Mathematical Appendix
Mathematical modeling of tumor therapy with oncolytic viruses:
Effects of parametric heterogeneity on cell dynamics
Georgy P. Karev1 , Artem S. Novozhilov1, Eugene V. Koonin1,*
1
National Center for Biotechnology Information, National Library of Medicine, National
Institutes of Health, Bethesda, MD 20894, USA.
*To whom correspondence should be addressed. Email koonin@ncbi.nlm.nih.gov
On Mathematical Theory of Inhomogeneous Community Models
Let us consider a model of a community consisting of m interacting populations in
the general form
dN j/dt = N jF j(N,a), j=1,…m,
where N j is the size of j-th population, N=(N1, …,N m). The growth rate F j can depend on
parameters and a=(a1,a2,…as) is the vector-parameter with the domain A. We suppose
here that every individual is characterized by its own value of vector-parameters a. Let
lj(t,a) be the density of the j-th population along the vector-parameter a at time t. In [1-3]
the following model of an inhomogeneous community was explored:
l j (t , a)
= F j(N,a) l j(t,a),
t
N j(t) =

(A.1)
l j(t,a)da .
A
We suppose here that the fitness of individuals in the j-th population, defined by the
formula Fj(N,a)=[dl j(t,a)/dt]/l j(t,a), depends on the parameter a and on total sizes of all
subpopulations but not on the parameter distribution inside the subpopulations.
This type of models can well describe, e. g., cooperative dynamics of populations
or resource-consumer model of inhomogeneous community, but are not applicable to
prey-predator or epidemiological problems, where the interactions between populations
should be “balanced” (e.g., the number of newly infected individuals should be equal to
decrease of the number of susceptible individuals). Nevertheless, the main ideas and
techniques used in [1] can be applied to “balanced” inhomogeneous community models
with slight modifications.
We do not develop here a general theory of such models; instead, we consider a
model of a biological community composed of two interacting populations depending on
parameters:
x(t , a )
 x(t , a )[ F1 ( X , Y , Eb (t ))  aG1 ( X , Y , Eb (t ))],
t
y (t , b)
 y (t , b)[ F2 ( X , Y , E a (t ))  bG2 ( X , Y , E a (t ))],
t
(A.2)
assuming that parameters a and b are distributed. Here x(t , a ) , y (t , b) are the population
X (t )   x(t , a)da , Y (t )   y(t , b)db ) are the total sizes of the cell
densities,
A
B
populations. Denote p1(t,a)=x(t,a)/X(t), p2(t,b)=y(t,b)/Y(t) the current probability density
functions of corresponding parameters in uninfected and infected populations
accordingly; then
Ea (t ) 
 ax(t, a)da X (t)   ap (t, a)da , E (t)   by(t, b)db Y (t)   bp (t, b)db
A
A
1
b
B
B
2
are the current mean values of the parameters a and b .
We suppose that the initial population sizes and initial distributions of the
parameters are given:
x(0, a)  X 0 p1 (0, a),
y(0, b)  Y0 p2 (0, b) .
The particular form of system (A.2) is motivated by the models explored in the paper,
which are special cases of (A.2).
Problem (A.2) can be reduced to the Cauchy problem for a certain system of ODE.
Let
Ma()=  exp(a)p1(0, a)da
A
Mb()=  exp(b)p2(0, b )db,
(A.3)
B
be the mgfs (moment generating functions) of the initial probability densities p1(0,a) and
p2(0,b).
Let us consider the system for auxiliary variables
dq10(t)/dt= q10(t)F1(X*,Y*, E1(t)), q11(0)=1;
(A.4)
dq11(t)/dt= G1(X*,Y*, E1(t)), q11(0)=0;
dq20(t)/dt= q20(t) F2(X*,Y*, E2(t)), q20(0)=1;
dq21(t)/dt= G2(X*,Y*, E2(t)), q21(0)=0.
The functions X*(t), Y*(t) and Ei(t) are defined by the formulas
E1 (t ) 
dM a ( )
1
,
1
M a (q 1 (t )) d   q11 (t )
X*(t)= X(0) q10(t) Ma(q11(t)),
E 2 (t ) 
dM b ( )
1
2
M b (q 1 (t )) d   q 21 (t )
(A.5)
Y*(t)= Y(0) q20(t) Mb(q21(t)).
Theorem 1. Let Cauchy problem (A.4)-(A.5) have a unique global solution {q ji(t)}
at t[0,T) where 0T. Then the functions
x(t,a) = x(0,a) q10(t) exp[a q11(t)],
y(t,b) = y(0,b) q20(t) exp[b q21(t)],
X(t) = X(0) q10(t) Ma(q11(t)),
Y(t) = Y(0) q20(t) Mb(q21(t))
satisfy system (A.2) at t[0,T).
(A.6)
Conversely, if x(t,a), y(t,b), and X(t)=

