# Forecasting Social Conflicts in Africa Using an Epidemic Type Aftershock Sequence Model

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## Abstract

**:**

## 1. Introduction and Motivation

## 2. Social Conflicts in Africa

## 3. The ETAS Model

#### 3.1. Representation

#### 3.2. Estimation

#### 3.3. Inference

## 4. Modeling and Forecasting Social Conflicts in Africa

#### 4.1. Parameter Estimates

#### 4.2. Residuals

#### 4.3. Further Results

#### 4.4. Forecasting for 2017

## 5. Conclusions and Further Research

#### Limitations and Future Research

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Magnitude Distribution of Social Conflicts

^{4}= 0.002%, which is about the frequency of the largest event. Hence, if we use 4 full magnitude steps, or 40 smaller steps, the frequency of the smallest and largest events follow the Gutenberg–Richter law. This means we will use a magnitude range of (${M}_{min}$, ${M}_{max}={M}_{min}+4$). Lastly, we have to determine ${M}_{min}$, which can be chosen rather arbitrarily. In earthquake studies, the minimum magnitude is often set at ${M}_{min}=3$, as this associates with an earthquake considered of reasonable size. We will also set ${M}_{min}=3$ which gives a magnitude range of $\{{M}_{min},{M}_{max}\}=\left\{3,7\right\}$. For this magnitude range, $x\cong 20.56\%$ and the set ${\{x/{\omega}^{n}\}}_{n=0}^{n=40}$ gives the percentages of events for each magnitude {3.0, 3.1, …, 7.0}.

**Table A1.**The magnitude distribution of the social conflict data. For each magnitude, there is the corresponding number of fatalities, the expected percentage of events, the observed percentage of events, the expected number of events and the observed number of events.

Magnitude | Fatalities | % Expected | % Observed | Expected | Observed |
---|---|---|---|---|---|

