Social distancing and COVID-19 spread

1. Introduction

We are interested in exploring the effect of “social distancing” to “spread of virus”. In particular, we wish to answer the following question: “What if some people do not follow the guideline for social distancing?” Since the society is connected to each other, we think that the effect of disobeying the social distancing might be more severe than we thought.

To check this idea, we make the following assumptions:

The population consists of two groups: One is following the social distancing. The other does not follow the social distancing. We call the former as “normal” group and the latter as “non-normal” group. Let \(W_1\) be the population proportion of the normal group and \(W_2\) be the population proportion of the non-normal group.
A person in the normal group meets \(m_1\) people daily on average. Also, a person in the non-normal group meets \(m_2\) people daily on average, where \(m_2\) is larger than \(m_1\). The actual numbers of the meetings are generated from Poisson distributions. The population average is \(m=W_1 m_1 + W_2 m_2\). We set \(m=4.4\).
If a carrir (= person with virus) meets someone at random, the virus will be tranfered with probability \(p=0.03\).
If someone get the virus then he/she will be available (without self-segregration) for ten day. After 10 days, he/she will be hospotalized.

In the first simulation, we consider three scenarios:

No violation of the social distancing.
Modest violation (20% meets \(m=10\) people daily)
Severe violation (5% meets \(m=31\) people daily)

2. Computation

Notation:

\(Q_t\): the number of new cases at the \(t\)-th date.
\(C_t\): cumulative number of cases at the \(t\)-th date.
\(n_{tj}\): realized number of meetings for group \(j\) at the \(t\)-th date.
\(a\): initial cases (we used \(a=10\))
\(W_j\): population proportion for group \(j\) (\(j=1\) normal group)
\(p_t\): infection rate for each meeting at the \(t\)-th date
\(N\): total population size (N=50,000,000)

Formula

We compute the number of cases recursively by the following formula.

\[ Q_t = \left(C_{t-1} - C_{t - 10}\right)\left(\sum_{j = 1}^2 W_j^*n_{tj}\right) p_{t-1} \]

where initial values are

\[ C_{-8} = C_{-7} = \cdots =C_0 = 0 \]

and \(W_j^*\) and \(n_{tj}\) are

\[ W_j^* = \frac{m_jW_j}{\sum_{j= 1}^2{m_jW_j}}, \ \ \ n_{tj} \sim Poi(m_j) \]

Also, the infection probability also changes over time: \[ p_t = p_0 \left(1 - \frac{C_t}{N} \right) \]

3. Simulation Study

Simulation Parameters

p_0 = 0.03 # Initial probability
N = 50000000 # Number of population
a = 10 # initial number of cases
B = 1000 # Number of Simulation
set.seed(1)

##### Complex Model ######

m = list(c(4.4),  c(3, 10), c(3, 31))
w = list(c(1),  c(0.8, 0.2), c(0.95, 0.05))
l = 50
res = mapply(resmat, m = m, w = w, MoreArgs = list(l = l, isSimple = FALSE))
yli = min(res[[length(m)]][[1]])
yui = max(res[[length(m)]][[1]])
colm = c(3, 2, 4, 0)
for(i in 1:length(m)){
  boxplot(res[[i]][[1]], outline = FALSE, #main = sprintf("m = %g, w = %g", m[[i]], w[[i]]), 
          col = colm[i], log = "y", ylim = c(yli, yui), ylab = "log-scale cases", xlab = "time")
  Csummary = apply(res[[i]][[1]], 2, function(x){summary(x)[2:5]})
  points(1:l, Csummary["Mean",], type = "l")
  legend("topleft", legend =  sprintf("m = %g, w = %g", m[[i]], w[[i]]))
}

par(mfrow = c(1,1))
for(i in length(m):1){
  boxplot(res[[i]][[1]], outline = FALSE, 
          col = colm[i], add = if(i != length(m)) TRUE else FALSE,
          main = "number of cases w/ different values of m and w", log = "y", ylab = "log-scale cases", xlab = "time")
  Csummary = apply(res[[i]][[1]], 2, function(x){summary(x)[2:5]})
  points(1:l, Csummary["Mean",], type = "l")
}
legend("topleft", legend = paste("m=", m,"w=", w), fill = colm, cex = 1)

Each box plot represents the median, 25%, 75% quantile of the cumulative cases and the solid line represents the mean of simulated results.

