S1 Text
Given the set of odes presented in the main text:
(1A) 𝑋 = 1 − 𝑆 − 𝑅1 − 𝑅2 − 𝑅3 − 𝑅1,2 − 𝑅1,3 − 𝑅2,3
(1B) 𝑆̇ = 𝜆𝑆 − 𝑐𝑆 + 𝛽𝑆𝑋 − (𝜏 + 𝛾)𝑆
(1C) 𝑅̇𝑖 = 𝜆𝑅𝑖 − 𝑅𝑖 (𝑐 + 𝛾) + 𝛽𝑅𝑖 𝑋 − 𝜏(𝜒𝑗 + 𝜒𝑘 + 𝜒𝑖 𝑑)𝑅𝑖 + 𝜏𝜒𝑖 𝑆𝑝𝑖
̇ = 𝜆𝑅 − 𝑅𝑗,𝑘 (𝑐 + 𝛾) + 𝛽𝑅𝑗,𝑘 𝑋 − 𝜏(𝜒𝑖 + (𝜒𝑗 + 𝜒𝑘 )𝑑)𝑅𝑗,𝑘 + (𝜒𝑘 +
(1D) 𝑅𝑗,𝑘
𝑗,𝑘
𝑗
𝑘
𝜒𝑗 𝑑𝜒𝑖,𝑘
)𝜏𝑅𝑗 𝑝𝑘 + (𝜒𝑗 + 𝜒𝑘 𝑑𝜒𝑖,𝑗 )𝜏𝑅𝑘 𝑝𝑗
̇
𝑉
𝑓
We will transform the equations 1B-1D, 𝑉̇ = 𝑓 to 𝑉+1 = 𝑉+1
𝑆̇
𝜆𝑆
𝑆
𝑆
=
+ 𝛽𝑋
− (𝑐 + 𝜏 + 𝛾)
𝑆+1
𝑆+1
1+𝑆
1+𝑆
𝜆𝑆
1
1
=
+ 𝛽𝑋 (1 −
) − (𝑐 + 𝜏 + 𝛾) (1 −
)
𝑆+1
1+𝑆
1+𝑆
𝜆𝑅𝑖
𝑅̇𝑖
𝑅𝑖
𝑅𝑖
𝑅𝑖
𝜏𝜒𝑖
=
− (𝑐 + 𝛾)
+ 𝛽𝑋
− 𝜏(𝜒𝑗 + 𝜒𝑘 + 𝜒𝑖 𝑑)
+
𝑆𝑝
𝑅𝑖 + 1 𝑅𝑖 + 1
1 + 𝑅𝑖
1 + 𝑅𝑖
1 + 𝑅𝑖 𝑅𝑖 + 1 𝑖
𝜆 𝑅𝑖
𝑅𝑖
𝑅𝑖
=
− (𝑐 + 𝛾) (1 −
) + 𝛽𝑋 (1 −
)
𝑅𝑖 + 1
1 + 𝑅𝑖
1 + 𝑅𝑖
𝑅𝑖
𝜏𝜒𝑖
− 𝜏(𝜒𝑗 + 𝜒𝑘 + 𝜒𝑖 𝑑) (1 −
)+
𝑆𝑝
1 + 𝑅𝑖
𝑅𝑖 + 1 𝑖
̇
𝜆𝑅𝑗,𝑘
𝑅𝑗,𝑘
𝑅𝑗,𝑘
𝑅𝑗,𝑘
𝑅𝑗,𝑘
=
− (𝑐 + 𝛾)
+ 𝛽𝑋
− 𝜏(𝜒𝑖 + (𝜒𝑗 + 𝜒𝑘 )𝑑)
𝑅𝑗,𝑘 + 1 𝑅𝑗,𝑘 + 1
1 + 𝑅𝑗,𝑘
1 + 𝑅𝑗,𝑘
1 + 𝑅𝑗,𝑘
𝜒𝑗 𝜏
𝜒𝑘 𝜏
+
𝑅𝑝 +
𝑅 𝑝
𝑅𝑗,𝑘 + 1 𝑗 𝑘 𝑅𝑗,𝑘 + 1 𝑘 𝑗
𝜆𝑅𝑗,𝑘
1
1
=
− (𝑐 + 𝛾) (1 −
) + 𝛽𝑋 (1 −
)
𝑅𝑗,𝑘 + 1
1 + 𝑅𝑗,𝑘
1 + 𝑅𝑗,𝑘
𝑘
𝜒𝑘 + 𝜒𝑗 𝑑𝜒𝑖,𝑘
1
− 𝜏(𝜒𝑖 + (𝜒𝑗 + 𝜒𝑘 )𝑑) (1 −
𝜏𝑅𝑗 𝑝𝑘
)+
1 + 𝑅𝑗,𝑘
𝑅𝑗,𝑘 + 1
𝑗
+
𝜒𝑗 + 𝜒𝑘 𝑑𝜒𝑖,𝑗
𝑅𝑗,𝑘 + 1
𝜏𝑅𝑘 𝑝𝑗
It was recently shown that the transformation 𝑒 𝑣(𝑡) = 𝑉(𝑡) can be useful for a first order
approximation of the equilibrium of similar dynamic systems, after some algebraic
manipulation [1]. However, the approximation holds for 𝑉(𝑡) ≈ 1, so for our set of
equations we will define 𝑒 𝑣(𝑡) = 𝑉(𝑡) + 1. Now we can use the identity 𝑣(𝑡) =
𝑉̇
ln(𝑉(𝑡) + 1) ⇒ 𝑉+1 = 𝑣̇ and rewrite the equations with our new, lower case variables, and
then use the approximation 𝑒 𝑧 ≈ 1 + 𝑧 :
𝑋 ≈ 1 − 𝑠 − 𝑟1 − 𝑟2 − 𝑟3 − 𝑟1,2 − 𝑟1,3 − 𝑟2,3