3 | 2 | 20.57 | 26.94 | 3565 | 4669 |

3.1 | 3 | 16.34 | 14.49 | 2832 | 2511 |

3.2 | 4 | 12.98 | 17.29 | 2250 | 2997 |

3.3 | 6 | 10.31 | 8.64 | 1787 | 1498 |

3.4 | 8 | 8.19 | 4.76 | 1419 | 825 |

3.5 | 10 | 6.50 | 10.00 | 1127 | 1733 |

3.6 | 10 | 5.17 | 0.00 | 896 | 0 |

3.7 | 12 | 4.10 | 3.91 | 711 | 677 |

3.8 | 15 | 3.26 | 3.64 | 565 | 631 |

3.9 | 19 | 2.59 | 2.02 | 449 | 351 |

4 | 22 | 2.06 | 1.77 | 357 | 306 |

4.1 | 27 | 1.63 | 1.29 | 283 | 224 |

4.2 | 31 | 1.30 | 1.07 | 225 | 186 |

4.3 | 38 | 1.03 | 0.93 | 179 | 162 |

4.4 | 45 | 0.82 | 0.33 | 142 | 58 |

4.5 | 50 | 0.65 | 0.85 | 113 | 147 |

4.6 | 58 | 0.52 | 0.44 | 90 | 76 |

4.7 | 67 | 0.41 | 0.34 | 71 | 59 |

4.8 | 78 | 0.33 | 0.27 | 57 | 47 |

4.9 | 97 | 0.26 | 0.05 | 45 | 9 |

5 | 100 | 0.21 | 0.33 | 36 | 57 |

5.1 | 107 | 0.16 | 0.13 | 28 | 22 |

5.2 | 129 | 0.13 | 0.07 | 22 | 13 |

5.3 | 150 | 0.10 | 0.00 | 18 | 0 |

5.4 | 183 | 0.08 | 0.07 | 14 | 12 |

5.5 | 204 | 0.07 | 0.05 | 11 | 8 |

5.6 | 248 | 0.05 | 0.05 | 9 | 8 |

5.7 | 310 | 0.04 | 0.03 | 7 | 6 |

5.8 | 349 | 0.03 | 0.00 | 6 | 0 |

5.9 | 400 | 0.03 | 0.05 | 4 | 8 |

6 | 400 | 0.02 | 0.00 | 4 | 0 |

6.1 | 409 | 0.02 | 0.02 | 3 | 3 |

6.2 | 458 | 0.01 | 0.01 | 2 | 2 |

6.3 | 590 | 0.01 | 0.00 | 2 | 0 |

6.4 | 597 | 0.01 | 0.02 | 1 | 4 |

6.5 | 597 | 0.01 | 0.00 | 1 | 0 |

6.6 | 597 | 0.01 | 0.00 | 1 | 0 |

6.7 | 598 | 0.00 | 0.01 | 1 | 1 |

6.8 | 598 | 0.00 | 0.00 | 1 | 0 |

6.9 | 599 | 0.00 | 0.00 | 0 | 0 |

7 | 600 | 0.00 | 0.01 | 0 | 1 |

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**Figure 2.**Map of social conflicts in Africa in the period 2012–2017 with at least two fatalities. Events with more than 100 fatalities are colored red.

**Figure 3.**The latitude and longitude of events over time. The red lines represent the latitude and longitude of the geographical center of Africa. In the latitude plot, points above the red line are more north and points below are more south. In the longitude plot, points above the red line are more east and points below the red line are more west.

**Figure 4.**Distribution of the number of events. The blue bars in the top figure show the number of events for each magnitude and the red line is the number of events corresponding to the Gutenberg–Richter law. In the bottom figure, the logarithm base 10 of the same data is shown. If the magnitude distribution follows the Gutenberg–Richter law, the blue bars in the bottom figure should correspond to the red line.

**Figure 5.**The five major regions of Africa: North, West, Central, East and South. Source: Wikimedia Commons, the free media repository.

**Figure 6.**Spatial background distributions of the model with the largest log-likelihood for the region North Africa. The area is divided into ${N}_{c}=4070$ cells. In these cells, the background intensity is assumed to be homogeneous. The cells give the probability of a background event occurring in that cell.

**Figure 7.**Spatial background distributions of the model with the largest log-likelihood for the region West Africa. The area is divided into ${N}_{c}=1102$ cells. In these cells, the background intensity is assumed to be homogeneous. The cells give the probability of a background event occurring in that cell.

**Figure 8.**Spatial background distributions of the model with the largest log-likelihood for the region Central Africa. The area is divided into ${N}_{c}=3330$ cells. In these cells, the background intensity is assumed to be homogeneous. The cells give the probability of a background event occurring in that cell.

**Figure 9.**Spatial background distributions of the model with the largest log-likelihood for the region East Africa. The area is divided into ${N}_{c}=4368$ cells. In these cells, the background intensity is assumed to be homogeneous. The cells give the probability of a background event occurring in that cell.

**Figure 10.**Results of the residual analysis for the North Africa catalog. In the top left (

**a**) figure the observed number of events is plotted against the transformed times from (11). In the top right figure (

**b**), the observed number of events (blue) is plotted against the expected number of events by the model (red). In the bottom two figures (

**c**) and (

**d**), the observed number of background and triggered events (blue) are compared with the number of events expected by the model (red).

**Figure 11.**Forecasts for North Africa. Top left: event catalog 2012–2016. Top right: City map with cities with 1000+ population. Bottom left: spatial background distribution estimated from the data from 2012–2016. Bottom right: forecast for the expected number of events with $M\ge 4$in each cell in the year 2017, together with the locations of the actual conflicts that have taken place with $M\ge 4$.

**Figure 12.**Forecasts for West Africa. Top left: event catalog 2012–2016. Top right: City map with cities with 1000+ population. Bottom left: spatial background distribution estimated from the data from 2012–2016. Bottom right: forecast for the expected number of events with $M\ge 4$in each cell in the year 2017, together with the locations of the actual conflicts that have taken place with $M\ge 4$.

**Figure 13.**Forecasts for Central Africa. Top left: event catalog 2012–2016. Top right: City map with cities with 1000+ population. Bottom left: spatial background distribution estimated from the data from 2012–2016. Bottom right: forecast for the expected number of events with $M\ge 4$in each cell in the year 2017, together with the locations of the actual conflicts that have taken place with $M\ge 4$.

**Figure 14.**Forecasts for East Africa. Top left: event catalog 2012–2016. Top right: City map with cities with 1000+ population. Bottom left: spatial background distribution estimated from the data from 2012–2016. Bottom right: forecast for the expected number of events with $M\ge 4$ in each cell in the year 2017, together with the locations of the actual conflicts that have taken place with $M\ge 4$.

