# Basic introduction to immunology and review of CTMC Magic by pas31212

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```									Theoretical immunology

Basic introduction to immunology and review of CTMC
Magic 042 – Lecture 2

ıs
Carmen Molina-Par´

Department of Applied Mathematics, School of Mathematics, University of Leeds

7th of October 2009
Theoretical immunology

Outline of the talk
Infections: viruses
Immunology
General Theory of Continuous Time Markov Chains (CTMC)
Transition probabilities
The Kolmogorov equations
Birth and death processes
Birth and death processes
Kolmogorov equations
Stationary probability distribution
Continuous time birth and death process with absorbing states
Probability of population extinction
Quasi-stationary probability distribution
Uniqueness and existence for the approximated quasi-stationary probability
distribution
Time to extinction
Theoretical immunology
Infections: viruses

Virus replication inside a cell
Theoretical immunology
Infections: viruses

There is a wide range of viral infections
Theoretical immunology
Infections: viruses

How viral particles can attach and gain entry into the cell
Theoretical immunology
Infections: viruses

Diﬀerent ways for viral uncoating
Theoretical immunology
Infections: viruses

Example: Epstein-Barr virus I

The Epstein-Barr virus (EBV) provides an example of human cancer linked
to a viral infection.
Infection by this DNA virus is so common that nearly 90% of adults in the
USA over the age of 40 have detectable levels of anti-EBV antibodies in
their blood.
EBV prefers to invade B cells of the adaptive immune system, especially
long-lived memory B cells.
Most people infected as children have few symptoms and are unaware that
they have been infected, but teenagers and young adults infected for the
ﬁrst time often develop mononucleosis (also called glandular fever), a
severe ﬂu-like disease that can lead to high fever, painful swelling of lymph
nodes, and fatigue that can persist for several months.
Theoretical immunology
Infections: viruses

Example: Epstein-Barr virus II

After symptoms subside, EBV can remain dormant in the B cells for life,
with its genome maintained in the B cell nucleus.
Some of the gene products encoded by the EBV genome inhibit apoptosis
and thereby presumably help to prevent the virus from being cleared from
the body.
Thus, when an infected B cell acquires cancer-promoting mutations, the
usual mechanism for eliminating precancerous cells by apoptosis is
inhibited, and a form of B cell cancer called Burkitt’s lymphoma may
develop.
Theoretical immunology
Immunology

Immunology (vertebrates) I

Innate immune system: ﬁrst line of defense.
Adaptive immune system: sophisticated defenses against infection.
Theoretical immunology
Immunology

Immunology (vertebrates) II

In vertebrates, the innate responses call the adaptive immune responses
into play, and both work together to eliminate the pathogens.
Innate immune responses are general defense reactions. Adaptive
responses are highly speciﬁc to the particular pathogen that induced them,
and they provide long-lasting protection.
molecules they produce.
These responses are destructive: they should be directed only against
foreign molecules and not against molecules of the host itself.
The adaptive immune system uses multiple mechanisms to avoid
damaging responses against self molecules.
Occasionally, however, these mechanisms fail, and the system turns
against the host, causing autoimmune diseases, which can be fatal.
Theoretical immunology
Immunology

Immunology (vertebrates) III

Self versus non-self or harmless versus dangerous
Many harmless foreign molecules enter the body, and it would be pointless
and potentially dangerous to mount adaptive immune responses against
them.
Allergic conditions such as hayfever and allergic asthma are examples of
adaptive immune responses against apparently harmless foreign molecules.
An individual normally avoids such inappropriate responses because the
innate immune system only calls adaptive immune responses into play
when it recognises conserved patterns of molecules that are speciﬁcally
The innate immune system can even distinguish between diﬀerent classes
of pathogens and recruit the most eﬀective form of adaptive immune
response to eliminate them.
Theoretical immunology
Immunology

