The activation of a specific immune response takes place in the lymphoid
organs such as the spleen. We present here a simplified model of the
proliferation of specific immune cells in the form of a single delay equation.
We show that the system can undergo switches in stability as the delay is
increased, and we interpret these results in the context of sustaining an
effective immune response to a dendritic cell vaccine.Comment: 7 pages, 5 figures. Presented at the 8th AIMS Conference on Dynamical
  Systems, Differential Equations and Applications that took place in Dresden,
  Germany, May 25-28, 201

Gallegos, Angela

Radunskaya, Ami

English

arXiv

The activation of a specific immune response takes place in the lymphoid organs such as the spleen. We present here a simplified model of the proliferation of specific immune cells in the form of a single delay equation. We show that the system can undergo switches in stability as the delay is increased, and we interpret these results in the context of sustaining an effective immune response to a dendritic cell vaccine

Radunskaya, Ami E

Keck Graduate Institute

Claremont Colleges
Scholarship @ Claremont
Pomona Faculty Publications and Research Pomona Faculty Scholarship
1-1-2011
Do Longer Delays Matter? The Effect of
Prolonging Delay in CTL Activation
Angela Gallegos
Loyola Marymount University
Ami E. Radunskaya
Pomona College
This Conference Proceeding is brought to you for free and open access by the Pomona Faculty Scholarship at Scholarship @ Claremont. It has been
accepted for inclusion in Pomona Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more
information, please contact scholarship@cuc.claremont.edu.
Recommended Citation
Gallegos, A. Radunskaya, “Do Longer Delays Matter? The Effect of Prolonging Delay inCTL Activation”. Proceedings of the 8th AIMS
International Conference, 2011. http://arxiv.org/abs/1007.0225
DO LONGER DELAYS MATTER?
THE EFFECT OF PROLONGING DELAY IN CTL ACTIVATION
Angela Gallegos
Department of Mathematics, Occidental College
Los Angeles, CA USA
Ami Radunskaya
Department of Mathematics, Pomona College
Claremont, CA USA
Abstract. The activation of a specific immune response takes place in the
lymphoid organs such as the spleen. We present here a simplified model of the
proliferation of specific immune cells in the form of a single delay equation. We
show that the system can undergo switches in stability as the delay is increased,
and we interpret these results in the context of sustaining an effective immune
response to a dendritic cell vaccine.
1. Introduction
Immunotherapy seeks to arm the patient with an increased number of effective
immune cells that are specifically trained to seek out and destroy harmful cells. One
target for immunotherapy that has earned a place in the clinical spotlight recently
is melanoma, since specific peptides have been isolated from melanoma cells that
can be used in cancer vaccines. A subset of immune cells, called dendritic cells,
are able to stimulate the production of immune effector cells that are capable of
recognizing and killing specific tumor cells. Researchers have been able to extract
these cells from patients and culture them ex vivo to create a vaccine that can boost
a patient’s response against their own cancerous cells, [2]. The success of clinical
trials of dendritic cell (DC) vaccines has resulted in the recent FDA approval of the
first cancer vaccine for prostate cancer, [1]
Despite promising clinical responses in vaccine trials, it remains difficult to pre-
dict which patients will actually respond to these vaccines and why, [4, 5]. This
may be due to the complicated kinetics of the immune response to the presenta-
tion of antigen by the dendritic cells. In this paper, we analyze the effect of the
delay between the time the DCs come into contact with the T-cell population and
the initiation of the expansion, or proliferating phase. We focus on the dynamics
describing the active effector T-cell population in the spleen, using a phenomenolog-
ical description that captures the effect of the delay. Our goal here is to isolate the
role of this delay term in order to analyze its effect on the sustainability of T-cell