# C_t; cumulative number of cases
sapply(sequence(length(m)), function(x){
  apply(res[[x]][[1]], 2, function(x){summary(x)["Mean"]})
})

##            [,1]       [,2]         [,3]
##  [1,]  10.00000   10.00000 1.000000e+01
##  [2,]  11.31070   11.82686 1.384738e+01
##  [3,]  12.78459   14.04300 1.919471e+01
##  [4,]  14.46107   16.63239 2.660584e+01
##  [5,]  16.39074   19.76856 3.694127e+01
##  [6,]  18.58268   23.39759 5.113935e+01
##  [7,]  21.05753   27.75163 7.072856e+01
##  [8,]  23.84220   32.95962 9.808159e+01
##  [9,]  26.97328   39.13853 1.360711e+02
## [10,]  30.51003   46.43636 1.890271e+02
## [11,]  33.30294   53.16631 2.586733e+02
## [12,]  36.14460   60.82199 3.536247e+02
## [13,]  39.20156   69.55686 4.832465e+02
## [14,]  42.56114   79.31739 6.599279e+02
## [15,]  46.01500   90.34830 8.981439e+02
## [16,]  49.68053  102.69661 1.223551e+03
## [17,]  53.49801  116.47462 1.664918e+03
## [18,]  57.39135  132.01049 2.268301e+03
## [19,]  61.44799  149.44952 3.088430e+03
## [20,]  65.47572  168.54804 4.202746e+03
## [21,]  69.68483  190.26594 5.727279e+03
## [22,]  73.98429  214.27490 7.812510e+03
## [23,]  78.57553  241.40735 1.063732e+04
## [24,]  83.36972  271.72228 1.449791e+04
## [25,]  88.27871  305.13862 1.977731e+04
## [26,]  93.61660  342.23186 2.689704e+04
## [27,]  98.96162  383.91214 3.675466e+04
## [28,] 104.34369  430.61915 5.017698e+04
## [29,] 110.14341  482.69170 6.826366e+04
## [30,] 115.93314  541.31581 9.287770e+04
## [31,] 122.05147  606.42073 1.266673e+05
## [32,] 128.48073  679.42897 1.727073e+05
## [33,] 135.05608  761.38316 2.345615e+05
## [34,] 141.81356  853.30550 3.183563e+05
## [35,] 148.85399  954.80860 4.324352e+05
## [36,] 156.02105 1070.30369 5.872461e+05
## [37,] 163.56219 1196.87993 7.964382e+05
## [38,] 171.37307 1339.98653 1.079610e+06
## [39,] 179.35982 1498.86696 1.460942e+06
## [40,] 187.81436 1674.10623 1.974924e+06
## [41,] 196.40267 1868.85313 2.653908e+06
## [42,] 205.60729 2085.51436 3.553834e+06
## [43,] 214.58563 2329.99521 4.729330e+06
## [44,] 224.45517 2607.21433 6.264454e+06
## [45,] 234.29028 2915.29653 8.211535e+06
## [46,] 244.51011 3257.67781 1.063363e+07
## [47,] 255.22960 3638.87324 1.353968e+07
## [48,] 266.16251 4071.92587 1.693918e+07
## [49,] 277.65303 4550.99143 2.072885e+07
## [50,] 289.33603 5082.92722 2.484114e+07

At each row, the expected numbers of total cases are presented for each scenario.

4. Further simulation Study

We also consider other scenarios for the 5% outlier groups:

##### Complex Model ######
m = list(c(3, 30), c(5, 10),  c(3, 20))
w = list(c(0.99, 0.01), c(0.95, 0.05), c(0.95, 0.05))

l = 50
res = mapply(resmat, m = m, w = w, MoreArgs = list(l = l, isSimple = FALSE))
yli = min(res[[length(m)]][[1]])
yui = max(res[[length(m)]][[1]])
colm = c(3, 2, 4, 0)
for(i in 1:length(m)){
  boxplot(res[[i]][[1]], outline = FALSE, #main = sprintf("m = %g, w = %g", m[[i]], w[[i]]), 
          col = colm[i], log = "y", ylim = c(yli, yui), ylab = "log-scale cases", xlab = "time")
  Csummary = apply(res[[i]][[1]], 2, function(x){summary(x)[2:5]})
  points(1:l, Csummary["Mean",], type = "l")
  legend("topleft", legend =  sprintf("m = %g, w = %g", m[[i]], w[[i]]))
}

par(mfrow = c(1,1))
for(i in length(m):1){
  boxplot(res[[i]][[1]], outline = FALSE, 
          col = colm[i], add = if(i != length(m)) TRUE else FALSE,
          main = "number of cases w/ different values of m and w", log = "y", ylab = "log-scale cases", xlab = "time")
  Csummary = apply(res[[i]][[1]], 2, function(x){summary(x)[2:5]})
  points(1:l, Csummary["Mean",], type = "l")
}
legend("topleft", legend = paste("m=", m,"w=", w), fill = colm, cex = 1)

Each box plot represents the median, 25%, 75% quantile of the cumulative cases and the solid line represents the mean of simulated results.

