Supporting Material
1. Defining the emergence rate of double resistant bacteria for HGT:
Define Ri to be the frequency of patients infected with bacteria resistant to antibiotic i {1, 2} .
When modeling constant probabilities of HGT, the emergence of double resistance, for a treatment
strategy U , is proportional to:
t1

E HGT  U  : R1  U  R 2  U  dt
t0
for a time interval of length t1  t 0 . The proportion constant includes the rate of a successful
bacterial transmission, and probability of HGT of genetic material coding for antibiotic resistance
between the different bacteria.
Let rs , rr be the probabilities of receiving genetic material encoding for antibiotic resistance in the
presence of antibiotics to which the bacteria are susceptible or resistant, respectively. Similarly, let
d s , dr be the probabilities of donating genetic material encoding for antibiotic resistance in the
presence of antibiotics to which the bacteria are susceptible or resistant, respectively. We will
assume that these probabilities are independent of the type of antibiotic the bacteria are resistant
to.  will be defined as the time delay of stress-induction;  will be the relative persistence of
double resistant bacteria when having no advantage over single resistant bacteria within the same
host; C will be defined as the rate of bacterial transmission between patients (all as defined in the
main text).
The term that describes the emergence of double-resistant bacteria for a strategy U will be
denoted  HGT (U ) . We will derive the explicit expressions of  HGT for each of the three strategies.
The emergence of double resistance in combining is straightforward, since all single resistant
bacteria are under antibiotic stress.
(1)
HGT (Combining )  Crs d s EHGT Combining 
For the cycling strategy we will define  as the antibiotic cycle length. We can divide the time to
1
 length segments, in which only one drug is used in the hospital unit. After α days the process is
2
repeated and the first antibiotic is applied again. We take t 0  0, t1  k , for some k  N, to avoid
bias. Since at each of these segments the antibiotic treatment is constant we get:
1
 HGT  Cycling  
C
4
(i 1)
α
2k 1 2
 
i 0
 rr d s  rr d s  rsd r  rsd r  R1  cycling  R 2  cycling  dt
i
α
2
Since only one antibiotic is used, one strain of bacteria will be stressed throughout each time
segment. The multiplier inside the integral arises from the transportation of bacteria from one
patient to another. The double resistant bacteria's probability of contributing to double resistance
emergence is changed according to the relative advantage they have over the host's bacterial
population (thus the multiplication by  ). Equation (2) is simplified to:
(2)
 HGT  Cycling  

C
 rr d s  rr d s  rsd r  rsd r 
4
(i 1)
α
2k 1 2
 
i 0
R1  cycling  R 2  cycling  dt
i
α
2
C
(rr d s  rs d r )(1   ) EHGT (cycling )
4
Since mixing is the only strategy in which patients are treated with different antibiotics it has a more complex
expression for the double resistance emergence. Under the mixing strategy bacteria can be transported
between patients treated with the same or different antibiotics and consequently  now also appears in the
dynamics. Therefore, for the mixing strategy we get:
 HGT  mixing   C
t1
 8 r (d   d (1   ))  r d   r d
1
s
s
r
r s
s r
 rr (d r  d s (1   )) 
t0
d s (rs  (1   )rr )  d s rr  d r rs  d r (rr  rs (1   ))  R1  mixing  R 2  mixing  dt
Which simplifies to:
(3)
1
8
d s (rs  (1   )rr )  d s rr  d r rs  d r (rr  rs (1   ))  EHGT ( Mixing )
 HGT  Cycling   C rs (d s  d r (1   ))  rr d s  rs d r  rr (d r  d s (1   )) 
1.1 Comparing the emergence of double resistance for different strategies under stress-induced
HGT
We will compare the influence of stress-induced HGT on the different strategies. Due to the
simplifications presented in equations 1-3, we can vary rr , rs , dr , d s ,  ,  independently of the EHGT
value. Thus we can compare  HGT for different strategies. Note that we can disregard C throughout
this section, since it appears as a constant multiplier in the emergence of double resistant bacteria
for all of the strategies.
2
Mixing Vs. Combining
When comparing mixing to combining we get the following expression:
 HGT (mixing )
 HGT (combining )