production.
The model we present is very simple, and is not intended to realistically capture
the entire cascade of immune events in the response to antigen presentation by
dendritic cells. Nor does this model describe the trafficking of immune cells between
1991 Mathematics Subject Classification. Primary: 34K60, 37N25; Secondary: 92B05.
Key words and phrases. Delay Differential Equations, Stability Analysis.
1
ar
X
iv
:1
00
7.
02
25
v1
  [
ma
th.
DS
]  
1 J
ul 
20
10
2 ANGELA GALLEGOS AND AMI RADUNSKAYA
compartments of the body, in particular between lymphoid organs, such as the
spleen, and the tumor site. These effects are crucial to the understanding of the
immune response and must be considered in the design of vaccine therapies. One
group of experiments that looks at the trafficking of DCs between compartments is
described in [6], and a mathematical model of DC trafficking and T-cell activation
is presented in [7]. Elsewhere we combine these ideas, along with previous work on
cell-lysis rates, [8], in a more realistic model that includes several types of immune
cells, and trafficking between compartments.
In section 2 we describe the model itself, and then we show stability switching
as a function of the delay in Section 3. We conclude with some discussion of the
model simulations and results in Section 4.
2. Description of the Mathematical Model
After encountering antigen, such as tumor cells or tumor peptides, dendritic cells
migrate to lymph organs such as the spleen. As they migrate, they mature and,
upon arrival in the spleen, they are able to activate the proliferation of T-cells that
have the ability to seek out and destroy (lyse) the specific target cells that provided
the antigen. This activation requires some contact time between the dendritic cells
and the naive T-cells; this connection time is called the “synaptic connection time.”
In our model, we represent this time as a delay, τ . Once proliferation is initiated,
new antigen-specific T-cells are created. A somewhat simplified description of the
fate of these new cells is as follows: The cells must go through a check-point during
which the new cells are tested for efficacy against the antigen. If they pass this
test, they either move into the blood stream to migrate to the target cells, they
become memory cells in order to protect against future challenges by the same
antigen, or they return to the proliferating compartment to produce more specific
T-cells. Proliferation continues only while certain cytokines are present. These
cytokines are produced by previously activated T-cells that have been in contact
with dendritic cells for the synaptic connection time, τ . When the dendritic cells
have done their job “educating” the T-cells, it is believed that they die, [2], and are
then “cleared” from the spleen compartment. When the DCs are no longer present,
cell proliferation ceases, and the activated T-cells either move into the blood stream
or become memory cells. This process is represented graphically in Figure 1.
In this model, the concentration of the dendritic cells at a given time, t, is
given by a(t), while the number of activated T-cells is given by x(t). The activa-
tion/proliferation rate is then given by
a(t− τ)f(x(t− τ))x(t),
where f is represents the “feedback” function. The “feedback” function depends
on the number of T-cells present in the proliferating compartment time τ earlier in
the presence of the DCs; therefore they are producing the necessary cytokines. The
current number of proliferating T-cells is represented by x(t). We assume that the
“feedback” function, f , is a positive function that increases to some maximum level,
r, and then decreases. The justification for this functional form is that activation by
DCs is maximized at a certain concentration of DCs in the spleen. The addition of
more DCs has the effect of keeping the activated cells in the area. If they are present
for too long, the effectiveness of the T-cells begins to diminish. This is sometimes
called the “decreasing potential hypothesis” in the literature, [9, 10]. One simple
function that captures these dynamics is given by:
PROLONGING DELAY IN CTL ACTIVATION 3
 