**Table 1.**Fatality quantiles of events in the Armed Conflict Location and Event Data (ACLED) dataset between 2012 and 2017 with at least 2 fatalities.

Quantile | Fatalities | Quantile | Fatalities |
---|---|---|---|

0 | 2 | 0.91 | 20 |

1 | 2 | 0.92 | 22 |

2 | 2 | 0.93 | 25 |

3 | 3 | 0.94 | 28 |

4 | 3 | 0.95 | 31 |

5 | 4 | 0.96 | 38 |

6 | 6 | 0.97 | 47 |

7 | 8 | 0.98 | 58 |

8 | 10 | 0.99 | 96 |

9 | 19 | 1 | 600 |

**Table 2.**The magnitude distribution of the social conflict data. For each magnitude there is the corresponding number of fatalities, the percentage of events expected by the Gutenberg–Richter law, the observed percentage of events, the expected absolute number of events and the observed absolute number of events. The full table appears in the Appendix A.

Magnitude | Fatalities | % Expected | % Observed | Expected | Observed |
---|---|---|---|---|---|

3 | 2 | 20.57 | 26.94 | 3565 | 4669 |

4 | 22 | 2.06 | 1.77 | 357 | 306 |

5 | 100 | 0.21 | 0.33 | 36 | 57 |

6 | 400 | 0.02 | 0.00 | 4 | 0 |

7 | 600 | 0.00 | 0.01 | 0 | 1 |

**Table 3.**Parameter restrictions in our Epidemic Type Aftershock Sequence (ETAS) model. These restrictions are based on the restrictions used for modeling earthquake occurrences and on values that are physically desirables, see [10].

Parameter | Lower Bound | Upper Bound |
---|---|---|

$\mu $ | 0 | 1 |

k | 0.001 | 0.1 |

p | 0.5 | 2 |

c | 1.00 × 10^{−5} | 0.1 |

$\alpha $ | 0 | 2 |

d | 0.01 | 1 |

q | 1 | 3 |

$\gamma $ | 0 | 2 |

**Table 4.**Number of conflict events in each region for the learning period (2012), study period (2013–2016) and forecasting period (2017) and in total (2012–2017).

North | West | Central | East | South | Total | |
---|---|---|---|---|---|---|

N learning period | 291 | 378 | 203 | 708 | 40 | 1620 |

N study period | 3518 | 2164 | 1834 | 4617 | 117 | 12250 |

N forecasting period | 612 | 617 | 589 | 1630 | 16 | 3464 |

N total | 4421 | 3159 | 2626 | 6955 | 173 | 17334 |

**Table 5.**Summary statistics of the magnitudes and number of fatalities in each region. The columns are the minimum (Min.), the first quantile (1st Q.), the median (Med.), Mean, third quantile (3rd Q.) and the maximum (Max.).

Region | Min. | 1st Q. | Med. | Mean | 3rd Q. | Max. | |
---|---|---|---|---|---|---|---|

Magnitude | |||||||

North | 3.0 | 3.0 | 3.2 | 3.4 | 3.5 | 6.1 | |

West | 3.0 | 3.1 | 3.2 | 3.4 | 3.7 | 7 | |

Central | 3.0 | 3.1 | 3.2 | 3.3 | 3.5 | 6.1 | |

East | 3.0 | 3.0 | 3.2 | 3.3 | 3.5 | 6.7 | |

Fatalities | |||||||

North | 2 | 2 | 4 | 10 | 10 | 411 | |

West | 2 | 3 | 5 | 13 | 12 | 600 | |

Central | 2 | 3 | 4 | 9 | 10 | 420 | |

East | 2 | 2 | 4 | 9 | 10 | 598 |

Region | b | Standard Error |
---|---|---|

North | 0.956 | 0.025 |

West | 0.854 | 0.024 |

Central | 0.990 | 0.034 |

East | 1.006 | 0.022 |

**Table 7.**Parameter estimates for North Africa. Estimation results for the eight parameters of the ETAS model, the log likelihood and the number of events for the North Africa conflict catalog. Estimates marked with a * are at, or very close to, their constraint values. The best estimate, that is, the estimate of the run with the highest log likelihood, is given for each parameter. Furthermore, the median and the 95% confidence intervals are given.