# C_t; cumulative number of cases
sapply(sequence(length(m)), function(x){
  apply(res[[x]][[1]], 2, function(x){summary(x)["Mean"]})
})

##             [,1]       [,2]        [,3]
##  [1,]   10.00000   10.00000    10.00000
##  [2,]   11.64691   11.64154    12.19817
##  [3,]   13.54720   13.52920    14.88076
##  [4,]   15.77245   15.74376    18.21144
##  [5,]   18.31728   18.35275    22.22590
##  [6,]   21.32774   21.38052    27.12713
##  [7,]   24.81731   24.90517    33.12132
##  [8,]   28.87238   29.04948    40.50012
##  [9,]   33.51660   33.70933    49.60793
## [10,]   39.01048   39.29479    60.68669
## [11,]   43.74628   44.17294    72.00421
## [12,]   49.02957   49.53764    85.20119
## [13,]   54.83341   55.41432   100.93959
## [14,]   61.20365   61.97356   119.25853
## [15,]   68.26214   69.12290   140.84221
## [16,]   75.92823   76.95933   166.05466
## [17,]   84.26204   85.51983   195.67648
## [18,]   93.37559   94.92948   230.43942
## [19,]  103.38551  104.84897   270.85706
## [20,]  114.04453  115.69599   317.52695
## [21,]  125.83909  127.49901   372.43305
## [22,]  138.42739  140.32563   436.82193
## [23,]  152.28831  154.54605   511.06729
## [24,]  167.22913  170.28561   597.85835
## [25,]  183.46060  187.12675   700.41612
## [26,]  201.08438  205.17820   820.45671
## [27,]  220.24249  224.74585   958.33928
## [28,]  241.19425  246.29622  1120.46910
## [29,]  264.08343  269.29116  1310.52808
## [30,]  288.55907  294.50692  1532.14482
## [31,]  315.21346  321.90641  1785.14406
## [32,]  344.30926  351.05699  2085.44398
## [33,]  375.82131  382.85723  2435.04830
## [34,]  410.27385  418.14378  2843.74335
## [35,]  447.22578  455.75022  3322.07659
## [36,]  487.15218  497.44123  3875.48344
## [37,]  531.61434  543.08111  4521.63686
## [38,]  579.67300  592.31657  5278.73166
## [39,]  631.36705  645.63976  6155.94086
## [40,]  688.51652  704.76837  7189.19816
## [41,]  748.61709  767.97750  8404.07102
## [42,]  815.30020  836.54767  9814.11096
## [43,]  888.69548  910.13925 11419.65936
## [44,]  967.26958  989.22339 13325.75059
## [45,] 1052.41680 1074.09801 15560.53027
## [46,] 1144.16016 1170.61506 18148.61975
## [47,] 1245.10064 1272.07110 21200.68291
## [48,] 1354.25982 1385.63197 24753.79771
## [49,] 1470.83192 1505.75411 28933.47531
## [50,] 1598.91414 1640.83817 33703.28541

5. Discussion

Findings (from first simulation)

Under scenario 1 (everybody follows the social distancing), the total cases increase only 29 times after 50 days.
Under scenario 2 (80% follows social distancing, but 20% make modest violation by meeting 10 people every day), the total cases increase 508 times after 50 days.
Under scenario 3 (95% follows social distancing, but 5% make severe violation by meeting 31 people every day), the total cases increase more than 248,000 times after 50 days.
On average, the above three scenarios have the same value of average meetings (\(m=4.4\)), but the existence of the outlying groups makes a big difference in the total number of cases.

Findings (from the second simulation)

Under Scenario 1 (95% follows \(m=5\) and 5% follows \(m=10\)), the total number of cases increases to 159 times.
Under scenorio 2 (99% follows \(m=3\) and 1% follows \(m=30\)), the total number of cases is about the same as the first scenario.
Under Scenario 3 (95% follows \(m=3\) and 5% follows \(m=20\)), the total number of cases increases to 3368 times.

Social distancing and COVID-19 spread

Jae-kwang Kim and Yonghyun Kwon

21 March, 2020

1. Introduction

2. Computation

Notation:

Formula

3. Simulation Study

4. Further simulation Study

5. Discussion

Findings (from first simulation)

Findings (from the second simulation)

For comments or questions, please contact "jkim@iastate.edu"

Social distancing and COVID-19 spread

Jae-kwang Kim and Yonghyun Kwon

21 March, 2020

1. Introduction

2. Computation

Notation:

Formula

3. Simulation Study

4. Further simulation Study

5. Discussion

Findings (from first simulation)

Findings (from the second simulation)

For comments or questions, please contact "jkim@iastate.edu"

Social distancing is critical!