Dendritic Cells 
Proliferating T-cells 
x(t) 
Check 
Point 
Delay, ! 
“ON” switch via 
positive feedback 
Returning 
fraction FAIL 
TEST 
PASS 
TEST 
Blood 
Memory 
Figure 1. A graphical representation of the process underlying
the mathematical model.
f(xτ ) =
rxτ
1 + x4τ
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
where xτ = x(t− τ) represents the population of active T-cells from time τ ago.
We assume that T-cells leave the proliferating compartment through death, mov-
ing into the blood stream or the memory compartment at an average per-cell rate,
µ > 0. Putting these pieces together results in the following single delay equation:
(1)
dx
dt
= a(t− τ) f(xτ )x− µx
where for simplicity:
a(t) =
{
1 when DCs are present
0 otherwise
.
By rescaling the time variable, we can assume that r = 1, (the decay parameter,
µ, must also be appropriately scaled). Simulated solutions are shown in Figures
3 and 4. These solutions show that changing the value of τ , as well as the initial
history (the values of x(t) for −τ < t < 0), can have a strong effect on the long-term
behavior of the system. We verify this in the next section by performing a stability
analysis.
3. Stability Anlaysis
When dendritic cells are present in the spleen, a(t) = 1, so that the differential
equation becomes:
(2)
dx
dt
=
xτ
1 + x4τ
x− µx.
4 ANGELA GALLEGOS AND AMI RADUNSKAYA
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−0.2
0
0.2
0.4
0.6
0.8
Two non−zero equilibria at the intersection of y = x( 1 −      µ x3) and y = µ
Figure 2. The non-zero equilibria of the DDE given in 2 are the
solutions of x(1 − µx3) = µ. These are positive for µ-values in
(0, 3
3/4
4 ).
3.1. Equilibria and stability when τ = 0. Equation 2 potentially has three
equilibria. One is the zero fixed point: xE = 0, and the other two are solutions to
x
1 + x4
= µ⇔ x(1− µx3) = µ.
These two solutions are depicted graphically in Figure 2. Note that the function
x−µx4 has a maximum which occurs at xMax = (4µ)−1/3. When the maximum of
this function is greater than µ the two additional equilibria will exist. This occurs
for µ ∈
(
0,
33/4
4
)
. We denote the greater of these two equilibria by x∗, and the
smaller by x−, if they exist.
If x∗ is stable, then proliferation is sustained, and active effector cells continue
to be produced and sent into the blood stream and into the memory compartment.
However, if x∗ is unstable, then production may die off and no new effector cells
will be produced, even when DCs are still present in the spleen. Our goal in this
section is to show that changing the value of the delay, τ , can change the stability
of x∗.
Linearization around an equilibrium, xE gives:
dz
dt
= αzτ + βz,
PROLONGING DELAY IN CTL ACTIVATION 5
where z = x−xE , α = x ∂f∂xτ , evaluated at the equilibrium xE , and β = f(xE)−µ.
The characteristic equation is:
λ = αe−λτ + β.
Evaluating α and β at x = xE gives:
α =
µ2
xE
(
1− 3x4E
)
, when xE = x
∗, x−, and α = 0 when xE = 0
β = 0, xE = x
∗, x−, and β = −µ, xE = 0.
When we consider the characteristic equation in the case of the equilibrium xE = 0,
we always obtain λ = −µ < 0, independent of τ . Thus we see that the equilibrium
at xE = 0 is always stable.
On the other hand,linearizing around the upper equilibrium at x∗ in the case
when τ = 0 gives:
λ = α =
µ2
x
(
1− 3x4) < 0⇒ x > 4µ/3.
Using the fact that (x∗)4 = x
∗
µ − 1, we see that stability of the upper equilibrium is
ensured if
(3) x∗ >
1
31/4
.
We saw earlier that the maximum of x − µx4 occurs at xMax = (4µ)−1/3, and
we know that x∗ > xMax. Because we assume µ <
33/4
4
, we see that Equation 3 is
satisfied. Therefore, when τ = 0, x∗ is stable.
Since τ = 0 corresponds to the ODE case, and x− lies between the two stable
equilibria xE = 0 and xE = x
∗, we know that x− is unstable. Sample trajectories
are shown in Figure 3.
3.2. Stability switching as τ increases. Changes in stability can occur when
the real part of λ switches sign, i.e. when λ is purely imaginary. In this case, the
characteristic equation becomes
(4) λ = iω = αe−iωτ + β.
At the upper equilibrium, x∗, β = 0. Equating the real and imaginary parts in
Equation 4 in this case gives:
α cos(ωτ) = 0,(5)
−α sin(ωτ) = ω.(6)
The first equation implies that ωτ = (2n + 1)pi2 . Substituting this into the second
equation gives:
ω = ±α⇒ τ = ± (2n+ 1)pi
2α
= ± (2n+ 1)pix
∗
2µ2(1− 3x∗4) .
For a fixed value of µ, this gives a sequence of possible switching times, at which
the equilibrium at x∗ can change stability. For example, taking µ = 12 , we see that
x∗ = 1, and possible switching times occur when
τ = (2n+ 1)pi, n ∈ Z.
Figure 4 shows numerical solutions for increasing values of τ . We see that the
equilibrium point at x∗ = 1 loses stability when τ = pi, when a limit cycle appears.
6 ANGELA GALLEGOS AND AMI RADUNSKAYA
0 5 10 15 20 25 30 35 40 45 50
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
Ac
tiv
at
ed
 T
−c
ell
s
x*
x−
xE = 0
Figure 3. Sample trajectories in the case τ = 0 showing two stable
equilibria at x∗ and 0, and an unstable equilibrium at x−. In these
simulations µ = .5.
Increasing τ destroys this limit cycle and, for larger values of τ , solutions that once
tended to x∗ converge to the zero equilibrium.
4. Discussion and Conclusions
We have presented a simplified model of the interaction between dendritic cells
and T-cells at the initiation of a specific immune response. The model contains one
delay that represents the synaptic connection time between dendritic cells and T-
cells which is necessary for the development of cytokines that stimulate proliferation.
There is experimental evidence that prolonged contact between dendritic cells and
T-cells results in a decrease in viable T-cell production. The model presented here
shows a change in stability of the high T-cell equilibrium as the delay is increased.
This equilibrium goes from being stable when the delay, τ , is small to being unstable
for larger values of τ . These results support the experimental evidence for the
“decreasing potential hypothesis”. The model also suggest a role for dendritic cell
therapies in boosting the immune defense against certain diseases such as cancer.
However, the model indicates that levels of dendritic cells in the spleen should not
be maintained at high levels for too long in order to optimize the expansion of active
effector T-cells.
Acknowledgments
We would like to thank the organizers of the 8th AIMS Conference on Dynam-
ical Systems, Differential Equations and Applications that took place in Dresden,
Germany, May 25-28, 2010, at which this work was presented.
PROLONGING DELAY IN CTL ACTIVATION 7
0 10 20 30 40 50 60 70
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time
Ac
tiv
at
ed
 T
−c
ell
s
 
