pandemic in the Dense Graph continues to spread
Figure 6.10: Emulation via HMM module of the dense, tree, and sparse graphs.
until all the people have been infected at least once. For the Tree with Multiple Branches, the pandemic ends at time 860 with 146 people infected at least once, and for the Sparse Graph, it ends at time 799 with only 116 people infected at least once.
6.5.3 The Effects of Superspreaders
In this experiment, we emulate the effect of superspreaders by accounting for high variability in the contact rate of individuals by applying the stochastic model (6.4) to Dataset 1. For the sake of graph simplicity, we reduce the original 23 provinces in the dataset down to two groups: Group 1, corresponding to highly popular locations, contains about 400 members to- tal, and Group 2 contains the remaining 915 individuals. The [βkj] matrix, substituting (6.3) for the interactions of people, are given by:
[βkj] :=
"
β0+σW(t) +ξN(t) 0.0021
0.01 0.005
#
(k,j)
,
where β0 = 0.03, σ = 0.01, λ = 0.0143, and ξi∼Unif[0.5,2]. The values of αk, γk are deter- mined by what was estimated previously from the HMM experiment of Section6.5.1.
0 50 100 150
0 100 200 300 400
0 50 100 150
0 10 20 30 40 50 60 70
0 50 100 150
0 0.5 1 1.5 2
Figure 6.11: The spread of virus visualized over the five compartments for two disjoint groups.
The rate of interactions in the Group 1 is subjected to stochastic noise.
The trajectories of both groups over time are shown in Figure 6.11. The initial number of people in each compartment are chosen to be I1(0) = 26, E1(0) = 30 for Group 1, and I2(0) = 45, E2(0) = 63 for Group 2. For both groups, the remaining individuals are all susceptible. We observe that Group 2 initially enjoys a reduction in the number of infectious individuals for the two months, but increases as a result of coming into contact with members of Group 1, despite the fact that interactions outside of Group 1 are kept at far less than the interactions within Group 1. This experiment verifies an intuitive hypothesis: as a result of the fragile stability, the pandemic ends more slowly than the time it would have taken without the emergence of superspreaders.