Immunology (vertebrates) IV

Antigens and immunisation
Any substance capable of eliciting an adaptive immune response is referred
to as an antigen (antibody generator).
Most of what we know about such responses has come from studies in
which an experimenter tricks the adaptive immune system of a laboratory
animal (usually a mouse) into responding to a harmless foreign molecule,
such as a foreign protein.
The trick involves injecting the harmless molecule together with
immunostimulants (usually microbial in origin) called adjuvants, which
activate the innate immune system. This trick is called immunisation.
Theoretical immunology
Immunology

Immune cells

Adaptive immune responses are carried out by white blood cells called
lymphocytes.
There are two broad classes of such responses:
antibody responses and
T cell-mediated immune responses.
Theoretical immunology
Immunology

B cells
B cells are activated to secrete antibodies, which are proteins called
immunoglobulins.
The antibodies circulate in the bloodstream and permeate the other body
ﬂuids, where they bind speciﬁcally to the foreign antigen that stimulated
their production.
Binding of antibody inactivates viruses and microbial toxins (such as
tetanus toxin or diphtheria toxin) by blocking their ability to bind to
receptors on host cells.
Antibody binding also marks invading pathogens for destruction, mainly by
making it easier for phagocytic cells of the innate immune system to ingest
them.
Theoretical immunology
Immunology

T cells
In T cell-mediated immune responses, activated T cells react directly
against a foreign antigen that is presented to them on the surface of a
host cell, which is therefore referred to as an antigen-presenting cell.
T cells can detect microbes hiding inside host cells and either kill the
infected cells or help the infected cells or other cells to eliminate the
microbes.
The T cell, for example, might kill a virus-infected host cell that has viral
antigens on its surface, thereby eliminating the infected cell before the
virus has had a chance to replicate.
In other cases, the T cell produces signal molecules that either activate
macrophages to destroy the microbes that they have phagocytosed or help
activate B cells to make antibodies against the microbes.
Theoretical immunology
Immunology

Macrophages and neutrophils I

The rapid innate immune responses to an infection depend largely on
pattern recognition receptors made by cells of the innate immune system.
These receptors recognise microbe-associated molecules that are not
present in the host organism, called microbe-associated immunostimulants.
Some of the pattern recognition receptors are present on the surface of
professional phagocytic cells (phagocytes) such as macrophages and
neutrophils, where they mediate the uptake of pathogens, which are then
delivered to lysosomes for destruction.
Others are secreted and bind to the surface of pathogens, marking them
for destruction by either phagocytes or a system of blood proteins
collectively called the complement system.
Still others, including the Toll-like receptors (TLRs), activate intracellular
signaling pathways that lead to the secretion of extracellular signal
molecules that promote inﬂammation and help activate adaptive immune
responses.
Theoretical immunology
Immunology

Macrophages and neutrophils II
Theoretical immunology
Immunology

Macrophages and neutrophils III
Theoretical immunology
Immunology

Dendritic cells I

The cells of the vertebrate innate immune system that respond to those
patterns and activate adaptive immune responses most eﬃciently are
dendritic cells.
Present in most tissues, dendritic cells express high levels of TLRs and
other pattern recognition receptors, and they function by presenting
microbial antigens to T cells in peripheral lymphoid organs.
They recognise and phagocytose invading microbes or their products or
fragments of infected cells at a site of infection and then migrate with
their prey to a nearby lymph node.
In other cases, they pick up microbes or their products directly in a
peripheral lym-phoid organ such as the spleen.
In either case, the microbial patterns activate the dendritic cells so that
they, in turn, can directly activate the T cells in peripheral lymphoid organs
to respond to the microbial antigens displayed on the dendritic cell surface.
Theoretical immunology
Immunology

Dendritic cells II

Once activated, some of the T cells then migrate to the site of infection,
where they help destroy the microbes.
Other activated T cells remain in the lymphoid organ, where they help
keep the dendritic cells active, help activate other T cells, and help
activate B cells to make antibodies against the microbial antigens.
Theoretical immunology
Immunology