x(t,a)da, Y(t) =  y(t,b)db satisfy system
A
B
(A.2) at t[0,T), then Cauchy problem (A.4) has a global solution {q ji (t)} at t[0,T) and
these functions can be written in the form (A.6) at t[0,T).
Proof.
Let {qij(t)} be a solution of Cauchy problem (A.4)-(A.5) at t[0,T). According to
system (A.4),
x(t , a )
= x(0, a) exp[a q11(t)] {dq10(t)/dt + q10(t)a dq11(t)/dt} =
t
x(t,a){F1(X*,Y*, E1(t)) + a G1(X*,Y*, E1(t))] ;
y (t , b)
= y(0, b) exp[bq21(t)] {dq20(t)/dt + q20(t)bdq21(t)/dt} =
t
y(t,b){F2(X*,Y*, E2(t)) + bG2(X*,Y*, E2(t))].
Next,

x(t,a)da=  x(0,a) q10(t) exp[a q11(t)] da=X(0) q10(t) Ma(q11(t));
A

A
y(t,b)db=  y(0,b) q20(t)exp[b q21(t)] db =Y(0) q20(t)Mb(q21(t)).
B
B
Hence, the functions defined by (A.6), are connected by the relations
X(t)=

x(t,a)da, Y(t)=
A

y(t;b)db.
B
To complete the proof of the first part of the theorem, we have to determine the
current parameter distributions, p1 (t , a) and p2 (t , b) , and show that Ei(t) defined by
(A.5)
are
equal
to
the
current
mean
values
of
the
parameters,
i.e.,
Ea(t)=  ap1 (t , a)da =E1(t) and Eb(t)=  bp2 (t , b)db =E2(t). For this, we need the following
A
B
Proposition 1. Let the Cauchy problem (A.4)-(A.5) have a unique solution {qji(t)} at
t[0,T). Then, for all t[0,T),
1) the current parameter distributions, p1 (t , a) and p2 (t , b) are equal to
p1 (t , a) = p1 (0, a) exp[a q11(t)] / Ma(q11(t)),
(A.7)
p2 (t , b) = p2 (0, b) exp[b q21(t)] / Mb(q21(t));
2) the moment generation functions of p1 (t , a) and p2 (t , b) are equal to
Ma(t; )= Ma(+q11(t)) / Ma(q11(t)),
(A.8)
Mb(t; )= Mb(+q21(t)) / Mb(q21(t))
where Ma(), Mb() are the mgf of initial distributions (A.3).
Proof.
By definition, p1 (t , a) = x(t,a)/X(t), p2 (t , b) = y(t, b)/Y(t). According to (A.6),
p1 (t , a) = x(0,a) q10(t) exp[a q11(t)] / X(t).
Further, X(t) = X(0) q10(t) Ma(q11(t)) by (A.5), so
p1 (t , a) = p1 (0, a) exp[a q11(t)] / Ma(q11(t)).
Similarly, p2 (t , b) = p2 (0, b) exp[b q21(t)] / Mb(q21(t)). Next,
Ma(t; )=