 
! = 0
! = 3.5
! = 6
Figure 4. As τ is increased, the equilibrium at x∗ = 1 loses stabil-
ity. First, periodic solutions appear for τ greater than pi. Further
increases in τ result in a complete loss of stability, and solutions
tend towards the zero equilibrium.
References
[1] Lauren Cox, FDA OKs Vaccine Therapy for Prostate Cancer http://abcnews.go.com/
Health ABC News Medical Unit, April 29 (2010).
[2] S.Pilon-Thomas, M. E. Verhaegen and J.J. Mule´ Dendritic Cell-Based Therapeutics for Breast
Cancer, Heart Disease 20 (2004), 65–71.
[3] N. Koike, S. Pilon-Thomas, and J.J. Mulee´, Nonmyeloblative Chemotherapy Followed by
T-cell Adoptive Transfer and Dendritic Cell-based Vaccination Results in Rejection of Es-
tablished Melanoma J. Immunotherapy 31 4, May (2008), 402–412.
[4] T. Boon, P.G. Coulie, B.J. Van den Eynde, and P. van der Bruggen, Human T Cell Response
Against Melanoma, The Annual Review of Immunology, 24, 175–208.
[5] U. Trefzer et al., Tumour-dendritic hybrid cell vaccination for the treatment of patients with
malignant melanoma: immunological effects and clinical results, Vaccine 23 (2005), 2367–
2373.
[6] O. Preynat-Seauve et al., Melanoma-infiltrating dendritic cells induce protective antitumor
responses mediated by T cells, Melanoma Research 17 (2007), 169–176.
[7] B. Ludewig et al., Determining control parameters for dendritic cell-cytotoxic T lymphocyte
interaction, European Journal of Immunology 34 (2004), 2407–2418.
[8] L.G. de Pillis, A. Radunskaya, and C. Wiseman, A validated mathematical model of cell-
mediated immune response to tumor invasion, Cancer Research 65, September (2005), 7950–
7958.
[9] N.S. Joshi and S.M. Kaech, Effector CD8 T Cell Development: A Balancing Act between
Memory Cell Potential and Terminal Differentiation, The Journal of Immunology 180 (2008),
1309–1315.
[10] V. Kalia et al., Differentiation of memory B and T cells, Current Opinion in Immunology 18
(2006) 256–264.
E-mail address: angela@oxy.edu
E-mail address: aer04747@pomona.edu