Dendritic cells III
Theoretical immunology
General Theory of Continuous Time Markov Chains (CTMC)

Continuous time Markov chains (CTMC)

Let {X(t)}, where t ∈ [0, ∞), be a collection of discrete random variables with
values in a ﬁnite {0, 1, 2, . . . , N} or inﬁnite {0, 1, 2, . . .} state space.
Deﬁnition
The stochastic process {X(t)}, where t ∈ [0, ∞), is called a continuous time
Markov chain if it satisﬁes the following condition: for any sequence of real
numbers satisfying 0 ≤ t0 < t1 < . . . < tn < tn+1

P (X(tn+1 ) = in+1 | X(t0 ) = i0 , . . . , X(tn ) = in ) = P (X(tn+1 ) = in+1 | X(tn ) = in ) . (1)

This is the Markov Property: the transition to state in+1 at time tn+1 depends
only on the value of the state at time tn and does not depend on the past.
Theoretical immunology
General Theory of Continuous Time Markov Chains (CTMC)
Transition probabilities

Transition probabilities I

Deﬁnition
For the random variables {X(s)} and {X(t)}, where s < t, we deﬁne the
transition probabilities as:

pji (t, s) = P{X(t) = j | X(s) = i} for i, j = 0, 1, 2, . . . .
Theoretical immunology
General Theory of Continuous Time Markov Chains (CTMC)
Transition probabilities

Transition probabilities II

Deﬁnition
We will say that the transition probabilities are stationary or homogeneous if
they do not depend explicitly on s or t, but depend only on the length of the
time interval, t − s.

pji (t − s) = P{X(t) = j | X(s) = i} = P{X(t − s) = j | X(0) = i} .

The transition probabilities have the property
+∞
X
pji (t) = 1 for t ≥ 0 , ∀i = 0, 1, 2, . . . .
j=0

The matrix of transition probabilities, or transition matrix, P, is given by

P = (pji (t)) ,              (2)

which is a stochastic matrix for all t ≥ 0.
Theoretical immunology
General Theory of Continuous Time Markov Chains (CTMC)
The Kolmogorov equations

The Kolmogorov equations

Theorem
The transition probabilities

pji (t + ∆t) = P{ X(t + ∆t) = j | X(0) = i}                  (3)

satisfy the forward and backward Kolmogorov equations.
+∞
X
pji (t + ∆t) =             pjk (∆t) pki (t) forward Kolmogorov equation ,
k=0
+∞
(4)
X
=         pjk (t) pki (∆t) backward Kolmogorov equation .
k=0
Theoretical immunology
General Theory of Continuous Time Markov Chains (CTMC)
The Kolmogorov equations

Proof of the forward Kolmogorov equations

+∞
X
pji (t + ∆t) =                 P{X(t + ∆t) = j, X(t) = k | X(0) = i}
k=0

here we make use of the conditional probability property to get
+∞
X
=         P{X(t + ∆t) = j | X(t) = k and X(0) = i}P{ X(t) = k | X(0) = i}
k=0

now we use the Markov property
+∞
X
=         P{X(t + ∆t) = j | X(t) = k}P{ X(t) = k | X(0) = i}
k=0

we now use the general deﬁnition for the transition probabilities to get
+∞
X
=         pjk (∆t) pki (t) .
k=0
+∞
X
We obtain the forward Kolmogorov equations:pji (t + ∆t) =                    pjk (∆t) pki (t) .
k=0
Theoretical immunology
General Theory of Continuous Time Markov Chains (CTMC)
The Kolmogorov equations

Backward Kolmogorov equations

Example
Homework: Derive the backward Kolmogorov equations:
+∞
X
pji (t + ∆t) =             pjk (t) pki (∆t) backward Kolmogorov equation .   (5)
k=0
Theoretical immunology
Birth and death processes
Birth and death processes

