Measles in Southwest Kansas

A Maximum Flow Application

Introduction

This project analyzes graphs using the Ford-Fulkerson Algorithm to calculate the maximum flow. With many real life applications, this project creates easy to understand visualizations of maximum flow. Originally developed to model generic networks, the program was expanded to include epidemic modeling and a specific case study: the 2025 Kansas measles outbreak.

Goals

The primary purpose of this project is to demonstrate the broad applicability of maximum flow analysis using Ford-Fulkerson algorithm, with a focus on epidemiological modeling. By building a versatile software tool, this project aims to:

Illustrate algorithmic versatility
Bridge computational theory and public health practice
Provide epidemic modeling capabilities
Analyze the 2025 Kansas measles outbreak
Support data-driven public health decision-making

Scope

The scope of this project encompasses:

Development of a general-purpose maximum flow calculator using the Ford-Fulkerson algorithm
Extension for epidemic modeling, repurposing the base network components to simulate disease transmission
Kansas-specific outbreak modeling using real county data and geographic proximity
Performance benchmarking across different network sizes and densities

Kansas Measles Outbreak Context

In 2024, there were no cases of measles in Kansas. In contrast, as of April 2025, there have been 37 cases of measles in Southwest Kansas, as reported by the Kansas Department of Health and Environment. This follows national trends of measles cases surging: from 285 in 2024 to 884 in 2025, as reported by the CDC. In Kansas, 30 of the 37 cases are residents under 18 years old.

Methodology

This project originated from coursework in CS404 Introduction to Algorithms & Complexity, where my foundational understanding of Ford-Fulkerson was established. From here, my interest in epidemiology inspired me to create a more specific algorithm for epidemics.

Data Sources

For the Kansas County measles application, I used two data sources:

Kansas Department of Health and Environment's Division of Public Health report on Kansas' 2025 Measles Outbreak
US Census' Annual Estimates of the Resident Population for Counties in Kansas

Program Creation

Development Tools & Libraries:

Language: Python
IDE & Version Control: PyCharm, GitHub
Libraries: networkx, matplotlib, numpy, collections.deque, random, time, and string
AI Tools for code generation: Claude.ai & ChatGPT

I started by creating a general ford-fulkerson algorithm for a graph network. From there, I implemented an inherited class for an epidemic network to model likelihood of infection. Finally, I programmed a more specific application for the Kansas measles outbreak to model the flow of measles between counties.

Content

Overview of Program

Program Architecture

1. Flow Network: A general graph-based flow simulator with predefined tests. Each graph has a set number of nodes, a minimum and maximum capacity for the graph's edges, and a density between 0-1 defining the likelihood of edges between the nodes, 0 meaning no nodes are connecting and 1 indicating all nodes are connected.

Small: 5 nodes, minimum capacity 1, maximum capacity 15, with density 0.4.
Medium: 8 nodes, minimum capacity 1, maximum capacity 20, with density 0.3.
Large: 12 nodes, minimum capacity 1, maximum capacity 30, with density 0.2.

2. Flow Network Performance Tests: An option to benchmark the performance of the maximum flow analysis with a graph of 10, 100, and 1,000 nodes, running each with density 0.1, 0.5, and 1.0.

3. Kansas Measles Network:

Nodes: Kansas Counties
Sources: Counties with confirmed cases
Sinks: Adjacent counties
Edge Weights: Based on adjacency, case count, and population
Objective: Analyze likelihood of disease spread to neighboring counties

4. Custom Random Network: Graph and flow created by user-defined vertex count, capacity, and density.

5. Epidemic Network: This extends the Flow Network with epidemiological framing:

Nodes: Individuals
Source: Original infection case
Sink: Individual being analyzed for likelihood of infections

Parameters: As currently implemented, the graph is randomly generated with the following settings:

num_vertices = 10  # Number of vertices in the epidemic network
min_capacity = 1
max_capacity = 20
density = 0.3  # 30% density for the network
infection_prob = 0.1  # Probability of infection spread between nodes

Visualization

The graphs are visualized using matplotlib. While the flow and epidemic networks are visualized with traditional graphs, the Kansas network is modeled to look more similar to the layout of the state map.

Results

Flow Network Predefined Tests

Small Test

Nodes: 5
Minimum capacity: 1
Maximum capacity: 15
Density: 0.4
Calculated max flow: 1

Medium Test

Nodes: 8
Minimum capacity: 1
Maximum capacity: 20
Density: 0.3
Calculated max flow: 21

Large Test

Nodes: 12
Minimum capacity: 1
Maximum capacity: 30
Density: 0.2
Calculated max flow: 18

Flow Network – Testing Scalability & Timing

Density	10 Nodes	100 Nodes	1,000 Nodes
0.1	0.000078 s	0.012732 s	9.959056 s
0.5	0.000256 s	0.035640 s	37.675265 s
1.0	0.000303 s	0.060656 s	68.298057 s

Epidemic Network Simulation

Maximum nodes infected: 35

Maximum Flow Analysis - County Transmission Risk

Analysis performed using Ford-Fulkerson Algorithm (Total analysis time: 0.04 seconds)

Top 5 At-Risk Counties

County	Risk Score
Seward	429.18
Meade	267.96
Kearny	247.42
Stanton	195.42
Pratt	139.68

