Recent Researches in Modern Medicine
Drug Resistant and Wild-type Strains Interaction:
Investigating Effects of Conversion Delays for Possible Control
Strategies
KORSUK SIRINUKUNWATTANA
YONGWIMON LENBURY*
NARDTIDA TUMRASVIN
Department of Mathematics, Faculty of Science
Mahidol University
Rama 6 Rd., Bangkok 10400
Centre of Excellence in Mathematics, CHE, 328 Si Ayutthaya Road, Bangkok
THAILAND
*Corresponding author: scylb@mahidol.ac.th
Abstract: - The increasing threat of drug resistance is compromising medical care worldwide. To provide
deeper understanding of possible measures to avoid the endemicity of drug resistant strains, many models have
been proposed and analyzed on the dynamics of co-circulating wild-type and drug resistant viruses. We aim to
add to these works by considering a model which incorporates the effects of delay in the evolution of resistant
strain, as well as the role of the immune response and the target cells availability on the suppression of the peak
load of resistant virus. We investigate the effect of delays in connection with the system’s dynamic behaviour,
persistence of the two strains and the system stability.
Key-Words: - Drug resistance, Cost of resistance, maturation delays effects, global stability, uniform
persistence.
Puttasontiphot et al. [6] investigated bacteriaantibiotic dynamics in a chemostat exposed to
antimicrobial selection pressure. To support the
model derivation, experimental data were collected,
first from a culture of Enterococcus faecalis ATCC
29212 and Serratia marcescens ATCC 43861
growing in 1% Mueller-Hinton Broth II in the
absence of antibiotics, and then from a culture of
Bacteroides Fragilis with BMS-284756 for the
antibiotic. Their full model involves the density of a
sensitive strain S , that of a resistant strain R , the
concentration of the limiting resource, or substrate,
and the effective antibiotic level.
We, on the other hand, consider a situation in
which the resource is abundant so that we may take
the interactions of the two strains to be independent
of the substrate concentration. We further
incorporate the delay  in the evolution of the
resistant strain as well as the time it takes before the
resistant members become mature enough to
reproduce or transmit the chromosomal elements
known as plasmids to affect the conversion of
sensitive to resistant strain. Our referenced model is
thus written as follows.
1 Introduction
Many drugs and antibiotics that were formally
effective in fighting infections are no longer
effective because of the development of resistant
strains which poses a serious threat to public health.
Many intervention measures have been devised in
order to limit the emergence and spread of
antimicrobial-resistant bacteria such as methicillinresistant Staphylococcus aurous, vancomysinresistant Enterococci, and multidrug-resistant
Gram-negative bacilli [1]. However, these
pathogens still recover and continue to spread at
dangerous pace. Viral infections like HIV-1 are able
to rapidly produce resistant strains, causing life-long
infection, and resistance to Influenza-A drugs is
rising. Understanding drug resistant infections is
therefore at the forefront of concentrated research
[1].
Many mass-action models [1-6] have been used
to study the activity of a virus or bacterial species
within a host, describing the interaction between
strains, the cells they infect and the attempts of the
body’s immune response to remove the infection. Of
particular interest is the impact of drug
administration on the dynamics of co-circulating
wild-type and drug resistant viruses. In 2007,
ISBN: 978-960-474-278-3
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Recent Researches in Modern Medicine
dS
  S S  t  - H  S  t   S  t  R  t -   - dS S  t 
dt
-kS  t   I -kS  t   I
(1)
dR
 R (r  R(t))R(t  )  H(S(t))S(t)R(t  )
dt
 d R R (t   )
(2)
We shall first give a result on uniform
persistence, then study the global stability in the
1
case that the cost of resistance parameter   is
r
relatively low. Finally, we investigate the effect of
delay  on the oscillatory behavior of the solution
when  is rather high (low capacity r ).
where the first term on the right of (1) is the growth
rate of the sensitive strain, the second term accounts
for the conversion of sensitive members to resistant
ones.
