Team:VIT Vellore/Model

Team VIT Vellore/Description

The Math Modeling of our project

Introduction

The phrase, “We estimate”, sounds a lot like, “We hope”, without backing it up with math. We estimate, not hope that our model works as designed. Now, our project is up against two very capable challengers, antibiotics and traditional phage therapy. They are both very capable in their own rights, but hey, our project is designed to be one better than both of them! Thus our math model comes to the rescue.
We designed our model equations based on our hypothesis as well as recent publications.









The table below contains the list of variables used and their descriptions:

Table of variables
Variable Description
A The number of Resistant Bacteria
N The number of Non-resistant Bacteria
T Total number of bacteria in the system at any given time
C Number of Bacteria-Phage complexes
V1 Number of Phages in the first generation i.e. before lysis
V2 Number of Phages in the second generation i.e. after lysis
M Units of available antibiotic in the system at any given time

We made the following assumptions


1. All rates are independent of each other.

2. Removal rate accounts for immune response as well as natural death rate.

3. Rate of growth and death of AMR and non-AMR are considered to be equal.

4. Rate of mutation is assumed to be a constant.

5. Phage particles are assumed not to attach to dead bacterial surface proteins.

6. It is assumed that AMR cannot be reversed.

7. Burst size is considered to be a constant.

8. Antibiotics do not affect AMR bacteria.

9. It is assumed that each bacterial surface membrane is susceptible to the attachment of only one phage particle.

10. It is assumed that 1 unit of antibiotic is capable of killing one non-AMR bacterial cell.


The table below contains the list of constants used, their values and descriptions.

Table of constants and references.
Number Constant Value Description Reference
1 R1 2 hr-1 Rate of natural replication of bacteria (E.coli in intestine) 6
2 R2 0.23 hr-1 Rate of natural death and removal of bacteria (E.coli) from the intestine 6
3 R3 1*10-8 hr-1 Rate of mutation from non-AMR to AMR bacteria Assumed Value
4 R4 2.12 hr-1 Rate of lysis of a bacteria by a phage particle 2
5 R5 10-8 hr-1 Rate of adsorption of phage particles on the bacterial surface. 5
6 R6 1.02*10-5 hr-1 The death rate of bacteria due to antibiotics. 3
7 R7 1.8 hr-1 Rate of phage degradation 5
8 R8 0.1386 hr-1 Rate of degradation and removal of antibiotic 7
9 A0 0.7*106.2 This value signifies the number of AMR bacteria in the human gut 1
10 N0 0.3*106.2 This value signifies the number of Non-AMR bacteria in the human gut 1
11 x 250 This value signifies the number of phage particles released per bacterial lysis Collaboration (ETH Zurich)
12 k 106.2 Carrying capacity of the human gut (for Enterobacteria) 3
13 V0 109 Initial number of phages in the system Assumed Value
14 M0 8*105 Units Initial Number of Units of Antibiotic in the system Assumed Value

Mathematical Model

ARM'D UP Therapy Equations


  1. dA/dt = (R1 - R2)*A*(k-T)/k + R3*N - R4*C
  2. dN/dt = (R1 - R2)*N*(k-T)/k – R3*N
  3. dT/dt = (R1 - R2)*(k-T)*(A+N)/k – R4*C
  4. dV1/dt = -R5*T*V1 - R7*V1
  5. dV2/dt = R4*x*C – R7*V2
  6. dC/dt = R5*A*V1 - R4*C

Conventional Phage Therapy Equations


  1. dA/dt = (R1 - R2)*A*(k-T)/k – R4*Ca + R3*N
  2. dN/dt = (R1 - R2)*N*(k-T)/k – R4*Cn - R3*N
  3. dT/dt = (R1 - R2)*(k-T)*(A+N)/k – R4*(Ca+Cn)
  4. dCa/dt = R5*A*V – R4*Ca
  5. dCn/dt = R5*N*V – R4*Cn
  6. dV/dt = R4*x*(Ca+Cn) – R5*T*V – R7*V

Antibiotic Therapy Equations


  1. dA/dt = (R1 - R2)*A*(k-T)/k + R3*N
  2. dN/dt = (R1 - R2)*N*(k-T)/k - R3*N - R6*N*M
  3. dT/dt = (R1 - R2)*(k-T)*(A+N)/k - R6*N*M
  4. dM/dt = -R6*N*M – R8*M

ARM'D UP + Antibiotic Therapy Equations


  1. dA/dt = (R1 - R2)*A*(k-T)/k + R3*N – R4*C
  2. dN/dt = (R1 - R2)*N*(k-T)/k – R3*N – R6*N*M
  3. dT/dt = (R1 - R2)*(k-T)*(A+N)/k – R4*C
  4. dV1/dt = -R5*T*V1 - R7*V1
  5. dV2/dt = R4*x*C – R7*V2
  6. dC/dt = R5*A*V1 - R4*C
  7. dM/dt = -R6*N*M – R8*M

The graphs obtained using our model indicate


  • In our system, the number of Non-AMR bacteria reaches its carrying capacity, while the number of AMR bacteria reaches zero.

  • It can be seen that Conventional Phage therapy causes the complete elimination of the whole species of a certain bacteria in the system.

  • Hence, here we can visualize how exactly our system differentiates bacteria based on the presence of resistance as opposed to phage therapy which doesn't.

  • In terms of application, we can use our system to kill off the resistant gut flora without affecting the non-resistant gut microbes.

  • From the model for antibiotics, it can be seen that all non-AMR bacteria are eliminated, while the AMR bacteria continue to grow towards the carrying capacity, which is very disadvantageous.




The table below contains the list of references.

Table of References
Reference number Reference
1 Yoshimasa Yamamoto, Ryuji Kawahara, Yoshihiro Fujiya, Tadahiro Sasaki, Itaru Hirai, Diep Thi Khong, Thang Nam Nguyen, Bai Xuan Nguyen, Wide dissemination of colistin-resistant Escherichia coli with the mobile resistance gene mcr in healthy residents in Vietnam, Journal of Antimicrobial Chemotherapy, Volume 74, Issue 2, February 2019, Pages 523–524
2 Wang I. N. (2006). Lysis timing and bacteriophage fitness. Genetics, 172(1), 17–26. doi:10.1534/genetics.105.045922
3 Levin, B. R., & Udekwu, K. I. (2010). Population dynamics of antibiotic treatment: a mathematical model and hypotheses for time-kill and continuous-culture experiments. Antimicrobial agents and chemotherapy, 54(8), 3414–3426. doi:10.1128/AAC.00381-10
4 Hill, M. J., & Drasar, B. S. (1975). The normal colonic bacterial flora. Gut, 16(4), 318–323. doi:10.1136/gut.16.4.318
5 Vidurupola, Sukhitha. (2018). Analysis of deterministic and stochastic mathematical models with resistant bacteria and bacteria debris for bacteriophage dynamics. Applied Mathematics and Computation. 316. 215-228. 10.1016/j.amc.2017.08.022.
6 Freter, R., Brickner, H., Fekete, J., Vickerman, M. M., & Carey, K. E. (1983). Survival and implantation of Escherichia coli in the intestinal tract. Infection and immunity, 39(2), 686–703.
7 Li, J. (2003). Steady-state pharmacokinetics of intravenous colistin methanesulphonate in patients with cystic fibrosis. Journal of Antimicrobial Chemotherapy, 52(6), 987–992. doi: 10.1093/jac/dkg468


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