Detailed Transmission Risk Analysis

Seward County

Total incoming transmission risk: 429.18

Source County	Risk Value	Percentage
Haskell	121.73	28.4%
Stevens	93.08	21.7%
Gray	58.53	13.6%
Grant	55.41	12.9%
Morton	45.62	10.6%
Finney	29.23	6.8%
Ford	12.79	3.0%
Kiowa	12.79	3.0%

Meade County

Total incoming transmission risk: 267.96

Source County	Risk Value	Percentage
Grant	42.63	15.9%
Gray	42.63	15.9%
Haskell	42.63	15.9%
Morton	42.63	15.9%
Stevens	42.63	15.9%
Finney	29.23	10.9%
Ford	12.79	4.8%
Kiowa	12.79	4.8%

Kearny County

Total incoming transmission risk: 247.42

Source County	Risk Value	Percentage
Grant	38.52	15.6%
Gray	38.52	15.6%
Haskell	38.52	15.6%
Morton	38.52	15.6%
Stevens	38.52	15.6%
Finney	29.23	11.8%
Ford	12.79	5.2%
Kiowa	12.79	5.2%

Stanton County

Total incoming transmission risk: 195.42

Source County	Risk Value	Percentage
Finney	28.31	14.5%
Grant	28.31	14.5%
Gray	28.31	14.5%
Haskell	28.31	14.5%
Morton	28.31	14.5%
Stevens	28.31	14.5%
Ford	12.79	6.5%
Kiowa	12.79	6.5%

Pratt County

Total incoming transmission risk: 139.68

Source County	Risk Value	Percentage
Kiowa	46.95	33.6%
Finney	13.25	9.5%
Ford	13.25	9.5%
Grant	13.25	9.5%
Gray	13.25	9.5%
Haskell	13.25	9.5%
Morton	13.25	9.5%
Stevens	13.25	9.5%

Edwards County

Total incoming transmission risk: 136.48

Source County	Risk Value	Percentage
Kiowa	43.76	32.1%
Finney	13.25	9.7%
Ford	13.25	9.7%
Grant	13.25	9.7%
Gray	13.25	9.7%
Haskell	13.25	9.7%
Morton	13.25	9.7%
Stevens	13.25	9.7%

Comanche County

Total incoming transmission risk: 131.00

Source County	Risk Value	Percentage
Kiowa	38.27	29.2%
Finney	13.25	10.1%
Ford	13.25	10.1%
Grant	13.25	10.1%
Gray	13.25	10.1%
Haskell	13.25	10.1%
Morton	13.25	10.1%
Stevens	13.25	10.1%

Hodgeman County

Total incoming transmission risk: 115.84

Source County	Risk Value	Percentage
Finney	15.04	13.0%
Gray	15.04	13.0%
Haskell	15.04	13.0%
Stevens	15.04	13.0%
Grant	15.04	13.0%
Morton	15.04	13.0%
Ford	12.79	11.0%
Kiowa	12.79	11.0%

Clark County

Total incoming transmission risk: 114.69

Source County	Risk Value	Percentage
Kiowa	21.96	19.1%
Finney	13.25	11.6%
Ford	13.25	11.6%
Grant	13.25	11.6%
Gray	13.25	11.6%
Haskell	13.25	11.6%
Morton	13.25	11.6%
Stevens	13.25	11.6%

Barber County

Total incoming transmission risk: 114.22

Source County	Risk Value	Percentage
Kiowa	21.49	18.8%
Finney	13.25	11.6%
Ford	13.25	11.6%
Grant	13.25	11.6%
Gray	13.25	11.6%
Haskell	13.25	11.6%
Morton	13.25	11.6%
Stevens	13.25	11.6%

Lower Risk Counties

County	Total Risk	Top Source
Scott	22.88	Equal distribution (12.5% each)
Ness	21.14	Equal distribution (12.5% each)
Lane	19.71	Equal distribution (12.5% each)
Hamilton	16.47	Equal distribution (12.5% each)

Major Transmission Paths

Source → Destination	Flow Value
Haskell → Seward	121.73
Kiowa → Pratt	46.95
Kiowa → Edwards	43.76
Grant → Meade	42.63
Kiowa → Comanche	38.27
Grant → Kearny	38.52
Finney → Stanton	28.31
Kiowa → Clark	21.96
Kiowa → Barber	21.49
Finney → Hodgeman	15.04

Conclusion

This program demonstrates the power and versatility of maximum flow analysis using Ford-Fulkerson. By modeling disease spread as a flow network, we can better understand potential outbreak trajectories and support public health resources.

Future Improvements

Epidemic Network: Add more user input to specify the desired network. This would allow for better analysis for disease outbreaks and enable reusability for the program.

Kansas Model: This model can be improved by:

Using transportation data, vaccination rates, and healthcare access to increase transmission estimation accuracy
Using geopandas for accurate geographic visualization
Modeling time-based progression of outbreaks for dynamic forecasting

While powerful, the algorithm becomes computationally expensive for large graphs (100+ nodes), highlighting a limitation in scalability.

References

Bureau, US Census. "County Population Totals and Components of Change: 2020-2024." Census.Gov, 12 Mar. 2025, www.census.gov/data/tables/time-series/demo/popest/2020s-counties-total.html

"Measles Cases and Outbreaks." Centers for Disease Control and Prevention, Centers for Disease Control and Prevention, www.cdc.gov/measles/data-research/index.html. Accessed 25 Apr. 2025.

"Measles Data." Measles Data | KDHE, KS, www.kdhe.ks.gov/2314/Measles-Data. Accessed 25 Apr. 2025.

catlewin/2025_Measels_in_SW_Kansas