Birth and death processes
A continuous time birth and death process is a CTMC, X(t), with either a
ﬁnite {0, 1, 2, . . . , N} or inﬁnite {0, 1, 2, . . . , } state space. We introduce the
notation:
∆X(t) = change in the state of the stochastic process from t to t + ∆t
(6)
= X(t + ∆t) − X(t) .
The inﬁnitesimal transition probabilities of a birth and death process are:
pi+ji (∆t) = P{∆X(t) = j | X(t) = i}
8
>λi ∆t + o(∆t)
>                           j =1,
>
>
>
<µi ∆t + o(∆t)              j = −1 ,              (7)
>
=
>1 − (λi + µi )∆t + o(∆t)
>                           j =0
>
>
>
>
:o(∆t)                      j = −1, 0, 1 ,
where we set λi = birth rate and µi = death rate, when the population is of size
i. λi , µi ≥ 0 and o(∆t) is the Landau order symbol:
o(∆t)
lim         =0.                              (8)
∆t→0    ∆t
Theoretical immunology
Birth and death processes
Kolmogorov equations

Forward Kolmogorov equations: birth and death process I
The forward Kolmogorov diﬀerential equations for pji (t) can be derived
directly from Equation (7).
Assuming ∆t is suﬃciently small, we consider the transition probability
pji (t + ∆t). This transition probability can be expressed in terms of the
transition probabilities at time t as follows:
pi+j i (t + ∆t) = pj−1 i (t)[λj−1 ∆t + o(∆t)] + pj+1 i (t)[µj+1 ∆t + o(∆t)]
+∞
X
+ pji (t)[1 − (λj + µj )∆t + o(∆t)] +                pj+k i (t)o(∆t)
k=−1,0,1

= pj−1 i (t)λj−1 ∆t + pj+1 i (t)µj+1 ∆t
+ pji (t)[1 − (λj + µj )∆t] + o(∆t) ,
(9)
which holds for all i and j in the state space with the exception of the
endpoints, j = 0 and j = N.
If j = 0 and assuming that µ0 = 0, then
p0i (t + ∆t) = p1i (t)µ1 ∆t + p0i (t)(1 − λ0 ∆t) + o(∆t) .           (10)
Theoretical immunology
Birth and death processes
Kolmogorov equations

Forward Kolmogorov equations: birth and death process II

In the case of a ﬁnite state space, where j = N is the maximum
population size and assuming that λN = 0, we have

pNi (t + ∆t) = pN−1i (t)λN−1 ∆t + pNi (t)(1 − µN ∆t) + o(∆t) ,        (11)

and pKN (t) = 0 for K > N.
We can now derive the forward Kolmogorov diﬀerential equations using
the transition probabilities of the previous slides.
We obtain from equation (9)
pji (t + ∆t) − pji (t)                                                     o(∆t)
= pj−1 i (t)λj−1 +pj+1 i (t)µj+1 −pji (t)(λi +µj )+       .
∆t                                                                 ∆t
Theoretical immunology
Birth and death processes
Kolmogorov equations

Forward Kolmogorov equations: birth and death process III

o(∆t)
We then take the limit as ∆t → 0, where lim∆t→0                ∆t
=0

dpji (t)          pji (t + ∆t) − pji (t)
= lim
dt       ∆t→0            ∆t                                      (12)
= pj−1 i (t)λj−1 + pj+1 i (t)µj+1 − pji (t)(λj + µj ) ,

for i ≥ 0 and 1 ≤ j ≤ N − 1.
Using equation (10) we obtain:
dp0i (t)
= p1i (t)µ1 − λ0 p0i (t) ,   for i ≥ 0 .           (13)
dt
Using equation (11) we obtain:

dpNi (t)
= pN−1i (t)λN−1 − µN pNi (t) ,      for i ≥ 0 .        (14)
dt
Theoretical immunology
Birth and death processes
Kolmogorov equations