1
rs (d s  d r (1   ))  rr d s  rs d r  rr (d r  d s (1   ))  d s (rs  (1   )rr )  d s rr  d r rs  d r (rr  rs (1   ))  EHGT (mixing )
8
rs d s EHGT (combining )
Since the dynamics determining the value of EHGT are very complex, we can explore the multiplier
EHGT (mixing )
 HGT (mixing )
and see if it can have a strong influence on
. The term
EHGT (combining )
 HGT (combining )
of
describing the ratio of EHGT coefficients is:
(4)
1
rs (d s  d r (1   ))  rr d s  rs d r  rr (d r  d s (1   ))  d s (rs  (1   )rr )  d s rr  d r rs  d r (rr  rs (1   )) 
8

rs d s

1
dr
rr
d r rr d r
r r d
d r
 (   (1   ))    (1   ) r  r  r   r ( r   (1   ))  
  (1   )   
8
ds
rs
d s rs d s
rs rs d s
d s rs


1
dr
rr
rr d r
d r rr
  
 2  (2   )  (2   )  (2   )   (2   )  2
8
ds
rs
rs d s
d s rs


1
dr
rr
d r rr
  
 2  (2   )   (2   )   (2   )  (2   )   2
8
ds
rs
d s rs


d
1
r 
d r
 2   r  r  (2   )(1   )  2 r r  
8 
d s rs

 d s rs 
Since 0    1, 0 
dr
r
 1, 0  r  1, 0    1 , the upper bound of this term is 1. The lower bound is more
ds
rs

complex. It is mainly dominated by max  ,



d r rr 
,  . If the response of the bacteria to stress is quick,
d s rs 

i.e.  is small, then the stress-induction mechanisms have a strong effect. In this case

 EHGT (mixing )
 d r  1 d r
  max  r , r   (( r r  1) ) 
.
 HGT (combining ) 
 EHGT (combining )
 d s rs  4 ds rs
 HGT (mixing )
Therefore for a large enough effect of stress on both donors and recipients of genetic material and a
small value of  ,
 HGT (mixing )
will tend to zero. If the response of the bacteria to stress is slow,
 HGT (combining )
3
i.e.  larger then
lower bound of
d r rr
, , then even when stress affects HGT drastically, the coefficient will have a
d s rs

.
4
Cycling Vs. Mixing
From equations 2 and 3 we know that
 HGT (mixing )

 HGT (cycling )
1
rs (d s  d r (1   ))  rr d s  rs dr  rr (d r  d s (1   ))  d s (rs  (1   )rr )  d s rr  d r rs  dr (rr  rs (1   ))  E
8
HGT ( mixing )
1
E
HGT (cycling )
(rr d s  rs d r  rr d s  rs d r )
4
We will compare the coefficient of
EHGT (mixing )
:
EHGT (cycling )
(5)
1
rs (d s  d r (1   ))  rr d s  rs d r  rr (d r  d s (1   ))  d s (rs  (1   )rr )  d s rr  d r rs  d r (rr  rs (1   )) 
8

1
(rr d s  rs d r  rr d s  rs d r )
4


d
r 
d r
  2   r  r  (2   )(1   )  2 r r   
d s rs
1 
 d s rs 
  
 

 rr d r 
2

   1   


rs d s 





d r
2 r r 


1
2
d s rs

 (2   ) 

r d 
2   rr d r 
    1     r  r  1   

  rs d s 

 rs d s 
This term has a lower bound of 1. When
dr
r
and r decrease, the expression increases. The
ds
rs
upper bound is mainly determined by the term :
4
2
.
 rr d r 
   1   
 rs d s 
 HGT (mixing )

 HGT (cycling )
Thus
 r d 
max  r , r 
 rs d s 
then

r d 
2 max  r , r  (1   )
 rs d s 
EHGT (mixing )
which means that if
EHGT (cycling )
 r d r 
 is small enough,  HGT (mixing )   HGT (cycling ) . If  is zero
 rs d s 
 , and max  r ,
 HGT (mixing ) EHGT (mixing )
.

