Continuous Influenza Virus Production in Cell Culture Shows a
Periodic Accumulation of Defective Interfering Particles
Timo Frensing, Frank Stefan Heldt, Antje Pflugmacher, Ilona Behrendt, Ingo Jordan,
Dietrich Flockerzi, Yvonne Genzel, Udo Reichl
Text S1: Theoretical analysis for the models (1) and (2)
1.
Scaled DIP-Model in 6D
We consider the 6D system (1) with the 8 positive parameters D , Tin ,  , k1 , k2 , k3 ,
k33 , k4 , and f  0 and are mostly interested in nonnegative initial values of the form
T  0  T0  0 , I d  0  I s  0  Ic  0  0 , Vs  0   Vs 0  0 and Vd  0  Vd 0  0 . We
investigate the case with f  0 . A small positive f won’t change much the dynamics
in R 6 0 but we could always take Vd  0  to be 0 . Note that the nonnegative orthants
R 6 0 and the one with I d  I c  Vd  0 are positive invariant.
We will provide a structural analysis of system (1) for arbitrary parameter values. In
particular, we will prove first that the structure of system (1) without DIPs
 I d  Ic  Vd  0 excludes a Hopf bifurcation to periodic solutions for D   . In a
second step, we show that, in contrast, system (1) with DIPs allows a Hopf
bifurcation to periodic oscillations for D   . To this end we reduce the number of
parameters by introducing the scalings
k
k
   k2  D  t and   1 col T , I s ,Vs  ,   1 col  I d , I c ,Vd 
(S1.1)
k2  D
k2  D
with   col 1 , 2 , 3  ,   col 1 ,2 ,3  and arrive at the 6D-system
d1
 u    3  1  13 ,
d
d 2
 13   2   23 ,
d
d 3
  2  1   2  1  2    3 ,
d
d1
   3 1  13 ,
d
d 2
 31   2   23 ,
d
d3
  32  1   2  1   2   3 ,
d
with the 5 parameters
k DT
u  1 in 2  0,
 k2  D 

k3
 0,
k2  D
3 
k33
 0,
k2  D
1  0  
k1T0
,
k2  D
2  0  0,
3  0  
(S1.2b)
(S1.2c)
1  0   0,
(S1.2d)
2  0  0,
(S1.2e)
3  0  

k1Vs 0
,
k2  D
(S1.2a)
D
k2  D

k1Vd 0
,
k2  D
(S1.2f)
 0,
(S1.3a)
k4  D
 0.
k2  D
(S1.3b)
In the new variables, we obtain a system of the form
  F    B    U  A     B   ,
  0   0 ,
(S1.4a)
  C  ,  ,
  0  0 ,
(S1.4b)
with U   u , 0, 0  and
T
     3  0

0
 0 0 1 




A    
3
1
0
B     0 0  2  ,
,
(S1.4c)




0
  1   2    
 3 3 0 


     3  0

1


C  ,   
3
1
2
.
(S1.4d)


0
 3  1   2  1  2    


The octant O   j  0, j  0 is positive invariant with the DIP-free reduced system
  F     U  A    ,
  0   0 .
(S1.5)
Moreover, the octant 1  0, 2  3  0, j  0 and the positive 1 -axis are positive
invariant. We observe in (S1.3) that the new parameters depend linearly on Tin , k3 ,
k4 , k33 and  whereas the dependence on D is rather involved.
2.
DIP-free Model in 3D
To test whether the oscillations in virus titers are caused by DIPs or the continuous
process mode, we use the reduced model for  neglecting defective particles from
 . In this 3D case, we prefer the notation  x, y, z T instead of 1 ,  2 , 3 T so that
system (S1.5) reads
kT
dx
x  0   x0  1 0 ,
 u    z  x,
(S2.1a)
k2  D
d
dy
y  0  0,
 xz  y ,
(S2.1b)
d
kV
dz
z  0   z0  1 s 0 ,
  y   x  y    z,
(S2.1c)
k2  D
d
with the 4 parameters u ,  ,  and  from (S1.3). The sign of  is not fixed. The
positive x-axis is positive invariant. For   0 one has the globally exponentially
stable equilibrium x  u / on it.
We turn to the nonnegative equilibria E  col  x, y, z  of (S2.1) on the boundary of
R 3 0 . We first note that x cannot be 0 because of u  0 (cf. (S2.1a)). In case
z  0  y (cf. (S2.1b)) one has x  u /  0 for   0 . There's no nonnegative
equilibrium of the form E  col  x,0,0 for   0 .
We now come to the positive equilibria E  col  x, y, z  of (S2.1). The equation x  0
can be rewritten as
u
x
  z ,
the equation z  0 can be rewritten as
x  xz     x,
The equation
(S2.2a)
i.e.,
    1  z  x ,
(S2.2b)
y  xz,
(S2.2c)
originating from y  0 , and the relations (S2.2a) and (S2.2b) can be seen as defining
equations for positive steady states parameterized by
x  0, z   0,  1 , u  0
and
(S2.2d)
  1.
Lemma 2.1 Just in case   0 , system (S2.1) allows a nonnegative equilibrium on
 u