Do Longer Delays Matter? The Effect of Prolonging Delay in CTL Activation

Scholarship@Claremont

Claremont CollegesScholarship @ ClaremontPomona Faculty Publications and Research Pomona Faculty Scholarship1-1-2011Do Longer Delays Matter? The Effect ofProlonging Delay in CTL ActivationAngela GallegosLoyola Marymount UniversityAmi E. RadunskayaPomona CollegeThis Conference Proceeding is brought to you for free and open access by the Pomona Faculty Scholarship at Scholarship @ Claremont. It has beenaccepted for inclusion in Pomona Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For moreinformation, please contact scholarship@cuc.claremont.edu.Recommended CitationGallegos, A. Radunskaya, “Do Longer Delays Matter? The Effect of Prolonging Delay inCTL Activation”. Proceedings of the 8th AIMSInternational Conference, 2011. http://arxiv.org/abs/1007.0225DO LONGER DELAYS MATTER?THE EFFECT OF PROLONGING DELAY IN CTL ACTIVATIONAngela GallegosDepartment of Mathematics, Occidental CollegeLos Angeles, CA USAAmi RadunskayaDepartment of Mathematics, Pomona CollegeClaremont, CA USAAbstract. The activation of a specific immune response takes place in thelymphoid organs such as the spleen. We present here a simplified model of theproliferation of specific immune cells in the form of a single delay equation. Weshow that the system can undergo switches in stability as the delay is increased,and we interpret these results in the context of sustaining an effective immuneresponse to a dendritic cell vaccine.1. IntroductionImmunotherapy seeks to arm the patient with an increased number of effectiveimmune cells that are specifically trained to seek out and destroy harmful cells. Onetarget for immunotherapy that has earned a place in the clinical spotlight recentlyis melanoma, since specific peptides have been isolated from melanoma cells thatcan be used in cancer vaccines. A subset of immune cells, called dendritic cells,are able to stimulate the production of immune effector cells that are capable ofrecognizing and killing specific tumor cells. Researchers have been able to extractthese cells from patients and culture them ex vivo to create a vaccine that can boosta patient’s response against their own cancerous cells, [2]. The success of clinicaltrials of dendritic cell (DC) vaccines has resulted in the recent FDA approval of thefirst cancer vaccine for prostate cancer, [1]Despite promising clinical responses in vaccine trials, it remains difficult to pre-dict which patients will actually respond to these vaccines and why, [4, 5]. Thismay be due to the complicated kinetics of the immune response to the presenta-tion of antigen by the dendritic cells. In this paper, we analyze the effect of thedelay between the time the DCs come into contact with the T-cell population andthe initiation of the expansion, or proliferating phase. We focus on the dynamicsdescribing the active effector T-cell population in the spleen, using a phenomenolog-ical description that captures the effect of the delay. Our goal here is to isolate therole of this delay term in order to analyze its effect on the sustainability of T-cellproduction.The model we present is very simple, and is not intended to realistically capturethe entire cascade of immune events in the response to antigen presentation bydendritic cells. Nor does this model describe the trafficking of immune cells between1991 Mathematics Subject Classification. Primary: 34K60, 37N25; Secondary: 92B05.Key words and phrases. Delay Differential Equations, Stability Analysis.1arXiv:1007.0225v1  [math.DS]  1 Jul 20102 ANGELA GALLEGOS AND AMI RADUNSKAYAcompartments of the body, in particular between lymphoid organs, such as thespleen, and the tumor site. These effects are crucial to the understanding of theimmune response and must be considered in the design of vaccine therapies. Onegroup of experiments that looks at the trafficking of DCs between compartments isdescribed in [6], and a