 HGT (cycling ) EHGT (cycling )
Combining Vs. Cycling
From equations 1 and 3 we know that
 HGT  combining 
EHGT  combining 
rs d s

1
 HGT  cycling 
(rr d s  rs d r  rr d s  rs d r ) EHGT  cycling 
4
Again, we will compare the coefficient of
EHGT  combining 
EHGT  cycling 
:
(6)
rs d s
4

1
(rr d s  rs dr  rr d s  rs dr )  rr  dr  1   
4
 rs d s 
This term has a lower bound of 1. We can also see that
 HGT  combining 

 HGT  cycling 
1
r d 
max  r , r 
 rs d s 
EHGT  combining 
EHGT  cycling 
Thus if HGT is influenced enough by stress HGT  combining   HGT  cycling 
2. SIM
Analogous to the stress induced HGT, we define r and s as the mutation rates when bacteria are
not under antibiotic stress, and when they are under antibiotic stress, respectively.
We will also define the following term, for a treatment strategy U :
EM U  :
t1
  R U   R U  dt
1
2
t0
5
The emergence of double-resistant bacteria for a strategy U will be denoted  SIM (U ) . We will
derive the explicit expressions of  SIM for each of the three strategies.
The emergence of double resistance under the mixing strategy will be:
(7)
SIM  Mixing  
1
 μr  μ s  EM (Mixing )
2
For combining, the term describing double resistance emergence will be:
(8)
SIM  Combining   μ s EM (Combinig )
The emergence of double resistance under cycling is a bit messier. If T is the antibiotic currently
used in the hospital we get:
 SIM  Cycling  
t1
R  
s {2T }
1


 r (1  {2T } )  R2  s {1T }  r (1  {1T } )
  dt
t0
We will divide the integral to time segments, as was defined in section 1.
 SIM  Cycling  
(i 1)
α
2 k 1 2
  R  
1
i 0
k 1

s {2T }
1
iα  
2
  R  
s {2T }
1


 r (1  {2T } )  R2  s {1T }  r (1  {1T } )


 r (1  {2T } )  R2 s {1T }  r (1  {1T } )
i 0
iα
k 1 iα  α
  R  
1
i 0
s {2T }
  dt
i
α
2


 r (1  {2T } )  R2 s {1T }  r (1  {1T } )
 dt 
 dt
1
iα  
2
Simplifying and assuming, without loss of generality, that each cycle starts with treatment with
antibiotic 1 we get:
 SIM  Cycling  
k 1
1
iα  
2
i 0
iα

k 1 iα  α
 μ r R1  Cycling    μ s R2  Cycling   dt  
i 0
We will define:
6

1
iα  
2
 μ s R1  Cycling   μ r R2  Cycling    dt
(9)
k 1 iα  α
 
A›
i 0
k 1 iα  α
 
B›
i 0
R1  Cycling  dt 
1
iα  
2
R2  Cycling  dt 
1
iα  
2
k 1
1
iα  
2
i 0
iα
k 1
1
iα  
2
i 0
iα
 
 
R2  Cycling  dt
R1  Cycling  dt
t1
Thus A  B 
  R Cycling   R Cycling   dt  E
1
2
M (Cycling )
t0
And thus we get:
(10)
k 1
1
iα  
2
i 0
iα

k 1 iα  α
 μ r R1  Cycling    μ s R2  Cycling   dt  

 k 1 iα  α
μs 
R1  Cycling  dt 
 i 0 1
 iα  
2

 
i 0
k 1
1
iα  
2
i 0
iα
 

 k 1 iα  α
μ r 
R2  Cycling  dt 
 i 0 1
 iα  
2

 

 μ s R1  Cycling   μ r R2  Cycling    dt 
1
iα  
2


R2  Cycling  dt  



k 1
1
iα  
2
i 0
iα
 


R1  Cycling  dt  



 μ s A  μ r B .
2.1 SIM: Comparing the emergence of double resistance for the different strategies
If A  B (A and B as defined in (9)) then:
1
μ s A  μ r B   μ s  μ r  A  B   2
7
1
1
1
1
μ s A  μ r B  μ s B  μ r A 
2
2
2
2
μ s  A  B   μr  A  B  
μ s  μ r which is true under mutation is stress-induced.
A  B is usually true since the frequency of a resistant strain is lower when the drug applied is the
one it is susceptible to. We have also verified this by simulating random parameter sets and
observing the values of A and B (results not shown).
Thus, assuming A  B and using inequality (18) and μ s  μ r , we get:
1
2
SIM  Cycling   μ s A  μ r B  (μ s  μ r )  A  B   μ s ( A  B)
Now we can see that:
(11)
t1
  R Cycling    
s {2T }
1
 SIM (Cycling ) t0

 SIM ( Mixing )


 r (1  {2T } )  R2  Cycling   s {1T }  r (1  {1T } )
1
 r  s 
2
  dt
 t  R1  Mixing   R2  Mixing  dt
t1