𝑠̇ ≈ 𝜆𝑆 (1 − 𝑠) − (𝑐 + 𝜏 + 𝛾)(𝑠) + 𝛽𝑠 = 𝑠(𝛽 − 𝜆𝑠 − 𝑐 − 𝜏 − 𝛾) + 𝜆𝑠
𝑟𝑖̇ ≈ 𝑟𝑖 (𝛽 − 𝜆𝑅𝑖 − 𝑐 − 𝛾 − 𝜏(𝜒𝑗 + 𝜒𝑘 + 𝜒𝑖 𝑑)) + 𝜆𝑅𝑖 + 𝜏𝜒𝑖 𝑝𝑖 𝑠
𝑗
𝑟𝑗,𝑘̇ ≈ 𝑟𝑗,𝑘 (𝛽 − 𝜆𝑅𝑗,𝑘 − 𝑐 − 𝛾 − 𝜏(𝜒𝑖 + (𝜒𝑗 + 𝜒𝑘 )𝑑) + 𝜆𝑅𝑗,𝑘 + 𝜏(𝜒𝑗 + 𝜒𝑘 𝑑𝜒𝑖,𝑗 )𝑝𝑗 𝑟𝑘 + 𝜏(𝜒𝑘
𝑘
+ 𝜒𝑗 𝑑𝜒𝑖,𝑘
)𝑝𝑘 𝑟𝑗
We used the assumption that most patients are uninfected with the specific bacteria we
1
2
model (𝑋 > ), to achieve a more accurate approximation, by replacing 𝑒 𝑥 ≈ 1 + 𝑥 with
𝑒 𝑥 ≈ 8 − Σ𝑣𝑖≠𝑥 (1 + 𝑣𝑖 ) (corollary (1)).Note that without loss of generality, this assumption
1
holds for any variable 𝑉 satisfying 𝑉 > 2 (of course, there could be only one such variable)
and the equations could be easily rewritten for such a case.
The solution for the stable state is simple for the susceptible population:
𝑠∗ =
𝜆𝑠
−𝛽 + 𝜆𝑠 + 𝑐 + 𝜏 + 𝛾
Each of the single resistant strains has a relatively simple solution as well:
𝑟𝑖∗ =
𝜆𝑅𝑖 + 𝜏𝜒𝑖 𝑝𝑖 𝑠 ∗
(−𝛽 + 𝜆𝑅𝑖 + 𝑐 + 𝛾 + 𝜏(𝜒𝑗 + 𝜒𝑘 + 𝜒𝑖 𝑑))
=
𝜆𝑅𝑖 +
𝜏𝜒𝑖 𝑝𝑖 𝜆𝑠
−𝛽 + 𝜆𝑠 + 𝑐 + 𝜏 + 𝛾
(−𝛽 + 𝜆𝑅𝑖 + 𝑐 + 𝛾 + 𝜏(𝜒𝑗 + 𝜒𝑘 + 𝜒𝑖 𝑑))
The solution for both double resistant strains is more complex:
𝑗
∗
𝑟𝑗,𝑘
=
𝑘
𝜆𝑅𝑗,𝑘 + 𝜏(𝜒𝑗 + 𝜒𝑘 𝑑𝜒𝑖,𝑗 )𝑝𝑗 𝑟𝑘∗ + 𝜏(𝜒𝑘 + 𝜒𝑗 + 𝑑𝜒𝑖,𝑘
)𝑝𝑘 𝑟𝑗∗
(−𝛽 + 𝜆𝑅𝑗,𝑘 + 𝑐 + 𝛾 + 𝜏(𝜒𝑖 + (𝜒𝑗 + 𝜒𝑘 )𝑑)
𝜆𝑅𝑘 +
𝜆𝑅𝑗,𝑘 + 𝜏𝜒𝑗 𝑝𝑗
=
(
𝑗
𝑘
𝜏(𝜒𝑘 + 𝜒𝑗 + 𝑑𝜒𝑖,𝑘
)𝑝𝑘 𝜆𝑠
−𝛽 + 𝜆𝑠 + 𝑐 + 𝜏 + 𝛾
(−𝛽 + 𝜆𝑅𝑘 + 𝑐 + 𝛾 + 𝜏(𝜒𝑗 + 𝜒𝑖 + 𝜒𝑘 𝑑))
+ 𝜏𝜒𝑘 𝑝𝑘
𝜆𝑅 𝑗 +
𝜏(𝜒𝑗 + 𝜒𝑘 + 𝑑𝜒𝑖,𝑗 )𝑝𝑗 𝜆𝑠
−𝛽 + 𝜆𝑠 + 𝑐 + 𝜏 + 𝛾
(−𝛽 + 𝜆𝑅𝑗 + 𝑐 + 𝛾 + 𝜏(𝜒𝑘 + 𝜒𝑖 + 𝜒𝑗 𝑑))
(
)
(−𝛽 + 𝜆𝑅𝑗,𝑘 + 𝑐 + 𝛾 + 𝜏(𝜒𝑖 + (𝜒𝑗 + 𝜒𝑘 )𝑑)
)
A necessary constraint for this approximation is that
  V  c     *Treatment V  , V  X . . Where Treatment V  is the fraction of
patients of class V treated.