exp(a) p1 (t , a) da= {
A

exp[(+q11(t))a] p1 (0, a) da}/ Ma(q11(t)) =
A
Ma(+q11(t)) / Ma(q11(t)).
Similarly, Mb(t; )= Mb(+q21(t)) / Mb(q21(t)), as desired.
Now we are able to complete the proof of part 1 of Theorem:
Ea(t)=
 ap (t, a)da = Ma(t; ) /│=0=[Ma(+q11(t)) /│=0] / Ma(q11(t)),
A
1
and the last expression coincides with the formula (A.5) for E1(t).
Similarly, Eb(t)=  Mb(t; ) /│=0=[Mb(+q21(t)) /│=0] / Mb(q21(t)).
Let us prove now part 2 of Theorem; let x(t, a), y(t,b), and X(t)=


x(t,a)da, Y(t) =
y(t,b)db satisfy system (A.2) at t[0,T), so that
x(t , a )
 x(t , a )[ F1 ( X , Y , Eb (t ))  aG1 ( X , Y , Eb (t ))],
t
y (t , b)
 y (t , b)[ F2 ( X , Y , E a (t ))  bG2 ( X , Y , E a (t ))],
t
where
Ea (t ) 
 ax(t, a)da X (t) , E (t)   by(t, b)db Y (t) .
A
b
B
Let us temporarily define the functions
t
q 0(t) = exp[  F1(X(s),Y(s), Eb(s)) ds],
1
t
q11(t)
0
0
t
q 0(t) = exp[  F2(X(s),Y(s), Ea(s)) ds],
2
0
=  G1(X(s),Y(s), Eb(s)) ds,
t
q21(t)
=  G2(X(s),Y(s), Ea(s)) ds.
0
Then
dq1 (t ) 1
dq11 (t )
x(t , a) 1
= 0
+
a
,
dt
dt q01 (t )
t x(t , a)
dq 2 (t ) 1
dq12 (t )
y (t , b) 1
= 0
+
b
,
dt q02 (t )
dt
t y (t , b)
hence
ln x(t,a) = lnq10(t) + aq11(t) +C1
ln y(t,b) = ln q20(t) + bq21(t) + C2
where C j do not depend on t.
Taking C1 =ln x(0,a), C2 =ln y(0,b) we obtain that
x(t,a) = x(0,a) q10(t) exp[aq11(t)],
y(t,b) = y(0,b) q20(t) exp[bq21(t)].
Hence
X(t)
Y(t)