0
1
 s  r  EM (Cycling ) EM (Cycling )
 2

1
EM ( Mixing )




E
(
Mixing
)
 r
s M
2
And
(12)
t1
 SIM (Cycling )
 SIM (Combining )
  R Cycling    
1

s {2T }


 r (1  {2T } )  R2  Cycling   s {1T }  r (1  {1T } )
t0
s
 t  R1 Combining   R2 Combining  dt
t1
0
 s EM (Cycling )
EM (Cycling )


 s EM (Combining ) EM (Combining )
And of course
(13)
1
( s  r ) EM ( Mixing )
EM ( Mixing )
 2

 SIM (Combining )
 s EM (Combining )
EM (Combining )
 SIM ( Mixing )
8
  dt

Thus we get the same direction of the effect of stress induced genetic variation on the effectiveness
of the strategies as we got in both forms of stress induced HGT.
As for lower bounds on the ratio, we can denote r  x , and s  ax for a  1 . Now we see that for A
and B as in (9):
(14)
 SIM (Cycling )

 SIM ( Mixing ) 1
s A  r B
 r  s   t  R1  Mixing   R2  Mixing   dt
2
t1

0
axA  x B
1
 x  ax 
2
 t  R1  Mixing   R2  Mixing  dt
t1

0







 EM (Cycling )
aA   B
A B
aA   B




1
1
1
  a  Em ( Mixing )    u  a( A  B)  EM ( Mixing )    a  ( A  B)  EM ( Mixing )

2
2

2

aA   B
When we take the limit of this expression for a   we get that the multiplier goes to
2A
. This
A B
means that the effect of SIM on the effectiveness of mixing relative to cycling is bounded by the
ratio
2A
. This ratio expresses the mean frequency of patients infected with single resistant
A B
bacteria when they are treated with a drug they are resistant to, relative to the mean of their
frequency through time. Note that for the special case of   0 stress induction does not affect
cycling. We will have

 A
 A B

 2

 EM (Cycling )

 EM ( Mixing )

for any amount of stress induction.
With combining the result is similar:
(15)
 aA   B  EM (Cycling )
 SIM (Cycling )


 SIM (Combining )  a( A  B)  EM (Combining )
9
When we take the limit of this expression for a   , or   0 for any value of a , the multiplier
goes to
A
. Using (14) and (15) we can easily see that the lower bound on the effect of SIM on
A B
the effectiveness of mixing relative to combining is half.
3. Robustness
In order to talk about the robustness of our results to changes in parameters, we will look at the
behavior of the frequency of patients with resistant bacteria at equilibrium. We can define a new
variable in our system of differential equations: R : R1  R2 , where R : R1  R2 . This reduces
the order of the polynomial we need to solve for to obtain the equilibrium values.
For the combining strategy the equilibrium reduces to a quadratic equation, where the solutions are
S
 s( R  R  R m  R  R 2 (4 (R   s)m  (    m   ) 2 ))
2R (R   s)
R
R  R  R m  R  R 2 (4 (R   s)m  (    m   ) 2 )
2 (R   s)
The only solution which is positive ( S and R are frequencies) is
S
R

 s     m    (4 ( r   s)m  (     m   ) 2 )
2  ( r   s )

R     m    (4 (R   s)m  (     m   ) 2 )


2 (R   s)
Moreover, we can see that Req  Seq
R
.
S
Now we want to find the values of R1 and R2 at equilibrium.
At equilibrium we have
Ri 
dRi
 0 , therefore:
dt
Ri m
(    m   X )
We also know that X  1  R  S  1  S
And thus Ri 
Ri1 m
(    m   (1  S (1 
R

 S  1  S (1  R )
S
S
R
)))
S
Ri m

(    m   (1  S (1 
10
R1  R 2
)))
S
And since S is independent of the ratio, we can see that Ri ( with a constant sum R1  R2 ) affects
the equilibrium of Ri linearly with a slope of one. In addition,
R1 R1
.

