Employing Fuzzy Logic to Analyze the Structure of Complex Biological and Epidemic Spreading Models
Abstract
:1. Introduction
2. Theoretical Framework
2.1. Basic Definitions
2.2. Models of Complex Networks
2.2.1. Erdös–Renyi Model
2.2.2. Barabási–Albert Model
2.3. Epidemic Models in Biological Networks
3. Fuzzy Epidemics
3.1. Transmitted Diseases in Networks
3.2. Branching Processes
3.3. Susceptible–Infectious–Susceptible Model (SIS)
3.4. Transient Contact Model
3.5. Network Fuzzy Logic
3.6. Contact Structure and Partnership Dynamics
- Single (uncoupled) susceptible (or sensitive) individuals (S);
- Single infected (or contaminated) individuals (I);
- Concordant negative couples (i.e., susceptible–susceptible, ) when both partners are susceptible;
- Discordant couples (i.e., susceptible–infected, );
- Concordant positive couples (i.e., infected–infected, ) when both partners are infectious.
4. HIV Transmission Simulation of Biological Network
4.1. Proposed Model
- Is the individual aware of carrying the infection or not?
- Has the syringe been given to more than one individual or not?
- Is the individual aware of being infected or not?
- For how long will the individual use the syringe?
- How many individuals will use clean (new) needles?
- How many individuals share simultaneously the same needle?
- How often are individuals tested by a physician?
4.2. Random Graph
Algorithm 1 Epidemic Transiently Contact Model. |
|
4.3. Implementation
5. Results
- AIDS-: users not infected by HIV.
- AIDS+: users infected by HIV and they know it (they have been tested).
- AIDS?: users who have no knowledge if they are (or not) infected by HIV and they have not passed the one year time limit in order to be tested.
5.1. Comparing All Four Scenarios
5.2. Discussion
6. Conclusions and Future Work
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Symbol | Meaning |
---|---|
k | Node’s Degree |
Degree Distribution | |
L | Average Shortest Path Length |
b | Betweenness Node |
T | Transitivity |
C | Clustering Coefficient |
Local Clustering Coefficient of Node i | |
r | Probability Distribution Function related to Fuzzy Logic Setting |
Universe of Discourse | |
F | Fuzzy Set |
u | Support Value |
Membership Function | |
x | Linguistic Variable |
The Set of Names of x | |
Semantic Rule |
Scenario | 1 | 2 | 3 | 4 |
---|---|---|---|---|
Number of Users | 300 | 300 | 500 | 500 |
Number of Tests per Year | 1 | 2 | 1 | 2 |