Usually, R j ,k  V and Treatment R j ,k  Treatment V  , V  X and we get that the
condition translates to
(1)

  R  c     i    j   k  d
j ,k

Proof of corollary (I):
We will prove that e x  8  Σvi  x e i  8  Σvi  x 1  vi  is a more accurate approximation
v
than e x  1  x .
This is equivalent to minimizing the error term:

 
 
e x  1  x   e x  8  Σvi  x evi  8  Σvi  x 1  vi   8  Σvi  x 1  vi 

 e x  1  x   Σ vi  x evi  Σ vi  x 1  vi 
 e x  1  x   Σvi  x evi  Σvi  x 1  vi 
*
 e x  Σvi  x evi  1  x   Σvi  x 1  vi 
 X  1  ΣVi  X Vi  7  1  ln 1  X   7  ΣVi  X ln 1  Vi 
 X  ΣVi  X Vi  ln 1  X   ΣVi  X ln 1  Vi 

However, ln 1  X   ΣVi  X ln 1  Vi   ln 1  X   ln 1  ΣVi  X Vi
**

In addition X  ln 1  X   ΣVi  X Vi  ln 1  ΣVi  X Vi
***




 X  ΣVi  X Vi  ln 1  X   ln 1  ΣVi  X Vi  ln 1  X   Vi  X ln 1  Vi  Q.E.D
*e z  1  z  z  0 , **log 1  z  is concanve, and therefore subadditive ,
***Under Assumption I and the fact that  z  log 1  z   is monotonically increasing.
References: 1. Fujii, Kazuyuki. "Comment on" Epidemiological modeling of online social
network dynamics"." arXiv preprint arXiv:1402.1225 (2014).
Below we show the approximation (dashed lines) against numerical integration results (solid
curves) for parameters
X  0.07, p1  p2  p3  0.07, c  0.1 ,   0.1, R  R  0.003, R  R  R  3*105 ,
1
2
3
1,3
2,3
R  0.0015, S  c  X  R  R  R  R  R  R ,   0.03, d  0.778
1,2
1
2
3
1,2
1,3
2,3
for mix3 and mix2 . Embedded plots are enlarged sections of the original plots, made for
the reader's convenience.
Figure. A
Figure. B