The function H ( S ) in this second term is the
response function of the sensitive population to the
conversion attempts of the resistant population. In
[6], the response was assumed to take the form of a
Holling type function:
r
(3)
H (S ) 
Kr  S
In this paper, we shall let H ( S ) be a general
continuous function and investigate the effect
different formulations of H ( S ) have on the
dynamic behavior of the model.
The third term on the right of (1) is the death
rate, while the fourth term is the killing rate of S by
administered drugs. We also assume that the host is
being infected by the sensitive strain at the constant
rate I.
The first term on the right of equation (2) is the
growth rate of the resistant strain. Here, we have
taken into account the “cost of resistance” which
corresponds to the phenomena that the resistant
strain is less fit in the absence of treatment due to
being out competed by the wild type strain [7].
According to this concept, naturally we expect the
wild type strain to be stable with a higher
transmission rate than the drug resistant strain.
Furthermore, it was suggested in [7] and [8] that a
decrease in the peak load of resistant virus could be
attributed to the cytotoxic lymphocytes’ response or
a lack of target cells that limits the resistant virus
from replicating. We incorporate this effect by using
the logistic growth term for the resistant strain, so
that its growth rate decreases as the current density
rises to its carrying capacity r .
The second term on the right of (2) corresponds
to the increase in the resistant population from
conversion of the sensitive members. The last term
is the death rate of the resistant population. The
evolution of this strain has an intrinsic delay of  in
time which accounts for the delay in the responses
of the host and the immune system to the resistant
strain.
ISBN: 978-960-474-278-3
2 Uniform Persistence
Frequently, the strictly positive solutions of
biological model eventually approach the boundary
of non-zero zone. Such a situation is interpreted as
extinction of populations. Thus, the question that
arises is of specifying the condition that each
initially strictly positive solution is at some positive
distance away from the boundary as time evolves
[9]. In this part, we provide the restrictions to
guarantee that each strictly positive solution is
uniformly bounded away from the boundary, in
other words, uniformly persistence.
Letting
f ( S )   R r  R  H ( S ) S ,
(4)
S  d S  k  0 ,
(5)
R  d R  0 ,
(6)
and
 R r  R ,
(7)
we investigate some properties of the solutions of
(1) and (2), and the equilibrium point ( Sb , Rb )
which, by definition, satisfies the following system:
I   H ( Sb ) Rb  S  S  Sb ,
(8)
1
Rb 
(9)
f ( Sb ) .
R
we investigate some properties of the solutions of
(1) and (2), and the equilibrium point ( Sb , Rb )
which, by definition, satisfies the following system:
I   H ( Sb ) Rb  S   S  Sb ,
(10)
Rb 
1
(11)
f ( Sb ) .
R
In what follows,  denotes the set of real
numbers. We now state and prove our first result.
Theorem 1
Let
H ()
be a non-decreasing
function, S   S , and  S , R  be a bounded positive
solution of (1) and (2). Define
Rm  lim inf R  t  ,
RM  lim sup R  t  ,
t 
S m  lim inf S  t  ,
t 
184
t 
S M  lim sup S  t  .
t 
Recent Researches in Modern Medicine
Then,
Rm  Rb  RM ,
(12)
S m  Sb  S M .
(13)
This contradiction implies that   0   Sb , and
thus RM  Rb . In addition, it follows that
and
S M    0   Sb .
The other half of the proof is the similar, but we
need to construct a full time solution in an
appropriate way to yield Rm  Rb and S m  Sb ■
f   is monotonically increasing from
Proof
the hypothesis that H   is non-decreasing. For any
bounded positive solution  S , R  of (1) and (2), we
Remark 2
It is physically meaningful that the
equilibrium is bounded in the range of all bounded
positive solutions, and the numbers of both
microbial strains in the body should adjust to some
levels and remain steady when we are healthy.
can construct a full time solution   ,   by using
an  -limit set of  S , R  such that
RM    0   max   t  ,
t
Rm  min   t  ,
Corollary 3
Theorem 1 yields
inequalities
1
1
f  S m   Rm  RM 
f  SM  ,
t
Sm    t   S M , t  .
(Please see [10-12] for details on full time solutions
and their applications.)