mathematical model of DC trafficking and T-cell activationis presented in [7]. Elsewhere we combine these ideas, along with previous work oncell-lysis rates, [8], in a more realistic model that includes several types of immunecells, and trafficking between compartments.In section 2 we describe the model itself, and then we show stability switchingas a function of the delay in Section 3. We conclude with some discussion of themodel simulations and results in Section 4.2. Description of the Mathematical ModelAfter encountering antigen, such as tumor cells or tumor peptides, dendritic cellsmigrate to lymph organs such as the spleen. As they migrate, they mature and,upon arrival in the spleen, they are able to activate the proliferation of T-cells thathave the ability to seek out and destroy (lyse) the specific target cells that providedthe antigen. This activation requires some contact time between the dendritic cellsand the naive T-cells; this connection time is called the “synaptic connection time.”In our model, we represent this time as a delay, τ . Once proliferation is initiated,new antigen-specific T-cells are created. A somewhat simplified description of thefate of these new cells is as follows: The cells must go through a check-point duringwhich the new cells are tested for efficacy against the antigen. If they pass thistest, they either move into the blood stream to migrate to the target cells, theybecome memory cells in order to protect against future challenges by the sameantigen, or they return to the proliferating compartment to produce more specificT-cells. Proliferation continues only while certain cytokines are present. Thesecytokines are produced by previously activated T-cells that have been in contactwith dendritic cells for the synaptic connection time, τ . When the dendritic cellshave done their job “educating” the T-cells, it is believed that they die, [2], and arethen “cleared” from the spleen compartment. When the DCs are no longer present,cell proliferation ceases, and the activated T-cells either move into the blood streamor become memory cells. This process is represented graphically in Figure 1.In this model, the concentration of the dendritic cells at a given time, t, isgiven by a(t), while the number of activated T-cells is given by x(t). The activa-tion/proliferation rate is then given bya(t− τ)f(x(t− τ))x(t),where f is represents the “feedback” function. The “feedback” function dependson the number of T-cells present in the proliferating compartment time τ earlier inthe presence of the DCs; therefore they are producing the necessary cytokines. Thecurrent number of proliferating T-cells is represented by x(t). We assume that the“feedback” function, f , is a positive function that increases to some maximum level,r, and then decreases. The justification for this functional form is that activation byDCs is maximized at a certain concentration of DCs in the spleen. The addition ofmore DCs has the effect of keeping the activated cells in the area. If they are presentfor too long, the effectiveness of the T-cells begins to diminish. This is sometimescalled the “decreasing potential hypothesis” in the literature, [9, 10]. One simplefunction that captures these dynamics is given by:PROLONGING DELAY IN CTL ACTIVATION 3 Dendritic Cells Proliferating T-cells x(t) Check Point Delay, ! “ON” switch via positive feedback Returning fraction FAIL TEST PASS TEST Blood Memory Figure 1. A graphical representation of the process underlyingthe mathematical model.f(xτ ) =rxτ1 + x4τ0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 200.10.20.30.40.50.60.7where xτ = x(t− τ) represents the population of active T-cells from time τ ago.We assume that T-cells leave the proliferating compartment through death, mov-ing into the blood stream or the memory compartment at an average per-cell rate,µ > 0. Putting these pieces together results in the following single delay equation:(1)dxdt= a(t− τ) f(xτ )x− µxwhere for simplicity:a(t) ={1 when DCs are present0 otherwise.By rescaling the time variable, we can assume that r = 1, (the decay parameter,µ, must also be appropriately scaled). Simulated solutions are shown in Figures3 and 4. These solutions show that changing the value of τ , as well as the initialhistory (the values of x(t) for −τ < t < 0), can have a strong effect on the long-termbehavior of the system. We verify this in the next section by performing a stabilityanalysis.3. Stability AnlaysisWhen dendritic cells are present in the spleen, a(t) = 1, so that the differentialequation becomes:(2)dxdt=xτ1 + x4τx− µx.4 ANGELA GALLEGOS AND AMI RADUNSKAYA0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.200.20.40.60.8Two non−zero equilibria at the intersection of y = x( 1 −      µ x3) and y = µFigure 2. The non-zero equilibria of the DDE given in 2 are thesolutions of x(1 − µx3) = µ. These are positive for µ-values in(0, 33/44 ).3.1. Equilibria and stability when τ = 0. Equation 2 potentially has threeequilibria. One is the zero fixed point: xE = 0, and the other two are solutions tox1 + x4= µ⇔ x(1− µx3) = µ.These two solutions are depicted graphically in Figure 2. Note that the functionx−µx4 has a maximum which occurs at xMax = (4µ)−1/3. When the maximum ofthis function is greater than µ the two additional equilibria will exist. This occursfor µ ∈(0,33/44). We denote the greater of these two equilibria by x∗, and thesmaller by x−, if they exist.If x∗ is stable, then proliferation is sustained, and active effector cells continueto be produced and sent into the blood stream and into the memory compartment.However, if x∗ is unstable, then production may die off and no new effector cellswill be produced, even when DCs are still present in the spleen. Our goal in thissection is to show that changing the value of the delay, τ , can change the stabilityof x∗.Linearization around an equilibrium, xE gives:dzdt= αzτ + βz,PROLONGING DELAY IN CTL ACTIVATION 5where z = x−xE , α = x ∂f∂xτ , evaluated at the equilibrium xE , and β = f(xE)−µ.The characteristic equation is:λ = αe−λτ + β.Evaluating α and β at x = xE gives:α =µ2xE(1− 3x4E), when xE = x∗, x−, and α = 0 when xE = 0β = 0, xE = x∗, x−, and β = −µ, xE = 0.When we consider the characteristic equation in the case of the equilibrium xE = 0,we always obtain λ = −µ < 0, independent of τ . Thus we see that the equilibriumat xE = 0 is always stable.On the other hand,linearizing around the upper equilibrium at x∗ in the casewhen τ = 0 gives:λ = α =µ2x(1− 3x4) < 0⇒ x > 4µ/3.Using the fact that (x∗)4 = x∗µ − 1, we see that stability of the upper equilibrium isensured if(3) x∗ >131/4.We saw earlier that the maximum of x − µx4 occurs at xMax = (4µ)−1/3, andwe know that x∗ > xMax. Because we assume µ <33/44, we see that Equation 3 issatisfied. Therefore, when τ = 0, x∗ is stable.Since τ = 0 corresponds to the ODE case, and x− lies between the two stableequilibria xE = 0 and xE = x∗, we know that x− is unstable. Sample trajectoriesare shown in Figure 3.3.2. Stability switching as τ increases. Changes in stability can occur whenthe real part of λ switches sign, i.e. when λ is purely imaginary. In this case, thecharacteristic equation becomes(4) λ = iω = αe−iωτ + β.At the upper equilibrium, x∗, β = 0. Equating the real and imaginary parts inEquation 4 in this case gives:α cos(ωτ) = 0,(5)−α sin(ωτ) = ω.