MTH 465 Final Project Paper
Kimberly Matsuda
November 15, 2018
The journal article “Simulation Model of Influenza Spread Based on the Small World Network” written by Fatima-Zohra Younsi et. al models the spread of a flu epidemic over time that occurred in the human population of Oran, Algeria in 2009 [3, p. 1]. As a college student on campus, this relates to me as the flu is one of the many diseases that can spread on campus. With so many students and faculty confined to one space, diseases can spread all throughout campus. Because I live on campus, I interact more with other students and faculty on campus than if I were a commuter; then I am more likely to be exposed to another person who has the flu and thus may get the disease. Regardless of location, the flu virus can be spread from one person to another when a healthy person is infected by the virus through contact from an infected person [3, p. 2]. Younsi et. al utilizes a combined SEIR (Susceptible-Exposed-Infected-Removed)-SW (Small World Network) model to model a flu epidemic in Oran, Algeria in order to understand how the flu virus spreads among a human population and provide recommendations to health experts on how to contain and reduce such epidemics [3, p. 1].
In order to model the population of Oran, Algeria, Younsi et. al first con-structs a small world network (SW) sub-model as part of the combined SEIR-SW model to represent the population [3, p. 4]. To construct the SW, a regular lattice of N vertices in which each vertex is connected to its k nearest neighbors is first constructed [3, p. 5]. Each vertex represents an individual of the
popu-lation and an edge between any two vertices signifies that those two individuals (vertices) share a social relationship [3, p. 4-5]. The regular lattice is then trans-formed into a Watts-Strogatz small world network by randomly rewiring some of its edges using a parameter Pw, where Pw is a real number between 0 and 1
and is the probability that any edge of the regular lattice will be rewired [3, p. 5]. For a given edge e of the regular lattice that has vertex i as its endpoint, where i = 1, 2, . . . , N , a real number C between 0 and 1 is randomly chosen [3, p. 5]. If Pw> C, then e is deleted and a new edge is added that connects
i to another vertex m, where m is randomly chosen such that it is not i, not one of i’s k nearest neighbors, or a vertex that is part of a new random edge in the small world network [3, p. 5]. These new random edges signify when two people randomly interact with each other, such as when two people happen to meet on a subway [3, p. 5]. The final result is a small world network, or SW, that represents a human population.
Once an SW of the human population is constructed, Younsi et. al utilizes a SEIR model, a widely-used model in the field of epidemiology, in order to model the spread of influenza through the population [3, p. 6]. At any time t, the SEIR model divides a population into four groups: Susceptible S(t), Exposed E(t), Infectious I(t), Removed R(t) [3, p. 6] (See Figure 1).
Figure 1: SEIR Model Diagram, [3, p. 7]
The model is also provided with initial conditions, S0, I0, E0, R0, where S0, I0, E0, R0
are the number of individuals in the Susceptible group, number of individuals in the Exposed group, number of individuals in the Infectious group, and number of individuals in the Removed group respectively at the initial time t0 [3, p.
7]. The model tracks the number of individuals in each group as they change over time as the disease spreads through the population [3, p. 7]. Individuals who are in the Susceptible group, also known as susceptible individuals, do not have the disease being studied, which in this case is influenza, but can become infected with it if they come into contact with an individual from the Infectious group, also known as an infectious individual [3, p. 6]. If a susceptible indi-vidual comes into contact with an infectious indiindi-vidual, there is a probability α such that the disease is transmitted to the susceptible individual [2]. If the susceptible individual is infected, that individual then moves to the Exposed group; the now exposed individual has the disease but cannot transmit it to a susceptible individual [2]. An exposed individual moves to the Infectious group and becomes an infectious individual at a rate β; in other words, an exposed individual remains in the Exposed group for an average time period of 1β before moving to the Infectious group [3, p. 7]. The now infectious individual can transmit the disease to a susceptible individual with probability α if the infec-tious individual comes into contact with the susceptible individual [2][3, p. 7]. An infectious individual eventually recovers from the disease and moves to the Removed group at a rate γ; in other words, an average time period of 1γ passes before the infectious individual no longer has the disease and is now a removed individual [3, p. 7]. The removed individual then cannot be reinfected with the disease and remains in the Removed group for the remainder of the time for which the SEIR model is modelling the human population [3, p. 7]. This model is what is used by Younsi et. al in modelling the spread of influenza through the population of Oran, Algeria.
In order to model the spread of influenza through the Algerian population, Younsi et. al applies the SEIR model to the SW constructed earlier. When applying the SEIR model to the SW, the number of vertices N of the SW is
taken to be the total population size; further, Younsi et. al assumes that N is fixed [3, p. 7]. Then at any time t, N = S + E + I + R, where S, E, I, R are the number of susceptible individuals, number of exposed individuals, number of infectious individuals, and number of removed individuals respectively [3, p. 7]. Here, the time t is a discrete variable and is measured in days, where the time period ∆t signifies that one day has passed [3, p. 7]. The initial time is set as t0= 0 and initial conditions are set as I(0) = I0, S0= N − I0, E0= 0, R0= 0
where I0> 0 [3, p. 7]. In other words, Younsi et. al’s model assumes that the
population starts with an I0 number of infectious individuals with the rest of
the population being susceptible individuals [3, p. 7]. Then, to apply the SEIR model to the SW, I0vertices are chosen randomly to be infectious while the rest
of the vertices are susceptible [3, p. 7]. The SW changes over time as vertices move from one group of the SEIR model to another, transforming the SW to a SEIR-SW graph [3, p. 7]. The number of infectious vertices are tracked at each time t in order to monitor how influenza spreads through the graph [3, p. 7]. As discussed before, influenza can spread from one person to another if those people have contact with one another and only infectious individuals can spread influenza to susceptible individuals [2] [3, p. 2]. Further, since an edge of the graph between any two vertices means that those vertices have contact with one another, an edge between an infectious vertex and susceptible vertex means that those vertices have contact with one another and the susceptible vertex may become infected by the infectious vertex [3, p. 7]. To understand the probability that a susceptible vertex is infected, consider a susceptible vertex 1 connected to two other susceptible vertices 3 and 4 and two infectious vertices 2 and 5 as in Figure 2. Here, vertices are colored to distinguish between susceptible and infectious vertices. A green vertex is susceptible while a red vertex is infectious.
Figure 2: Small-Scale Model of Probability of Susceptible Vertex Becoming Infected
Since α is the probability such that an infectious vertex transmits the disease to a susceptible vertex if the two vertices have contact with one another, then there is a 1 − α probability that the disease will not be transmitted [3, p. 7]. Further, since susceptible vertices are not infected, susceptible vertices cannot transmit the disease to other susceptible vertices. Then there is a 1 probability that the disease is not transmitted from a susceptible vertex to another susceptible vertex. These corresponding probabilities are shown in Figure 3.
Figure 3: Small-Scale Model of Probability of Susceptible Vertex Becoming Infected (Probabilities of No Infection to Vertex 1 Displayed)
Under the assumption that contact between any two vertices does not affect whether or not the disease is transmitted due to contact between another pair of vertices, by the rules of probability of independent events, the probability that the disease is not transmitted to vertex 1 from any of its contacts between
vertices 2, 3, 4, 5, denoted as P (No Infection to Vertex 1), is
P (No Infection to Vertex 1) = (1 − α) · 1 · 1 · (1 − α) = (1 − α)2
Then the probability that vertex 1 is infected, denoted as P (Infection to Vertex 1), is
P (Infection to Vertex 1) = 1 − P (No Infection to Vertex 1) = 1 − (1 − α)2
Further, more generally, the probability that a susceptible vertex vs becomes
infected, denoted as P (Infection to vs), at a time t is
P (Infection to vs) = 1 − (1 − α)ki
where ki is the number of infectious neighbors of vs at time t [3, p. 7]. If vs
is infected, then vs becomes an exposed vertex ve [3, p. 7]. The time that ve
spends in the exposed group, denoted as te, follows an exponential distribution,
Exp(β1) [3, p. 7]. In other words, a random number from this distribution rounded to the nearest integer is generated for each veto be the amount of time
that ve is in the Exposed group before it moves to the Infectious group [3, p.
7]. Further, if tl is the time that vs becomes ve, then ve will remain in the
Exposed group until t = tl+ te [3, p. 7]. After this time period has passed, ve
becomes an infectious vertex vi[3, p. 7]. vi eventually recovers from the disease
after a time period ti, which follows an exponential distribution Exp(γ1), and
is randomly chosen similar to te; vi is now a removed vertex vr [3, p. 7]. Once
a vertex becomes vr, the vertex cannot return to the susceptible group [3, p.
8]. In other words, the vertex cannot be reinfected with the disease and is permanently immune [3, p. 8]. The current time tlis then set to tl+ 1 and the
SEIR-SW graph at time tl has changed to a new SEIR-SW graph at tl+ 1 [3,
p. 8]. The previous processes are applied to all vertices at each t until there are no more infectious vertices [3, p. 8]. The resulting graphs at each t and plot of the number of infectious vertices at each t can then illustrate how the disease spreads through the graph over time.
To see how the SEIR-SW model of Younsi et. al works in practice, a few small-scale models were developed in the programming language Mathematica. The function CreateSW was first used to construct a SW (See Appendix A). A regular lattice of n = 10 vertices, where each vertex was connected to its k = 4 nearest neighbors was first constructed. Then some of the regular lattice’s edges were rewired, where p = 0.1 is the probability that any of its edges were rewired. The resulting SW is shown in Figure 4.