Accordingly, from (2), we have
0    0    R  r  RM     
H    0    0     R    ,
R
f   0   .
R
First, we will show that RM  Rb , that is
RM    0  
1
R
f   0   
1
R
(15)
Proof
We initially verify (18) by constructing a
full time solution  S, R  such that
Rm  R  0   min R  t  ,
t
RM  max R  t  ,
t
Sm  S  t   S M , t   .
Hence, Sm  S  0  , and, from (2), we arrive at
prove that   0   Sb .
For the sake of contradiction, we assume
that   0   Sb .
Consequently, RM  Rb .
Since
1
R
We construct another full time solution f , e
f  S m   Rm .
(20)
From the proof of Theorem 1,   0   S M , which
Sm    0  , it follows that S m  Sb .
implies

1
f  SM  ,
R
in (15). Then, it is clear that
RM 
by using  -limit set of  S , R  such that
Sm  f  0   minf  t  ,
1
t
S M  max f  t  ,
R
t
Rm  e  t   RM t   .
Again, it follows that f 0  0 .
f  S m   Rm  RM 
(21)
1
R
f  SM  .
In order to verify (19), we again construct a full
time solution S , R  such that
S M  S  0   max S  t  ,
t
(16)
Sm  min S  t  ,
Since we already have Sm  S  0   Sb , and RM  Rb ,
(16) becomes
I  s  s  H  Sm e     Sm
ISBN: 978-960-474-278-3
(19)
Since f   is monotonically increasing, we need to
 s   s  H  Sb  Rb  Sb  I .
(18)
(14)
f  Sb  .
From (1), we have
I  s  s  H  Sm e     Sm ,
R
S  S  H  SM  Rm  SM  I  S  S  H  Sm  RM  Sm
and
1
following
t
Rm  R  t   RM , t   .
Since Rm  R    , we obtain
S  S  H  SM  Rm  SM  S  S  H  SM  R    SM
I .
(22)
(17)
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Recent Researches in Modern Medicine
By the definition of R  t  , R    RM , and from
(16), we have
I  s   s  H  S m  R     S m
 S   S  H  S m  RM  S m .
S  S  H  SM  Rb  SM  S  S  H  SM  Rm  SM  I
.
(26)
Since H   is non-decreasing,
I  S  S  H  Sb  Rb  Sb  S  S  H  SM  Rb  SM ,
(23)
According to (22) and (23), we therefore obtain
(27)
Hence, S M  Sb .
Now, to show that iv) implies
suppose S M  Sb . From (11) and (18), we have
1
1
Rb 
f  Sb  
f  S M   RM .
R
R
Again, since RM  Rb , we have RM  Rb . ■
S  S  H  SM  Sm  SM  I  S  S  H  Sm  RM  Sm .
■
Remark 4
Under the assumptions of Theorem
1, uniform persistence of the system (1)-(2)
physically represents the fact that our body will not
be free of infection. It is reported that the bacterial
populations colonize the mammalian digestive tract
since birth, assisting in that life form’s efficient
digestion and nutrients’ absorption [13].
Lemma 5
By the assumptions in Theorem 1,
the following conditions are equivalent.
i) RM  Rb
ii) S m  Sb
iii) Rm  Rb
Remark 6
Every
non-constant
periodic
solution of (1) and (2) must oscillate around the
basal level  Sb , Rb  ; otherwise, one of the cases in
Lemma 5 is satisfied which forces all strictly
bounded positive solutions to converge to  Sb , Rb  .
iv) S M  Sb
3 Global Stability in the Case  S   S
Proof To prove that i) implies ii), assume
that RM  Rb . Then from the second inequality in
(18),
I  S   S  H  S m  RM  S m
 S   S  H  S m  Rb  S m .
i),
In this section, we consider the case where the
composite removal rate S  d S  k of the sensitive
population is greater than its growth rate S . We
first state and prove the following theorem on the
stability of the equilibrium  Sb , Rb  .