Generator matrix Q I
We deﬁne the generator matrix Q as follows. Assume the transition
probabilities pij (t) are continuous and diﬀerentiable for t ≥ 0 and at t = 0
they satisfy:
pji (0) = 0 , j = i and pji (0) = 1 .
We deﬁne for j = i
pji (∆t) − pji (0)        pji (∆t)
qji = lim                        = lim           .
∆t→0+          ∆t           ∆t→0+    ∆t
Note that all qji ≥ 0 since pji (∆t) ≥ 0.
We deﬁne
pii (∆t) − pii (0)        pii (∆t) − 1
qii = lim                        = lim               .
∆t→0+          ∆t           ∆t→0+      ∆t
We have
+∞
X                                         +∞
X                      +∞
X
pji (∆t) = 1 ⇒ 1 − pii (∆t) =             pji (∆t) =             [qji ∆t + o(∆t)] .
j=0                                     j=0,j=i                j=0,j=i
Theoretical immunology
Birth and death processes
Kolmogorov equations

Generator matrix Q II

This implies
P+∞                                 +∞
−    j=0,j=i   [qji ∆t + o(∆t)]         X
qii = lim                                     =−             qji .
∆t→0+                  ∆t
j=0,j=i

We have
qii ≤ 0 .
It then follows that

pji (∆t) = δji + qji ∆t + o(∆t) .

The generator matrix Q has entries given by qji and qii as deﬁned above.
Theoretical immunology
Birth and death processes
Stationary probability distribution

Stationary probability distribution I

Deﬁnition
A positive stationary probability distribution can be deﬁned for a general
continuous time birth and death chain:

π = (π0 , π1 , π2 , . . . , )T ,

where the transition probability matrix, P. and generator matrix, Q, satisfy:
+∞
X
Qπ = 0 ,        P(t)π = π ,                πi = 1 ,   and πi ≥ 0 ,
i=0

for t ≥ 0 and i = 0, 1, 2, . . .
Theoretical immunology
Birth and death processes
Stationary probability distribution

Stationary probability distribution II

Theorem
Suppose the continuous time Markov chain {X(t)}, t ≥ 0, is a general birth
and death chain satisfying (7). If the state space is inﬁnite {0, 1, 2, . . .}, a
unique positive stationary probability distribution π exists iﬀ µi > 0 and
λi−1 > 0 for i = 1, 2, . . . , and
+∞
X λ0 λ1 . . . λi−1
< +∞ .                    (15)
i=1
µ1 µ2 . . . µ i

The stationary probability distribution is given by
λ0 λ1 . . . λi−1
πi =                    π0 ,     for i = 1, 2, . . .   (16)
µ1 µ2 . . . µ i
and
1
π0 =         P+∞ λ0 λ1 ...λi−1 .                (17)
1+      i=1     µ1 µ2 ...µi
Theoretical immunology
Birth and death processes
Stationary probability distribution

Stationary probability distribution III

Theorem
If the state space is ﬁnite {0, 1, 2, . . . , N}, then a unique positive stationary
probability distribution π exists if and only if

µi > 0 ,   and   λi−1 > 0 ,

for i = 1, 2, . . . , N. The stationary probability distribution is given by
Equations (16) and (17), where the index i and the summation on i extend
from 1 to N.
Theoretical immunology
Birth and death processes
Stationary probability distribution

Stationary probability distribution: proof I
The stationary probability distribution π satisﬁes Qπ = 0, which can be
written as

0     =     λi−1 πi−1 − (λi + µi )πi + µi+1 πi+1 ,   i = 1, 2, . . .   (18)
0     =     −λ0 π0 + µ1 π1 ,                                           (19)
P+∞
and i=0 πi = 1.
Solving these equations, we ﬁnd for π1
λ0
π1 =      π0 .
µ1
We can also show that for π2
λ0 λ1
π2 =         π0 .
µ1 µ2
We continue by induction on πi . Assume πj has been deﬁned for
j = 1, 2, . . . , i.
λ0 λ1 . . . λi−1
πi =                    π0 .
µ1 µ2 . . . µ i
Theoretical immunology
Birth and death processes
Stationary probability distribution