Figure 4: Small World Network (SW), 10 vertices, 4 neighbors, probability of rewiring 0.1
This SW was then input into the function CreateSEIRSW (See Appendix B). The parameters I0 = 2, α = 0.25, β = 2, γ = 4 were also input into this
func-tion. Then at t = 0, two vertices were chosen at random to be infectious while the rest of the vertices became susceptible. Following the same procedures as those outlined by Younsi et. al, the graph changes over time as vertices move
through the SEIR model. The evolving graph at each t is shown in Figure 5. For all SEIR-SW graphs, the vertices are colored to signify which group each vertex belongs to in the SEIR model. Green vertices are susceptible vertices, yellow vertices are exposed vertices, red vertices are infectious vertices, and blue vertices are removed vertices.
Figure 5: SEIR-SW Models at each t; Top left is t = 0 and from left to right is increasing t
To further see how the disease spreads through the population over time, similar to Younsi et. al, the number of infectious vertices was tracked and plotted at each t as shown in Figure 6.
Figure 6: Number of infectious vertices of Figure 4 at each time step
Plots like that in Figure 6 are useful for illustrating the behavior of an epidemic over time. In this case, Figure 6 shows that the number of infectious vertices increased until t = 15 and then decreased until the number of infectious vertices reaches 0 at t = 22. Now consider an SEIR-SW model on a slightly larger scale. As shown in Figure 7, an SW was constructed from a regular lattice that had n = 50 vertices that were each connected to their k = 10 nearest neighbors and then had its edges rewired with probability p = 0.1.
Figure 7: Small World Network (SW), 50 vertices, 10 neighbors, probability of rewiring 0.1
Figure 4, the spread of disease through the SW in Figure 7 is once again mon-itored. The initial SEIR-SW graph at t = 0, the SEIR-SW graph at the final time, and a plot of the number of infectious vertices over time are shown in Figure 8.
(a) SEIR-SW at t = 0 (b) SEIR-SW at final t
(c) Number of infectious vertices at each t
Figure 8: SEIR-SW graphs at t = 0 and final t and plot of number of infectious vertices at each t using Figure 7 graph
Similar to the graph at the final time in Figure 5, Figure 8b shows that every vertex was infected as every vertex is blue, indicating that every vertex recovered from the disease. Realistically, this represents the worst case scenario for an epidemic in that every individual of a population was infected. However, it is possible that some individuals are not infected during an epidemic. Another trial was conducted using the same SW in Figure 7 and the same values for the parameters I0, α, β, γ as before and the resulting initial and final graphs and
plot of infectious vertices are shown in Figure 9.
(a) SEIR-SW at t = 0 (b) SEIR-SW at final t
(c) Number of infectious vertices at each t
Figure 9: SEIR-SW graphs at t = 0 and final t and plot of number of infectious vertices at each t using Figure 7 graph
As Figure 9b shows, not every vertex was infected as there is one susceptible vertex. However, this is not much different from the worse case scenario of every vertex being infected. A more interesting case would be where there is more than a few susceptible vertices in the final graph. This would suggest changing one or more of the parameters I0, α, β, γ for the same SW in Figure 7. Consider
the parameter α. In order to increase the chances of obtaining a final graph such that there are more than a few susceptible vertices, this would suggest decreasing α, which thus decreases the probability of transmitting the disease from infectious vertices to susceptible vertices. So, another trial was conducted using the same SW in Figure 7 and same values for I0, β, γ as before, but α was
decreased from 0.25 = 14 to 0.0625 = 161. Figure 10 shows the initial and final graph and plot of infectious vertices at each t for this trial.
(a) SEIR-SW at t = 0 (b) SEIR-SW at final t
(c) Number of infectious vertices at each t
Figure 10: SEIR-SW graphs at t = 0 and final t and plot of number of infectious vertices at each t using Figure 7 graph and α = 0.0625
Unlike the final graphs in Figures 8b, 9b, the final graph in Figure 10b has a larger number of susceptible vertices. In addition, notice the differences in the plots of the number of infectious vertices at each t in Figures 9c and 10c. The number of infectious vertices in Figure 10c initially relatively increases at a slower rate than that in Figures 8c and 9c. Simply changing the parameter α changes the behavior of the disease significantly. Similar experiments were conducted by Younsi et. al using the SEIR-SW model except on a much larger scale in order to understand how influenza spreads through the population and understand what factors influence the spread of influenza.