R2 R2
We can analyze the equilibrium in the same manner with mixing, however the expression for the
equilibrium values there is very long. Suffice to say that we can find a solution for the frequency of
X that does not depend on the ratio
R1
R2
but only the sum R1  R2 (since we only used R ) and
once more get that the equilibrium values Ri 
Ri m
(    m   X )
are dependent on Ri in the same way
as in combining.
1
2
1
2
For cycling, in times i    t  i   antibiotic 2 is used (T  2) and in times i    t  i  
antibiotic 1 is used (T  1) for i  N . The frequencies of the bacterial strains after the drug has been
switched and the system has stabilized are:
R1 m

1
, i    t  i  

2
 (    m   X )
R1 (t )  
R1 m
1

 (  m   X ) , i    t  i  2 

R2 m

1
, i    t  i  

(




m


X
)
2

R2 (t )  
R2 m
1

 (  m   X ) , i  2   t  i  

Analogously to the other strategies, we can rewrite the equations so the value of X at equilibrium
will not be dependent on
R1
R2
. Thus, if equilibrium is reached, we again have linear dependence on
resistant strains entrance rates for a constant entrance rate sum. For our cycling lengths (200 days
for each cycle) the system reached equilibrium approximately 10 days after a drug switch.
We have explored the interactions between parameters in the SIM model as well. As we have shown
above, the bound on the effect of stress on mutation in cycling depends on the number of patients
with resistant bacteria treated with effective antibiotics, relative to all the resistant patients, namely
 A 
5
 A  B  . By simulating 10 random sets of parameters we saw that one of the most influential


parameters affecting
A
is  . The intuitive explanation is that as clearance rate due to antibiotic
A B
usage increases, the fraction of patients currently treated will quickly reduce. However, we must
11
note that the value of  does not affects the values of ESIM for different strategies in the same
manner.
12
Table S1
The mean values of the proportion of infected patients, emergence of double resistance through
mutation, and emergence of double resistance through HGT are given. The means are taken over
104 random sets of parameters (the sampling distributions are given in the main text).
Figure S1. Double resistance emergence ratio under mixing and cycling when assuming SIM.
The ratio of double resistance emergence under mixing to double resistance emergence under
cycling, (
 SIM (mixing )

), is shown as a function of mutation increase under stress, s , and of
 SIM (cycling )
r
antibiotic clearance time  (panel A) or entrance ratio,
m
(panel B). We can see that for a wide
m
1
2
range of parameters, especially when mutation rates are strongly increased by stress, the ratio is
substantially higher than one (any color different than dark blue in the figure), meaning that cycling
is a better strategy for inhibiting double resistance. The emergence is taken over 600 days. The
length of each cycle in the cycling strategy is 200 days. Other parameter values (except when
varied):
  0.9,   0.03, m  0.1, s  0.1, R  0.1, R  0.1,   0.5,   0.5 .
1
2
Figure S2. Double resistance emergence ratio under mixing and cycling when assuming stressinduced HGT.
The ratio of double resistance emergence under mixing to double resistance emergence under
cycling (
 HGT (mixing )
) is plotted as a function of HGT increase under stress,  , and of antibiotic
 HGT (cycling )
clearance time  (panel A) or entrance ratio,
m
(panel B). We can see that for a wide range of
m
1
2
parameters, especially when HGT rates are strongly increased by stress, the ratio is substantially
higher than one (any color different than dark blue in the figure), meaning that cycling is a better
strategy for inhibiting double resistance. The emergence is taken over 600 days. The length of each
cycle in the cycling strategy is 200 days. Other parameter values (except when varied):
  0.9,   0.03, m  0.1, s  0.1, R  0.1, R  0.1,   0.5,   0.5,   0.5 .
1
2
13
Table S1
Infection
SIM:
SIM:
Cycling
0.73238
2.6699 10
7
4.9089 10
7
2.4460 10
6
2.6699 10
6
2.9803 10
14
Mixing
0.73226
2.6992 10
7
4.9077 10
7
2.4784 10
6
2.6992 10
6
2.9796 10
14
5.4174 10
Combining
0.72580
4.8427 10
7
4.8427 10
7
4.8427 10
6
4.8427 10
6
5.2766 10
14
5.2766 10
  0.1
s
1
r
SIM:
SIM:
  0.1
s
 10
r
 1
s
1
r
HGT:
 1
s
 10
r
HGT:
14
HGT:
 1
 1
  0.1
 1
HGT:
 0
  10
4.0640 10
13
4.0640 10
13
14
4.0630 10
13
1.5643 10
12
14
5.2766 10
12
5.2766 10
12
5.4187 10
14
 1
  10