Stationary probability distribution: proof II
We now solve for πi+1 :

µi+1 πi+1 = (λi + µi )πi − λi−1 πi−1
h λ λ . . . λ (λ + µ )        λ0 λ1 . . . λi−1 i
0 1          i−1   i   i
=                             −                    π0
µ1 µ2 . . . µ i      µ1 µ2 . . . µi−1
λ0 λ1 . . . λi−1 λi
πi+1 =                      π0 .
µ1 µ2 . . . µi+1

We now apply the condition +∞ πi = 1
P
i=0

+∞
X                     +∞
X λ0 λ1 . . . λi−1
πi = 1 − π0 =                       π0
i=1                   i=1
µ1 µ2 . . . µ i

“   +∞
X λ0 λ1 . . . λi−1 ”
⇒1= 1+                     π0
i=1
µ1 µ2 . . . µ i
1
⇒ π0 =        P+∞ λ0 λ1 ...λi−1 .
1+     i=1   µ1 µ2 ...µi
Theoretical immunology
Birth and death processes
Stationary probability distribution

Stationary probability distribution: proof III

A unique positive stationary distribution exists if and only if the following
summation is positive and ﬁnite:
+∞
X λ0 λ1 . . . λi−1
0<                       < +∞ .                  (20)
i=1
µ1 µ2 . . . µ i

If λi = 0 for some i and µi > 0 for i ≥ 1, a stationary distribution still
exists but it is not positive.
If λ0 = 0 and µi > 0 for i ≥ 1 then we ﬁnd that π0 = 1 and πi = 0 for
i ≥ 1.
Theoretical immunology
Continuous time birth and death process with absorbing states
Probability of population extinction

Continuous time birth and death process with absorbing states

Let us assume that a continuous time birth and death process has
λ0 = 0 = µ0 .
We have just shown that a positive stationary probability distribution does
not exist.
Furthermore, the zero state is absorbing and eventually the total
population will become extinct as t → +∞.

lim p0 (t) = 1 .
t→+∞

This implies that extinction is certain with probability one.
Theoretical immunology
Continuous time birth and death process with absorbing states
Probability of population extinction

Probability of population extinction

Theorem
Let µ0 = 0 = λ0 in a general birth and death chain with X(0) = m ≥ 1.
Suppose µi > 0 and λi > 0 for i = 1, 2, . . .. Then, if
+∞
X µ1 µ2 . . . µ i
= +∞ ,              (21)
i=1
λ1 λ2 . . . λi

we have limt→+∞ p0 (t) = 1 , and if
+∞
X µ1 µ2 . . . µ i
< +∞ ,              (22)
i=1
λ1 λ2 . . . λi

then we have                                                P+∞     µ1 µ2 ...µi
i=m λ λ ...λ
lim p0 (t) =            P+∞1 µ2 µ2 ...µi .
1
i
(23)
t→+∞                  1+     i=1 λ1 λ2 ...λi
Theoretical immunology
Continuous time birth and death process with absorbing states
Quasi-stationary probability distribution

Quasi-stationary probability distribution I

In birth and death models, when state zero is absorbing, there is no
stationary probability distribution.
Although limt→+∞ p0 (t) = 1, prior to reaching extinction we may ﬁnd that
the probability distribution of X(t) can be approximately stationary for a
long period of time, specially if the expected time to extinction is relatively
large.
This approximate stationary distribution is known as the quasi-stationary
probability distribution.
We denote the quasi-stationary probability distribution associated with
X(t) as:

qi (t) = P(X(t) = i | non-extinction)
pi (t)                               (24)
=               ,   for i = 1, 2, . . . .
1 − p0 (t)
Theoretical immunology
Continuous time birth and death process with absorbing states
Quasi-stationary probability distribution

Quasi-stationary probability distribution II

The quasi-stationary distribution probabilities satisfy the following system
of diﬀerential equations:
dqi (t)      1 dpi         pi      1 dp0
=             +
dt       1 − p0 dt      1 − p0 1 − p0 dt                                (25)
= λi−1 qi−1 − (λi + µi )qi + µi+1 qi+1 + qi (µ1 q1 ) . for i ≥ 1

We make use of equations (13) and (14) (with λ0 = 0 = µ0 ) to obtain:
dq1      1 dp1         p1       1 dp0
=              +
dt    1 − p0 dt      1 − p0 1 − p0 dt         (26)
= q2 µ2 − q1 (λ1 + µ1 ) + q1 (q1 µ1 ) .