Features | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Percentage of Fuzziness | 0% Fuzzy | 10% Fuzzy | 30% Fuzzy | 50% Fuzzy | ||||||||
Shared Syringe | 10 | 0 | 7 | 9 | 0 | 7 | 7 | 0 | 6 | 5 | 0 | 4 |
New Syringe | 0 | 10 | 3 | 0 | 9 | 2 | 0 | 7 | 1 | 0 | 5 | 1 |
Fuzzy | 0 | 0 | 0 | 1 | 1 | 1 | 3 | 3 | 3 | 5 | 5 | 5 |
Option of Illness | 100 | 390 | 1390 | 2390 | 3930 |
---|---|---|---|---|---|
0% Fuzzy—1st example | |||||
AIDS- | 298 | 290 | 240 | 142 | 126 |
AIDS+ | 0 | 2 | 41 | 138 | 174 |
AIDS? | 2 | 8 | 19 | 20 | 0 |
0% Fuzzy—2nd example | |||||
AIDS- | 298 | 298 | 298 | 298 | 298 |
AIDS+ | 0 | 2 | 2 | 2 | 2 |
AIDS? | 2 | 0 | 0 | 0 | 0 |
0% Fuzzy—3rd example | |||||
AIDS- | 297 | 288 | 200 | 135 | 119 |
AIDS+ | 0 | 5 | 63 | 156 | 181 |
AIDS? | 3 | 7 | 37 | 9 | 0 |
10% Fuzzy—4th example | |||||
AIDS- | 293 | 288 | 191 | 135 | 130 |
AIDS+ | 3 | 5 | 74 | 162 | 170 |
AIDS? | 4 | 7 | 35 | 3 | 0 |
10% Fuzzy—5th example | |||||
AIDS- | 298 | 298 | 298 | 298 | 298 |
AIDS+ | 0 | 2 | 2 | 2 | 2 |
AIDS? | 2 | 0 | 0 | 0 | 0 |
10% Fuzzy—6th example | |||||
AIDS- | 296 | 285 | 217 | 148 | 128 |
AIDS+ | 0 | 9 | 55 | 141 | 172 |
AIDS? | 4 | 6 | 28 | 11 | 0 |
30% Fuzzy—7th example | |||||
AIDS- | 297 | 288 | 234 | 178 | 145 |
AIDS+ | 0 | 4 | 54 | 109 | 155 |
AIDS? | 3 | 8 | 12 | 13 | 0 |
30% Fuzzy—8th example | |||||
AIDS- | 298 | 298 | 298 | 298 | 298 |
AIDS+ | 0 | 2 | 2 | 2 | 2 |
AIDS? | 2 | 0 | 0 | 0 | 0 |
30% Fuzzy—9th example | |||||
AIDS- | 292 | 290 | 258 | 189 | 138 |
AIDS+ | 2 | 4 | 32 | 88 | 162 |
AIDS? | 4 | 6 | 10 | 23 | 0 |
50% Fuzzy—10th example | |||||
AIDS- | 294 | 285 | 213 | 153 | 136 |
AIDS+ | 2 | 5 | 64 | 134 | 164 |
AIDS? | 4 | 10 | 23 | 13 | 0 |
50% Fuzz—11th example | |||||
AIDS- | 298 | 298 | 298 | 298 | 298 |
AIDS+ | 0 | 2 | 2 | 2 | 2 |
AIDS? | 2 | 0 | 0 | 0 | 0 |
50% Fuzzy—12th example | |||||
AIDS- | 296 | 294 | 277 | 264 | 210 |
AIDS+ | 0 | 2 | 16 | 31 | 90 |
AIDS? | 4 | 4 | 7 | 5 | 0 |
Option of Illness | 100 | 390 | 1390 | 2390 | 3930 |
---|---|---|---|---|---|
0% Fuzzy—1st example | |||||
AIDS- | 295 | 294 | 290 | 284 | 284 |
AIDS+ | 2 | 4 | 4 | 16 | 16 |
AIDS? | 3 | 4 | 6 | 0 | 0 |
0% Fuzzy—2nd example | |||||
AIDS- | 298 | 298 | 298 | 298 | 298 |
AIDS+ | 0 | 2 | 2 | 2 | 2 |
AIDS? | 2 | 0 | 0 | 0 | 0 |
0% Fuzzy—3rd example | |||||
AIDS- | 295 | 292 | 291 | 291 | 291 |
AIDS+ | 0 | 5 | 9 | 9 | 9 |
AIDS? | 5 | 3 | 0 | 0 | 0 |
10% Fuzzy—4th example | |||||
AIDS- | 295 | 290 | 258 | 254 | 254 |
AIDS+ | 1 | 4 | 39 | 46 | 46 |
AIDS? | 4 | 6 | 3 | 0 | 0 |
10% Fuzzy—5th example | |||||
AIDS- | 298 | 298 | 298 | 298 | 298 |