Younsi et. al utilizes the SEIR-SW model as described before to simulate the number of influenza cases in Oran, Algeria in 2009 over time and understand how the parameters α, β, and γ and the properties and structure of the SW affect the spread of influenza through the population [3, p. 8]. In Oran, Algeria, in 2009, two influenza epidemics occurred, where the first occurred from Week 34 to Week 43 and the second occurred from Week 44 to Week 52 [3, p. 8]. Data on the number of influenza cases over both waves was provided by the Public Health Department of Oran; graphs of this data are shown in Figure 11 [3, p. 8].
Figure 11: Time Series of Influenza Cases (First and Second Waves), Oran, Algeria 2009 [3, p. 8]
9]. These parameters were either derived from the data or estimated based on the data [3, p. 9].
Table 1: Estimated parameter values for SEIR-SW Model [3, p. 9]
Parameters Value(First Wave) Value(Second Wave)
S0 1350 5020 I0 8 157 k 8 10 Pw 0.1 0.1 α 0.16 0.13 β 2.2 2 γ 4.5 4
These parameter values were then input into the algorithm to construct the SEIR-SW model and track the number of infectious individuals over time [3, p. 9]. The graphs for the number of infectious individuals over time for both the first and second epidemic waves are shown in Figure 12 [3, p. 9].
Figure 12: Simulated Influenza Cases using the SEIR-SW Model (First and Second Waves), Oran, Algeria 2009 [3, p. 9]
While the shapes of the graphs for the simulated epidemic waves and the epi-demic waves from the data are not identical, both sets of graphs share similar behavior as to how the number of infectious individuals changed over time [3, p. 9]. For the first wave, both the simulated graph and the data graph illustrate that the number of infectious individuals increased until about Week 39 − 40, and then started to decrease [3, p. 8-9]. For the second wave, both the simulated graph and the data graph illustrate that the number of infectious individuals increased until about Week 48 − 49, and then continually decreased until the number of infectious individuals was 0 at Week 52 [3, p. 8-9]. The SEIR-SW
model thus is accurate for modelling the behavior of an influenza epidemic over a period of time [3, p. 8-9]. It was further found from the simulated epidemic waves that the parameters α, β, γ, k of the SEIR-SW model have a significant impact on the spread of the epidemic over time [3, p. 9]. Increasing one or more of these parameters caused the epidemic to “spread more rapidly, and vice versa” [3, p. 9]. Younsi et. al also investigated the properties and structure of the SEIR-SW graphs from the simulations to determine if they affect the spread of influenza [3, p. 10]. The values of the properties of the SEIR-SW graphs from both epidemic waves are given in Table 2.
Table 2: Small World Network Analysis for the two influenza epidemic waves [3, p. 10]
Properties of Social Network First Wave Second Wave
Number of vertices 10864 51770
Number of edges 8 157
Average path length 3.369 3.668
Clustering coefficient 0.386 0.383
Diameter 5 5
Average degree distribution 8 10
Density 0.012 0.004
Modularity(communities) 0.723(17) 0.759(32)
The modularity of a graph is a measure of how strongly the vertices can be separated into groups, clusters, or communities where the vertices are densely interconnected, while the average path length is the average shortest distance between pairs of vertices [1] [3, p. 6]. In addition to the modularity and average path length, the diameter and clustering coefficient was about the same for both epidemic waves despite the differences in the number of vertices and edges [3, p. 10]. However, the number of communities was greater for the second wave than for the first wave, which is due to the larger size of the network in the second
wave, or the larger number of vertices and edges, and the larger average degree distribution of the second wave [3, p. 10]. The larger number of communities in the second wave signifies that more vertices are densely connected to each other, so susceptible vertices are more likely to come into contact with infec-tious vertices and get influenza [3, p. 11]. Further, the epidemic spreads more rapidly as the size of the network and the average vertex degree k increases [3, p. 11]. Furthermore, Younsi et. al recommends that k be kept as low as possible during an epidemic; in other words, individuals should reduce contact with each other as much as possible to reduce the spread of influenza [3, p. 11]. Younsi et. al further concludes from the results that the structure and properties of a network representing a population in combination with the parameters α, β, γ of influenza are significant variables in modeling and predicting an epidemic; further, the proposed SEIR-SW model is accurate in this prediction as demon-strated by Figures 11 and 12 [3, p. 12]. Younsi et. al finally concludes that knowledge of this information can be useful for public health experts and deci-sion makers to take more effective preventive measures in an influenza epidemic [3, p. 12].
References
[1] “Modularity.” Wikipedia, 14 Nov. 2018. Web. 24 Nov. 2018.
[2] “SEIR and SEIRS models.” Institute for Disease Modeling, n.d. Web. 03 Nov. 2018.
[3] Younsi, Fatima-Zohra, et. al. “SEIR-SW, Simulation Model of Influenza Spread Based on the Small World Network.” Tsinghua Science and Tech-nology, vol. 20, no. 5, 2015, pp. 460-473. Web. 01 Nov. 2018.