the boundary of R 3 0 , namely E0  col   , 0, 0  . System (S2.1) possesses a unique
 

positive equilibrium E1  col  x, y, z  with
 1 u     ,  1 ,


u
y  xz,
,
  1 
u
where the side conditions can be rewritten in terms of the parameters as
  1 u :  .
  1 and   1    
min
0 z 
x

3.
(S2.3a)
(S2.3b)
Bifurcation Analysis in 3D
The Jacobian of (S2.1) at a nonnegative equilibrium E is given by
  z

0
x


Jac  E    z
1
x

  z   z   x  y    
(S3.1b)
in terms of 6 variables which are related by the equilibrium conditions in (S2.2).
Lemma 3.1 The Jacobian J 0 : Jac  E0  at the boundary equilibrium E0 possesses
three negative eigenvalues for    min and exactly two negative eigenvalues for
   min ,0 . A transcritical stationary bifurcation gives rise to the positive equilibrium
E1 for    min . The Jacobian J  : Jac  E1  at the positive equilibrium E1 possesses
three negative eigenvalues for  close to  min . For negative  's, the real parts of the
eigenvalues of J  stay negative, so that there does not occur a Hopf bifurcation to
periodic solutions in the nonlinear system (S2.1) at E1 for negative  's.
Proof:
1) For negative  , one has
u /  
 0

J 0 : Jac  E    0 1
u /   .
 0  u /    
(S3.2)
It has the negative eigenvalue  with the x-axis as eigenspace. The  2  2 -block
possesses the negative trace 1    u / 
and the determinant     1 u /  .
Thus one encounters the eigenvalue 0 exactly for    min . A transcritical stationary
bifurcation happens giving rise to the positive equilibrium: For  's with (S2.3b) there
exists the positive equilibrium E1 emanating at    min from E0 (which exists for all
negative  's).
2) At positive equilibria E1  col  x, y, z  , the Jacobian (S3.1) can be rewritten as
0
 x    z
0
x 
 u / x



J  : Jac  E1    z
1
x  z
1
x 
(S3.3)
  z
  z  x    z   z  x 
in terms of 5 variables. We seek conditions for pure imaginary eigenvalues of J  . So
we investigate the characteristic polynomial
    : det  I  J     3   2 A2   A1  A0
(S3.4a)
with positive coefficients
A2   z     x  1, A1   z   x  1 , A0  xz  1  z  .
(S3.4b)
Thus the Jacobian J  does not possess a nonnegative real eigenvalue and it has the
negative determinant  A0 . Hence, for sufficiently small positive x , J  has 3 negative
eigenvalues. We investigate whether these eigenvalues can cross the imaginary axis
at some nonzero points i . In order to have a nonzero pure imaginary eigenvalue
i one has to solve the necessary and sufficient conditions
0  Re(   i  , 0  Im(   i  .
!
!
This amounts to solving  2  A1  A0 / A2 , i.e.,
u
u
(S3.5)
   x  1  x  1   z    x  1   z   x  1   xz   1  z  .
x
x
Here, we consider the variables u ,  , x and z from (S2.2d) as parameters defining
an equilibrium via the equations (S2.2a)-(S2.2c).
3) We show that, given an admissible solution z  Z1/2  x, u, a  of the quadratic
equation (S3.5) (wrt. z ), the corresponding   z  u / x from (S2.2a) is necessarily
u
nonnegative. To this end, we scale z via z  
introducing the new positive
x
variable  so that equation (S3.5) is equivalent to
q  u,  , x,   :  u  x  x  1   x  1   x    1 x   u   0 .
Because of

q  u ,  , x,    q  0,  , x,    x  2 x 2   2     x  1   x

(S3.6)
(S3.7)
a necessary condition for having an admissible solution of (S3.5) is   2 . Hence,
equation (S3.5) does not have an admissible solution for which  is negative
    0,1  .
Remark 3.2 (Hopf bifurcation to periodic oscillations)
The right-hand side of (S3.5) represents a u -independent quadratic polynomial in z
 1 
 0, 1 with maximal value x 
 . The left-hand side of
 2 
(S3.5) is independent of z and is tending to 0 for u  0 . Thus, for sufficiently small
u , one has explicit formulae for the two zeros z  Z1/2  x, u,   of (S3.5): So given
2
which is positive on
x  0 ,   1 and a sufficiently small u  0 one first has solutions z  Z1/2  x, u,   and
u
then y  xz  0 ,   z 
and    1  z  x  0 for the equilibrium values (cf.
x
(S2.2)). The condition   2 from (S3.7) can be written as
u u !u
u
  