AIDS+ | 0 | 2 | 2 | 2 | 2 |
AIDS? | 2 | 0 | 0 | 0 | 0 |
10% Fuzzy—6th example | |||||
AIDS- | 296 | 292 | 273 | 273 | 273 |
AIDS+ | 1 | 3 | 27 | 27 | 27 |
AIDS? | 3 | 5 | 0 | 0 | 0 |
30% Fuzzy—7th example | |||||
AIDS- | 296 | 288 | 261 | 244 | 244 |
AIDS+ | 0 | 5 | 35 | 56 | 56 |
AIDS? | 4 | 7 | 4 | 0 | 0 |
30% Fuzzy—8th example | |||||
AIDS- | 298 | 298 | 298 | 298 | 298 |
AIDS+ | 0 | 2 | 2 | 2 | 2 |
AIDS? | 2 | 0 | 0 | 0 | 0 |
30% Fuzzy—9th example | |||||
AIDS- | 294 | 288 | 278 | 278 | 278 |
AIDS+ | 2 | 5 | 22 | 22 | 22 |
AIDS? | 4 | 7 | 0 | 0 | 0 |
50% Fuzzy—10th example | |||||
AIDS- | 295 | 288 | 259 | 253 | 253 |
AIDS+ | 0 | 5 | 36 | 47 | 47 |
AIDS? | 5 | 7 | 5 | 0 | 0 |
50% Fuzzy—11th example | |||||
AIDS- | 298 | 298 | 298 | 298 | 298 |
AIDS+ | 0 | 2 | 2 | 2 | 2 |
AIDS? | 2 | 0 | 0 | 0 | 0 |
50% Fuzzy—12th example | |||||
AIDS- | 296 | 294 | 293 | 293 | 293 |
AIDS+ | 0 | 4 | 7 | 7 | 7 |
AIDS? | 4 | 2 | 0 | 0 | 0 |
Option of Illness | 100 | 390 | 1390 | 2390 | 3930 |
---|---|---|---|---|---|
0% Fuzzy—1st example | |||||
AIDS- | 491 | 489 | 385 | 260 | 211 |
AIDS+ | 2 | 3 | 78 | 214 | 289 |
AIDS? | 7 | 8 | 37 | 26 | 0 |
0% Fuzzy—2nd example | |||||
AIDS- | 497 | 497 | 497 | 497 | 497 |
AIDS+ | 0 | 3 | 3 | 3 | 3 |
AIDS? | 3 | 0 | 0 | 0 | 0 |
0% Fuzzy—3rd example | |||||
AIDS- | 491 | 483 | 312 | 190 | 172 |
AIDS+ | 1 | 4 | 131 | 294 | 328 |
AIDS? | 8 | 13 | 57 | 16 | 0 |
10% Fuzzy—4th example | |||||
AIDS- | 492 | 482 | 384 | 235 | 219 |
AIDS+ | 2 | 5 | 78 | 247 | 281 |
AIDS? | 6 | 13 | 38 | 18 | 0 |
10% Fuzzy—5th example | |||||
AIDS- | 497 | 497 | 497 | 497 | 497 |
AIDS+ | 0 | 3 | 3 | 3 | 3 |
AIDS? | 3 | 0 | 0 | 0 | 0 |
10% Fuzzy—6th example | |||||
AIDS- | 491 | 477 | 292 | 183 | 171 |
AIDS+ | 1 | 5 | 141 | 311 | 329 |
AIDS? | 8 | 18 | 67 | 6 | 0 |
30% Fuzzy—7th example | |||||
AIDS- | 491 | 479 | 283 | 169 | 173 |
AIDS+ | 1 | 4 | 140 | 322 | 329 |
AIDS? | 8 | 17 | 77 | 9 | 0 |
30% Fuzzy—8th example | |||||
AIDS- | 497 | 497 | 497 | 497 | 497 |
AIDS+ | 0 | 3 | 3 | 3 | 3 |
AIDS? | 3 | 0 | 0 | 0 | 0 |
30% Fuzzy—9th example | |||||
AIDS- | 492 | 475 | 294 | 231 | 223 |
AIDS+ | 1 | 8 | 151 | 261 | 275 |
AIDS? | 7 | 17 | 55 | 8 | 0 |
50% Fuzzy—10th example | |||||
AIDS- | 491 | 483 | 312 | 190 | 172 |
AIDS+ | 1 | 4 | 131 | 294 | 328 |
AIDS? | 8 | 13 | 57 | 16 | 0 |
50% Fuzzy—11th example | |||||
AIDS- | 497 | 497 | 497 | 497 | 497 |
AIDS+ | 0 | 3 | 3 | 3 | 3 |
AIDS? | 3 | 0 | 0 | 0 | 0 |
50% Fuzzy—12th example | |||||
AIDS- | 494 | 487 | 326 | 241 | 221 |