x(t,a)da=  x(0,a) q10(t) exp[aq11(t)] da=X(0) q10(t) Ma(q11(t)),
y(t,b)db=

y(0,b) q20(t) exp[bq21(t)]db =Y(0) q20(t) Mb(p21(t)) at t[0,T).
From the definition, {qij(t)} is a solution of Cauchy problem (A.4)-(A.5) for t[0,T).
The proof is completed.
Corollary 1. System (A.2) is equivalent to the system (A.7) for the current parameter
distributions and the following non-autonomous system of ODEs for the total population
sizes:
dX
 X [ F1 ( X , Y , E b (t ))  E a (t )G1 ( X , Y , Eb (t ))],
dt
dY
 Y [ F2 ( X , Y , E a (t ))  E b (t )G 2 ( X , Y , E a (t ))],
dt
X (0)  X 0 , Y (0)  Y0
where the mean parameter values are
(A.9)
E a (t ) 
dM a ( )
1
,
M a (q1 (t )) d   q ( t )
1
Eb (t ) 
dM b ( )
1
M b (q 2 (t )) d   q
(A.10)
2 (t )
and Ma(λ) and Mb(λ) are the mgf of the given initial pdfs p1 (0, a) and p2 (0, b) . The
auxiliary variables q1(t), q2(t) can be found from the system
dq1
 G1 ( X , Y , Ea (t )),
dt
dq2
 G2 ( X , Y , Eb (t )),
dt
q1 (0)  0, q2 (0)  0 .
(A.11)
Proof.
It is enough to prove that system (A.7), (A.9)-(A.11) is equivalent to (A.4)-(A.5).
Let Cauchy problem (A.4)-(A.5) have a unique global solution {q ji(t)} at t[0,T).
Then, according to Theorem 1, functions (A.6) solve (A.2); it is easy to check that X(t) =
X(0) q10(t) Ma(q11(t)), Y(t) = Y(0) q20(t) Mb(q21(t)) solve (A.9) and the definitions of the
auxiliary variables and mean values of the parameters coincide with (A.10)-(A.11). Next,
if the current parameter distributions p1 (t , a) , p2 (t , b) are defined by (A.7), then
x(t, a)  X (t ) p1 (t , a), y(t , b)  Y (t ) p2 (t , b) solve (A.2).
Conversely, let X(t), Y(t), q1(t), q2(t) solve (A.9)-(A.11) at t[0,T). Let q11(t) = q1(t),
q21(t) = q2(t) and
t
t
0
0
q10(t)=exp[  F1(X(s),Y(s),Eb(s))ds], q20(t)=exp[  F2(X(s),Y(s),Ea(s))ds].
Then {qji(t)} is a solution of Cauchy problem (A.4)-(A.5); according to Proposition 1,
the mgfs of the current distributions are given by (A.8) and mean parameter values (A.10)
coincide with that defined by formulas (A.5). Q.E.D.
Corollary 2. The current mean parameter values that are determined by formulas (A.5)
satisfy the equations
dEa (t )
  a2 (t )G1 ( X , Y , Eb (t )),
dt
dEb (t )
  b2 (t )G2` ( X , Y , Ea (t )) ,
dt
(A.12)
where  a2 (t )  0 and  b2 (t )  0 are the current variances of the parameter distributions.
Proof.
Differentiating
(A.5),
using
(A.8),
(A.11),
and
the
fact
that
d 2 M (t ,  )
 (t ) 
 E 2 (t ) , we obtain the necessary equations. Q.E.D.
2
d
 0
2
Example 1. Let parameter a have the Γ-distribution at the initial instant with
coefficients s, η, k:
P(0,a)= sk (a - )k-1exp[-(a - )s]/(k),
with the mean =  + k /s, the variance  2 = k /s2 and the mgf Ma() = exp() /(1-/s) k.
Then
Ma(t; )= Ma(+q1(t)) / Ma(q1(t))= exp() /(1-/(s- q1(t)) k ,
i.e., the parameter a has again the Γ-distribution with coefficients s-q1(t), η, k at any
moment t<T such that the solution q1(t)< s exists for t  [0, T ) . Next,
M (+q1(t)) /│=0 / M(q1(t))=k/(s-q1(t))+.
Hence, the current mean of the parameter is
E a (t ) 
k

s  q1 (t )
where the auxiliary variable q1(t) is defined by (A.11).
(A.13)
As a particular inhomogeneous community model let us consider system (7):
x(t ,  )
x(t ,  )Y (t )
 x(t ,  )1  ( X (t )  Y (t ))  
,
t
X (t )  Y (t )
E  (t ) X (t ) y (t ,  )
y (t ,  )
 y (t ,  )1  ( X (t )  Y (t ))  
 y (t ,  ).
t
X (t )  Y (t )
(A.14)
Suppose that, at the initial moment, both parameters β and δ have the Γ-distribution
with coefficients s1, η1, k1 and s2, η2, k2, respectively. Then the current mean parameter
values are given by (A.13) and, finally, system (A.14) is reduced to the following system
of ODEs:
dq1
Y

,
dt
X Y
dq 2
 1,
dt
k1
dX
XY
 X (1  ( X  Y ))  (
 1 )
,
dt
s1  q1 (t )
X Y
k1
k2
dY
XY
 Y (1  ( X  Y ))  (
 1 )
(
  2 )Y ,
dt
s1  q1 (t )
X Y
s 2  q 2 (t )
X (0)  X 0 , Y (0)  Y0 , q1 (0)  0, q 2 (0)  0.
References:
1.
Karev GP: Heterogeneity effects in population dynamics. Doklady Mathematics
2000, 62(1):141-144.
2.
Karev GP: Dynamics of inhomogeneous populations and global demography
models. Journal of Biological Systems 2005, 13(1):83-104.
3.
Novozhilov AS: Analysis of a generalized population predator-prey model with a
parameter distributed normally over the individuals in the predator population.
Journal of Computer and Systems Sciences International 2004, 43(3):378-382.