If the state space is ﬁnite, we have
dqN     1 dpN         pN     1 dp0
=            +
dt   1 − p0 dt     1 − p0 1 − p0 dt          (27)
= qN−1 λN−1 − qN µN + qN (q1 µ1 ) .
Theoretical immunology
Continuous time birth and death process with absorbing states
Quasi-stationary probability distribution

Approximation to the quasi-stationary probability distribution I

The quasi-stationary probability distribution can be approximated by
making the assumption µ1 = 0.
˜
We thus have dq = Qq, with dt

−λ1
0                                                 1
µ2          0       ...
B           λ1          −λ2 − µ2     µ3       ...        C
˜
Q=B
B
0             λ2       −λ3 − µ3   ...
C
C
.              .          .
@                                                        A
.              .          .      ..
.              .          .           . .

This new continuous time birth and death process will have a unique
positive stationary probability distribution given by π = (˜1 , π2 , . . .)T .
˜    π ˜
Theoretical immunology
Continuous time birth and death process with absorbing states
Uniqueness and existence for the approximated quasi-stationary probability distribution

Approximation to the quasi-stationary probability distribution II

The stationary probability distribution must satisfy
+∞
λ1 λ2 . . . λi−1                          X
˜
πi =                       ˜
π1 ,            and            ˜
πi = 1 .   (28)
µ2 µ3 . . . µ i                          i=1

Therefore, a unique positive stationary probability distribution exists for
˜
the system dq = Qq, if:
dt

+∞
X λ1 λ2 . . . λi−1
< +∞ .                             (29)
i=1
µ2 µ3 . . . µ i

˜
The probability distribution π is an approximation to the exact
quasi-stationary probability distribution.
Theoretical immunology
Continuous time birth and death process with absorbing states
Uniqueness and existence for the approximated quasi-stationary probability distribution

Approximation to the quasi-stationary probability distribution III
˜           π
The stationary probability distribution π satisﬁes Q˜ = 0, which can be
written as

0      =      λi−1 πi−1 − (λi + µi )˜i + µi+1 πi+1 ,
˜                π         ˜                    i = 2, 3, . . . ,   (30)
0      =      −λ1 π1 + µ2 π2 ,
˜       ˜                                                            (31)
P+∞
and         i=1   ˜
πi = 1.
˜
We now solve these equations and ﬁnd π2 :
λ1
˜
π2 =         ˜
π1 .
µ2
˜
For π3 we have
µ3 π3 = (λ2 + µ2 )˜2 − λ1 π1
˜                π       ˜
h (λ + µ )            i
2      2
=               λ1 − λ1 π1˜
µ2
λ1 λ2
˜
π3 =       ˜
π1 .
µ2 µ3
Theoretical immunology
Continuous time birth and death process with absorbing states
Uniqueness and existence for the approximated quasi-stationary probability distribution

Approximation to the quasi-stationary probability distribution IV

˜               ˜
We now continue by induction on πi . Assume πj has been deﬁned for
j = 2, 3, . . . , i.
λ1 λ2 . . . λi−1
˜
πi =                   ˜
π1 .
µ2 µ3 . . . µ i
˜
We can now write for πi+1 :