AIDS+ | 0 | 3 | 114 | 246 | 279 |
AIDS? | 6 | 10 | 60 | 13 | 0 |
Option of Illness | 100 | 390 | 1390 | 2390 | 3930 |
---|---|---|---|---|---|
0% Fuzzy—1st example | |||||
AIDS- | 487 | 477 | 417 | 391 | 376 |
AIDS+ | 3 | 13 | 74 | 106 | 123 |
AIDS? | 10 | 10 | 9 | 2 | 0 |
0% Fuzzy—2nd example | |||||
AIDS- | 497 | 497 | 497 | 497 | 497 |
AIDS+ | 0 | 3 | 3 | 3 | 3 |
AIDS? | 3 | 0 | 0 | 0 | 0 |
0% Fuzzy—3rd example | |||||
AIDS- | 486 | 481 | 413 | 343 | 318 |
AIDS+ | 4 | 14 | 70 | 141 | 182 |
AIDS? | 10 | 5 | 17 | 16 | 0 |
10% Fuzzy—4th example | |||||
AIDS- | 488 | 487 | 446 | 412 | 395 |
AIDS+ | 4 | 9 | 45 | 80 | 105 |
AIDS? | 8 | 4 | 9 | 8 | 0 |
10% Fuzzy—5th example | |||||
AIDS- | 497 | 497 | 497 | 497 | 497 |
AIDS+ | 0 | 3 | 3 | 3 | 3 |
AIDS? | 3 | 0 | 0 | 0 | 0 |
10% Fuzzy—6th example | |||||
AIDS- | 493 | 485 | 435 | 365 | 331 |
AIDS+ | 0 | 10 | 57 | 115 | 169 |
AIDS? | 7 | 5 | 8 | 17 | 0 |
30% Fuzzy—7th example | |||||
AIDS- | 488 | 485 | 448 | 422 | 400 |
AIDS+ | 5 | 10 | 43 | 73 | 100 |
AIDS? | 7 | 5 | 9 | 5 | 0 |
30% Fuzzy—8th example | |||||
AIDS- | 497 | 497 | 497 | 497 | 497 |
AIDS+ | 0 | 3 | 3 | 3 | 3 |
AIDS? | 3 | 0 | 0 | 0 | 0 |
30% Fuzzy—9th example | |||||
AIDS- | 489 | 486 | 420 | 352 | 340 |
AIDS+ | 3 | 9 | 62 | 140 | 160 |
AIDS? | 8 | 5 | 18 | 8 | 0 |
50% Fuzzy—10th example | |||||
AIDS- | 492 | 478 | 417 | 392 | 380 |
AIDS+ | 2 | 14 | 74 | 105 | 120 |
AIDS? | 6 | 8 | 9 | 3 | 0 |
Fuzzy—11th example | |||||
AIDS- | 497 | 497 | 497 | 497 | 497 |
AIDS+ | 0 | 3 | 3 | 3 | 3 |
AIDS? | 3 | 0 | 0 | 0 | 0 |
50% Fuzzy—12th example | |||||
AIDS- | 492 | 486 | 461 | 454 | 443 |
AIDS+ | 0 | 9 | 36 | 45 | 57 |
AIDS? | 8 | 5 | 3 | 1 | 0 |
Option of Illness | 100 | 390 | 1390 | 2390 | 3930 | 100 | 390 | 1390 | 2390 | 3930 |
---|---|---|---|---|---|---|---|---|---|---|
1st scenario—1 test—300 users | 3rd scenario—1 test—500 users | |||||||||
AIDS- | 297 | 288 | 200 | 135 | 119 | 491 | 483 | 312 | 190 | 172 |
AIDS+ | 0 | 5 | 63 | 156 | 181 | 1 | 4 | 131 | 294 | 328 |
AIDS? | 3 | 7 | 37 | 9 | 0 | 8 | 13 | 57 | 16 | 0 |
2nd scenario—2 tests—300 users | 4th scenario—2 tests—500 users | |||||||||
AIDS- | 295 | 292 | 291 | 291 | 291 | 486 | 481 | 413 | 343 | 318 |
AIDS+ | 0 | 5 | 9 | 9 | 9 | 4 | 14 | 70 | 141 | 182 |
AIDS? | 5 | 3 | 0 | 0 | 0 | 10 | 5 | 17 | 16 | 0 |
Option of Illness | 100 | 390 | 1390 | 2390 | 3930 | 100 | 390 | 1390 | 2390 | 3930 |
---|---|---|---|---|---|---|---|---|---|---|
1st scenario—1 test—300 users | 3rd scenario—1 test—500 users | |||||||||