(6)The first equation implies that ωτ = (2n + 1)pi2 . Substituting this into the secondequation gives:ω = ±α⇒ τ = ± (2n+ 1)pi2α= ± (2n+ 1)pix∗2µ2(1− 3x∗4) .For a fixed value of µ, this gives a sequence of possible switching times, at whichthe equilibrium at x∗ can change stability. For example, taking µ = 12 , we see thatx∗ = 1, and possible switching times occur whenτ = (2n+ 1)pi, n ∈ Z.Figure 4 shows numerical solutions for increasing values of τ . We see that theequilibrium point at x∗ = 1 loses stability when τ = pi, when a limit cycle appears.6 ANGELA GALLEGOS AND AMI RADUNSKAYA0 5 10 15 20 25 30 35 40 45 5000.10.20.30.40.50.60.70.80.91TimeActivated T−cellsx*x−xE = 0Figure 3. Sample trajectories in the case τ = 0 showing two stableequilibria at x∗ and 0, and an unstable equilibrium at x−. In thesesimulations µ = .5.Increasing τ destroys this limit cycle and, for larger values of τ , solutions that oncetended to x∗ converge to the zero equilibrium.4. Discussion and ConclusionsWe have presented a simplified model of the interaction between dendritic cellsand T-cells at the initiation of a specific immune response. The model contains onedelay that represents the synaptic connection time between dendritic cells and T-cells which is necessary for the development of cytokines that stimulate proliferation.There is experimental evidence that prolonged contact between dendritic cells andT-cells results in a decrease in viable T-cell production. The model presented hereshows a change in stability of the high T-cell equilibrium as the delay is increased.This equilibrium goes from being stable when the delay, τ , is small to being unstablefor larger values of τ . These results support the experimental evidence for the“decreasing potential hypothesis”. The model also suggest a role for dendritic celltherapies in boosting the immune defense against certain diseases such as cancer.However, the model indicates that levels of dendritic cells in the spleen should notbe maintained at high levels for too long in order to optimize the expansion of activeeffector T-cells.AcknowledgmentsWe would like to thank the organizers of the 8th AIMS Conference on Dynam-ical Systems, Differential Equations and Applications that took place in Dresden,Germany, May 25-28, 2010, at which this work was presented.PROLONGING DELAY IN CTL ACTIVATION 70 10 20 30 40 50 60 7000.20.40.60.811.21.4TimeActivated T−cells  ! = 0! = 3.5! = 6Figure 4. As τ is increased, the equilibrium at x∗ = 1 loses stabil-ity. First, periodic solutions appear for τ greater than pi. Furtherincreases in τ result in a complete loss of stability, and solutionstend towards the zero equilibrium.References[1] Lauren Cox, FDA OKs Vaccine Therapy for Prostate Cancer http://abcnews.go.com/Health ABC News Medical Unit, April 29 (2010).[2] S.Pilon-Thomas, M. E. Verhaegen and J.J. Mule´ Dendritic Cell-Based Therapeutics for BreastCancer, Heart Disease 20 (2004), 65–71.[3] N. Koike, S. Pilon-Thomas, and J.J. Mulee´, Nonmyeloblative Chemotherapy Followed byT-cell Adoptive Transfer and Dendritic Cell-based Vaccination Results in Rejection of Es-tablished Melanoma J. Immunotherapy 31 4, May (2008), 402–412.[4] T. Boon, P.G. Coulie, B.J. Van den Eynde, and P. van der Bruggen, Human T Cell ResponseAgainst Melanoma, The Annual Review of Immunology, 24, 175–208.[5] U. Trefzer et al., Tumour-dendritic hybrid cell vaccination for the treatment of patients withmalignant melanoma: immunological effects and clinical results, Vaccine 23 (2005), 2367–2373.[6] O. Preynat-Seauve et al., Melanoma-infiltrating dendritic cells induce protective antitumorresponses mediated by T cells, Melanoma Research 17 (2007), 169–176.[7] B. Ludewig et al., Determining control parameters for dendritic cell-cytotoxic T lymphocyteinteraction, European Journal of Immunology 34 (2004), 2407–2418.[8] L.G. de Pillis, A. Radunskaya, and C. Wiseman, A validated mathematical model of cell-mediated immune response to tumor invasion, Cancer Research 65, September (2005), 7950–7958.[9] N.S. Joshi and S.M. Kaech, Effector CD8 T Cell Development: A Balancing Act betweenMemory Cell Potential and Terminal Differentiation, The Journal of Immunology 180 (2008),1309–1315.[10] V. Kalia et al., Differentiation of memory B and T cells, Current Opinion in Immunology 18(2006) 256–264.E-mail address: angela@oxy.eduE-mail address: aer04747@pomona.edu

Do Longer Delays Matter? The Effect of Prolonging Delay in CTL Activation

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