(24)
Since H   is non-decreasing, it follows that
Theorem 7 Let S   S , and  S , R  be a bounded
positive solution of (1) and (2). Define
1
(28)
L1 
sup f ( S ) ,
S  S  H  Sm  Rb  Sm  S  S  H  Sb  Rb  Sb  I
.
(25)
Consequently, by (24) and (25), Sm  Sb .
To show that ii) implies iii), we suppose Sm  Sb .
R
R  such that,
We construct a full time solution  S,
as before,
Rm  R  0   min R  t  ,
L2 
1
R
L3 
t
RM  max R  t  ,
t
Sm  S  t   S M , t   .
It follows from (20) that
1
1
Rb 
f  Sb  
f  S m   Rm
R
R
However, we have Rm  Rb . Hence, we must have
Rm  Rb .
To show that iii) implies iv), we consider the first
inequality of (19),
S  Sb ,  
sup f ( S ) ,
S  0, Sb 
H  Sb  S b
S   S 
.
(29)
(30)
Then,
RM  Rb  L1  S M  Sb  ,
(31)
Rb  Rm  L2  Sb  Sm  ,
(32)
Sb  Sm  L3  RM  Rb  ,
(33)
S M  Sb  L3  Rb  Rm  .
(34)
If L L L  1 then every bounded positive solution of
(1) and (2) converges to a positive equilibrium.
2
1 2 3
Proof
Note that although f   is monotonically
increasing, (28) is well-defined because S  t  is
ISBN: 978-960-474-278-3
186
Recent Researches in Modern Medicine
One crucial underlying assumption that ensures
persistence and stability is that the “cost of
resistance”  is low enough so that its inverse, the
carrying capacity r , is high enough for  R r to
exceed the death rate R of the resistant strain. If
the cost of resistance is sufficiently high then the
persistence or stability of the system could be lost,
leading to extinction of one strain or both, unless
certain conditions on the delay  are satisfied.
The main advantage of modelling and analysis is
that we may quantitatively investigate the possible
strategies for therapy and control in terms of the
complex dynamics of competing bacterial or viral
strains. Clinically, it is difficult to assess pharmacodynamic effects of therapy regimens due to the
complexity in repeatedly determining the viral load
at the site of infection and antibiotic concentrations
during the dosing interval [14]. Using dynamic
models of sensitive-resistant strains interactions can
overcome these difficulties. Through the above
model development and analysis, we gain insightful
information that could prove useful in designing
empiric therapy and monitoring strategies.
bounded. Let ( S , R ) be a bounded positive solution
of the system (1) and (2). By constructing a full time
solution   ,   such that
RM    0   max   t  ,
t
Rm  min   t  ,
t
Sm    t   S M , t   ,
we again have RM    0  
  0   Sb .
1
R
f    0   , and
The mean value theorem implies that
f    0    f  Sb 
 sup f ( S ) . As a result,
  0   Sb
S Sb ,  
RM  Rb 
1
R
 f   0    f  S  
b
 L1    0   Sb   L1  S M  Sb  .
We now construct a full time solution f , e
such that
Sm  f  0   minf  t  ,

t
S M  max f  t  ,
5 Acknowledgement
t
This research work is partially supported by the
National Center for Genetic Engineering and
Biotechnology, and the Centre of Excellence in
Mathematics, CHE, Thailand, and Mahidol
University.
Rm  e  t   RM t   .
With (10) and (16), we have
1
Sb  Sm 
 H  Sm e    Sm  H  Sb  Rb Sb 
S  S 

H  Sb  Sb
S   S
 RM
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 Rb   L3  RM  Rb  .
In the same way, (32) and (34) can be proven.
Hence, by (31)-(34),
RM  Rm  L1 L2 L23  RM  Rm  .
(35)
If L1 L2 L23  1 , then RM  Rm  0 . By Lemma 5, the
solution ( R, S ) converges to ( Sb , Rb ) .
4 Conclusion
We have shown that the system is uniformly
persistent when the response function H   is a
non-decreasing function of S . Further, the
equilibrium state ( Sb , Rb ) is globally stable in the
case that the removal rate S  d S  k of the
sensitive strain is greater than its specific growth
rate.
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