AIDS- | 296 | 285 | 217 | 148 | 128 | 491 | 477 | 292 | 183 | 171 |
AIDS+ | 0 | 9 | 55 | 141 | 172 | 1 | 5 | 141 | 311 | 329 |
AIDS? | 4 | 6 | 28 | 11 | 0 | 8 | 18 | 67 | 6 | 0 |
2nd scenario—2 tests—300 users | 4th scenario—2 tests—500 users | |||||||||
AIDS- | 296 | 292 | 273 | 273 | 273 | 493 | 485 | 435 | 365 | 331 |
AIDS+ | 1 | 3 | 27 | 27 | 27 | 0 | 10 | 57 | 115 | 169 |
AIDS? | 3 | 5 | 0 | 0 | 0 | 7 | 5 | 8 | 17 | 0 |
Option of Illness | 100 | 390 | 1390 | 2390 | 3930 | 100 | 390 | 1390 | 2390 | 3930 |
---|---|---|---|---|---|---|---|---|---|---|
1st scenario—1 test—300 users | 3rd scenario—1 test—500 users | |||||||||
AIDS- | 292 | 290 | 258 | 189 | 138 | 492 | 475 | 294 | 231 | 223 |
AIDS+ | 2 | 4 | 32 | 88 | 162 | 1 | 8 | 151 | 261 | 275 |
AIDS? | 4 | 6 | 10 | 23 | 0 | 7 | 17 | 55 | 8 | 0 |
2nd scenario—2 tests—300 users | 4th scenario—2 tests—500 users | |||||||||
AIDS- | 294 | 288 | 278 | 278 | 278 | 489 | 486 | 420 | 352 | 340 |
AIDS+ | 2 | 5 | 22 | 22 | 22 | 3 | 9 | 62 | 140 | 160 |
AIDS? | 4 | 7 | 0 | 0 | 0 | 8 | 5 | 18 | 8 | 0 |
Option of Illness | 100 | 390 | 1390 | 2390 | 3930 | 100 | 390 | 1390 | 2390 | 3930 |
---|---|---|---|---|---|---|---|---|---|---|
1st scenario—1 test—300 users | 3rd scenario—1 test—500 users | |||||||||
AIDS- | 296 | 294 | 277 | 264 | 210 | 494 | 487 | 326 | 241 | 221 |
AIDS+ | 0 | 2 | 16 | 31 | 90 | 0 | 3 | 114 | 246 | 279 |
AIDS? | 4 | 4 | 7 | 5 | 0 | 6 | 10 | 60 | 13 | 0 |
2nd scenario—2 tests—300 users | 4th scenario—2 tests—500 users | |||||||||
AIDS- | 296 | 294 | 293 | 293 | 293 | 492 | 486 | 461 | 454 | 443 |
AIDS+ | 0 | 4 | 7 | 7 | 7 | 0 | 9 | 36 | 45 | 57 |
AIDS? | 4 | 2 | 0 | 0 | 0 | 8 | 5 | 3 | 1 | 0 |
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Lefevr, N.; Kanavos, A.; Gerogiannis, V.C.; Iliadis, L.; Pintelas, P. Employing Fuzzy Logic to Analyze the Structure of Complex Biological and Epidemic Spreading Models. Mathematics 2021, 9, 977. https://doi.org/10.3390/math9090977
Lefevr N, Kanavos A, Gerogiannis VC, Iliadis L, Pintelas P. Employing Fuzzy Logic to Analyze the Structure of Complex Biological and Epidemic Spreading Models. Mathematics. 2021; 9(9):977. https://doi.org/10.3390/math9090977
Chicago/Turabian StyleLefevr, Nickie, Andreas Kanavos, Vassilis C. Gerogiannis, Lazaros Iliadis, and Panagiotis Pintelas. 2021. "Employing Fuzzy Logic to Analyze the Structure of Complex Biological and Epidemic Spreading Models" Mathematics 9, no. 9: 977. https://doi.org/10.3390/math9090977