  1    0.
x x x u
So, recalling (S2.3a), we arrive at the necessary conditions
1
  1 and   1      1
2 /u
 
(S3.8)
asking for a sufficiently small u in case of 0     1 / 2 . In the original
parameters, (S3.8) is given by
 k3  k2  D  Dk1Tin    k  k
D
(S3.9)
3
2
 k2  D  k4  D   2k1DTin
asking in particular for D   and k2  D  k3 .
There are explicit - simple but tedious - formulae for sufficient conditions. The period
near the bifurcation point is 2 /  in first approximation. The amplitude of the
oscillation near the bifurcation point has an explicit approximation formula [1]. For a
numerical example one might take
1
x  1,  2, u  9  82.08 , z  0.1, y  0.1,  0.1  u,   0.9
(S3.10)
6
and vary  near 2 or u near 0.01 .

4.

Periodic oscillations in the 6D DIP-model
We investigate the 6D system (S1.2) in case of   0 , f  0 and Vd  0  0 . We turn
to the determination of positive equilibria E2 of the 6-dimensional system (S1.2) and
parameterize the equilibrium values by 1 , 3 , u ,  and 3 . With  :   3 , the
equations (S1.2a), (S1.2b) and (S1.2d) provide

u
u

v   23 , and    3 ,  2  1 3 , 1  1 3 .
(S4.1a)
1
1
1  3

The remaining equations (S1.2c), (S1.2e) and (S1.2f) then lead to
 


2   1 3  2 3 ,   2  1   2  1  2  ,
(S4.1b)
3
 

and finally to
3 
3 2


2 3
3 1  3   
(S4.1c)
reducing on   0 to
3  3c :
u
.
3 1  u 
(S4.2)
It is a tedious task to derive the conditions that guarantee the realizability of these
parameter values in terms of the original parameters in system (1). We note that the
determinant of C  ,  vanishes along (S4.1) for   R 30 . Moreover, taking 3  0 in
(S4.1),
the
positive
E2  R 30  R 30
equilibrium
tends
to
the
equilibrium
E1  0  R 30  R 30 . This limiting equilibrium E1  0 is critical in the sense that its
Jacobian possesses 0 as an eigenvalue (giving rise to the bifurcating E2 ).
The over-all stability in R 6 0 of E1 is determined by the eigenvalues of the Jacobian of
(S1.4a) which is in block-triangular form
 F  
å 
J     
(S4.3)

C  , 0  
 0
where we have suppressed the remaining parameters. The eigenvalues of the upper
left block F   are in the left half-plane of C by Lemma 3.1 since  has been
assumed to be negative. We note that the matrix C  ,0 in (S4.3) is an exponentially
stable
matrix
exactly
E1  0  1 ,  2 , 3   0
T
O   j  0, j  0 ,
for
[2].
C  ,0
 3   3c
First,
at
because
any
of
the
boundary
positive
equilibrium
invariance
of
cannot possess non-real eigenvalues. Secondly,
considering  3   3c   , there is exactly one eigenvalue of C  ,0
passing
transversally from the left to the right half-plane in C when  passes from negative
to positive values. This follows easily from the representation
det   I  C  , 0     3  d 2 2  d1  d 0
(S4.4a)
of the characteristic polynomial with the coefficients
d2  1    1  0, d1  1     u  313 , d0   u  33 u  1  .
(S4.4b)
The relations in (S4.4) show that C  ,0 possesses the eigenvalue 0  0 3  that
is vanishing at  3   3c with a positive  3c -derivative at  3c [0'  3c   0 because of
u
  u ] . The values given by (S4.1) with   0 thus provide a critical point
 1
in the space of parameters  3c ,  c , u c ,  c ,  c and states 1c ,  2c , 3c which induces a
transcritical steady-state bifurcation to the exponentially stable positive equilibrium
E2 .
 3c13 
Varying, for example, u away from criticality may entail a subsequent Hopf
bifurcation from the positive equilibrium E2 at some parameter value u Hopf . Based on
numerical simulations, this seems indeed to be the case.
Supplementary references
1. Kuznets︡︠ ov IUA (1998) Elements of applied bifurcation theory. New York: Springer. 591 p.
2. Berman A, Plemmons RJ (1994) Nonnegative matrices in the mathematical sciences. Philadelphia:
Society for Industrial and Applied Mathematics. 340 p.