Parameter | Best Value | Median | 95% Confidence Interval |
---|---|---|---|

$\mu $ | 4.60e-1 | 5.05e-1 | {4.17e-1, 5.45e-1} |

k | 3.76e-2 | 3.58e-2 | {2.57e-2, 6.13e-2} |

p | 7.46e-1 | 7.43e-1 | {6.79e-1, 8.53e-1} |

c | 1.00e-5 * | 1.04e-5 | {1.00e-5, 9.91e-2} |

$\alpha $ | 0.39e-1 | 2.48e-1 | {5.21e-3, 5.37e-1} |

d | 1.00e-2 * | 1.00e-2 | {1.00e-2, 1.00e-2} |

q | 3.00e0 * | 3.00e0 | {2.98e0, 3.00e0} |

$\gamma $ | 6.94e-4 * | 3.05e-3 | {1.58e-5, 9.04e-3} |

LogL | 1.33.e3 | 1.31e3 | {6.36e2, 1.33e3} |

Events | 3.56e3 | 3.63e3 | {3.35e3, 3.75e3} |

**Table 8.**Parameter estimates for West Africa. Estimation results for the eight parameters of the ETAS model, the log likelihood and the number of events for the West Africa conflict catalog. Estimates marked with a * are at, or very close to, their constraint values. The best estimate, that is, the estimate of the run with the highest log likelihood, is given for each parameter. Furthermore, the median and the 95% confidence intervals are given.

Parameter | Best Value | Median | 95% Confidence Interval |
---|---|---|---|

$\mu $ | 5.02e-1 | 5.21e-1 | {4.13e-1, 5.40e-1} |

k | 2.26e-2 | 2.24e-2 | {1.75e-2, 3.02e-2} |

p | 7.0e1-1 | 7.22e-1 | {6.61e-1, 7.51e-1} |

c | 1.00e-5 * | 1.04e-5 | {1.00e-5, 1.15e-2} |

$\alpha $ | 2.46e-1 | 4.76e-1 | {4.45e-2, 5.62e-1} |

d | 1.00e-2 * | 1.00e-2 | {1.00e-2, 1.00e-2} |

q | 3.00e0 * | 3.00e0 | {2.98e0, 3.00e0} |

$\gamma $ | 1.97-4 * | 2.15e-3 | {2.54e-5, 6.74e-3} |

LogL | −6.16e3 | −6.17e3 | {−6.20e2, −6.16e3} |

Events | 2.16e3 | 2.26e3 | {2.04e3, 2.31e3} |

**Table 9.**Parameter estimates for Central Africa. Estimation results for the eight parameters of the ETAS model, the log likelihood and the number of events for the Central Africa conflict catalog. Estimates marked with a * are at, or very close to, their constraint values. The best estimate, that is, the estimate of the run with the highest log likelihood, is given for each parameter. Furthermore, the median and the 95% confidence intervals are given.

Parameter | Best Value | Median | 95% Confidence Interval |
---|---|---|---|

$\mu $ | 3.79e-1 | 3.59e-1 | {3.01e-1, 4.34e-1} |

k | 3.52e-2 | 3.80e-2 | {2.02e-2, 4.29e-2} |

p | 7.95e-1 | 7.96e-1 | {7.48e-1, 8.28e-1} |

c | 1.00e-5 * | 1.00e-5 | {1.00e-5, 1.15e-2} |

$\alpha $ | 3.03e-1 | 1.86e-1 | {4.49e-2, 8.27e-1} |

d | 1.00e-2 * | 1.00e-2 | {1.00e-2, 1.00e-2} |

q | 3.00e0 * | 2.99e0 | {2.96e0, 3.00e0} |

$\gamma $ | 3.21-5 * | 5.65e-3 | {8.02e-5, 2.39e-2} |

LogL | −3.13e3 | −3.15e3 | {−3.30e2, −3.13e3} |

Events | 1.89e3 | 1.90e3 | {1.68e3, 2.00e3} |

**Table 10.**Parameter estimates for East Africa. Estimation results for the 8 parameters of the ETAS model, the log likelihood and the number of events for the East Africa conflict catalog. Estimates marked with a * are at, or very close to, their constraint values. The best estimate, that is, the estimate of the run with the highest log likelihood, is given for each parameter. Furthermore, the median and the 95% confidence intervals are given.

Parameter | Best Value | Median | 95% Confidence Interval |
---|---|---|---|

$\mu $ | 8.38e-1 | 8.54e-1 | {7.69e-1, 9.56e-1} |

k | 3.07e-2 | 2.96e-2 | {2.25e-2, 3.88e-2} |

p | 7.14e-1 | 6.95e-1 | {6.56e-1, 7.80e-1} |

c | 1.00e-5 * | 1.05e-5 | {1.00e-5, 1.19e-5} |

$\alpha $ | 8.93e-2 | 6.13e-1 | {1.09e-2, 5.29e-1} |

d | 1.00e-2 * | 1.00e-2 | {1.00e-2, 1.00e-2} |

q | 3.00e0 * | 3.00e0 | {2.97e0, 3.00e0} |

$\gamma $ | 8.18-5 * | 6.57e-3 | {8.18e-5, 1.50e-2} |

LogL | −8.39e3 | −8.42e3 | {−3.30e2, -3.13e3} |

Events | 4.59e3 | 4.68e3 | {1.68e3, 2.00e3} |

Region | $\mathit{\mu}$ | k | p | $\mathit{\alpha}$ | Events |
---|---|---|---|---|---|

North | 0.460 | 0.0376 | 0.746 | 0.139 | 3518 |

West | 0.502 | 0.0226 | 0.701 | 0.246 | 2164 |

Central | 0.379 | 0.0352 | 0.795 | 0.303 | 1834 |

East | 0.838 | 0.0307 | 0.714 | 0.0893 | 4617 |

**Table 12.**Total number of expected and observed events, background events and triggered events for each region. The expected numbers of events are calculated using (12) to (14).

Region | All | Background | Triggered | |||
---|---|---|---|---|---|---|

Expected | Observed | Expected | Observed | Expected | Observed | |

North | 3558 | 3516 | 671 | 691 | 2887 | 2824 |

West | 2160 | 2163 | 733 | 705 | 1427 | 1457 |

Central | 1889 | 1834 | 553 | 525 | 1337 | 1308 |

East | 4589 | 4616 | 1223 | 1255 | 3365 | 3361 |

**Table 13.**Results for the Number of Events’ test for the different regions. The estimated ETAS model for each region is used to simulate ${N}_{SIM}=100$ catalogs. The median and 95% confidence interval are calculated for the simulated catalogs, as well as the relative difference (% dif.). Furthermore, the probability (Prob.) that we observe the number of events in the event catalog or more events is in the final column.

Region | Median | Observed | % Dif. | 95% Confidence Interval | Prob. |
---|---|---|---|---|---|

North | 3101 | 3516 | −11.8% | {2737, 3592} | 0.064 |

West | 1933 | 2163 | −10.6% | {1774, 2107} | 0.019 |

Central | 1861 | 1834 | +1.5% | {1652, 2076} | 0.62 |

East | 4506 | 4616 | −2.4% | {4090, 4852} | 0.34 |

**Table 14.**Forecasts for the number of events with magnitude $M\ge 4$ in 2017. The average number of these events in the period 2012–2016 is given, the number in 2016, the number predicted by the model and the observed number of events in 2017.

Region | Average | Last Year | Model | Observed |
---|---|---|---|---|

North | 54.2 | 58 | 16.7 | 17 |

West | 61.8 | 39 | 18.3 | 22 |

Central | 23.2 | 13 | 13.8 | 39 |

East | 58.4 | 50 | 30.5 | 69 |

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## Share and Cite

**MDPI and ACS Style**

van den Hengel, G.; Franses, P.H.
Forecasting Social Conflicts in Africa Using an Epidemic Type Aftershock Sequence Model. *Forecasting* **2020**, *2*, 284-308.
https://doi.org/10.3390/forecast2030016

**AMA Style**

van den Hengel G, Franses PH.
Forecasting Social Conflicts in Africa Using an Epidemic Type Aftershock Sequence Model. *Forecasting*. 2020; 2(3):284-308.
https://doi.org/10.3390/forecast2030016

**Chicago/Turabian Style**

van den Hengel, Gilian, and Philip Hans Franses.
2020. "Forecasting Social Conflicts in Africa Using an Epidemic Type Aftershock Sequence Model" *Forecasting* 2, no. 3: 284-308.
https://doi.org/10.3390/forecast2030016