µi+1 πi+1 = (λi + µi )˜i − λi−1 πi−1
˜                  π            ˜
h λ λ . . . λ (λ + µ )         λ1 λ2 . . . λi−1 i
1 2         i−1    i    i
=                              −                    ˜
π1
µ2 µ3 . . . µ i       µ2 µ3 . . . µi−1
λ1 λ2 . . . λi−1 λi
˜
πi+1 =                      ˜
π1 .
µ2 µ3 . . . µi+1
Theoretical immunology
Continuous time birth and death process with absorbing states
Uniqueness and existence for the approximated quasi-stationary probability distribution

Approximation to the quasi-stationary probability distribution V

˜
We have solved by induction in terms of π1 .
˜
We now must solve for π1 .
We make use of the condition
+∞
X
˜
πi = 1 .
i=1

We ﬁnd
+∞
X                               +∞
X λ1 λ2 . . . λi−1
πi = 1 − π1 =
˜        ˜                                    ˜
π1
i=2                             i=2
µ2 µ3 . . . µ i
“   +∞
X λ1 λ2 . . . λi−1 ”
⇒1= 1+                      ˜
π1
i=2
µ2 µ3 . . . µ i
1
˜
π1 =             P+∞ λ1 λ2 ...λi−1 .
1+         i=2      µ2 µ3 ...µi
Theoretical immunology
Continuous time birth and death process with absorbing states
Time to extinction

Expected times to extinction

Suppose {X(t)}, t ≥ 0, is a continuous time birth and death chain with
X(0) = m ≥ 1, satisfying λ0 = 0 = µ0 and λi > 0 and µi > 0 for
i = 1, 2, . . ..
Furthermore, we assume that limt→∞ p0 (t) = 1.
The expected time to extinction τm = E(τ0,m ) satisﬁes:
81      P∞ λ λ ...λ
< µ1 + i=2 1 µ2 ...µi−1 , m = 1 ,
τm =               h 1 iP                    i                       (32)
λ1 ...λi−1
:τ1 + m−1 µ1 ...µs ∞
P
s=1   λ1 ...λs i=s+1 µ1 ...µi    , m = 2, 3, . . . .
Theoretical immunology
Continuous time birth and death process with absorbing states
Time to extinction

Expected times to extinction: proof I

If we let zi = τi − τi+1 ≤ 0, we have
1         λi             µi
τi =             +         τi+1 +         τi−1 .
λi + µi   λi + µi        λi + µi
We now subtract τi from both sides of the previous equation to obtain
1         λi                     µi
0=      +         (τi+1 − τi ) +         (τi−1 − τi ) ,
λi + µi   λi + µi                λi + µi
1    µi
τi − τi+1 =    + (τi−1 − τi ) ,                                        (33)
λi   λi
1    µi
zi =    + zi−1 .
λi   λi
Theoretical immunology
Continuous time birth and death process with absorbing states
Time to extinction

Expected times to extinction: proof II

We continue by induction to obtain
1          µm              µ2 . . . µ m    µ1 . . . µ m
zm =        +           + ... +                  +              z0
λm       λm λm−1            λ1 . . . λm     λ1 . . . λm
hλ ...λ
µ1 . . . µ m 1       m−1                1      i
=                            + ... +        + z0
λ1 . . . λm µ1 . . . µm               µ1
m
µ1 . . . µm h X λ1 . . . λi−1         i                     (34)
=                               − τ1 ,        as z0 = −τ1
λ1 . . . λm i=1 µ1 . . . µi
m
λ1 . . . λm       1    X λ1 . . . λi−1
⇒                    zm =    +                  − τ1 .
µ1 . . . µ m      µ1   i=2
µ1 . . . µ i
Theoretical immunology
Continuous time birth and death process with absorbing states
Time to extinction

Expected times to extinction: proof III

λ1 ...λi−1
P∞
Suppose            i=2
µ1 ...µi
= +∞ ⇒ τ1 = +∞ .
But since {τi }∞ is a non-decreasing sequence
i=1                                            τm = +∞.
P λ1 ...λi−1
Suppose ∞ µ1 ...µi < +∞ ⇒ τ1 < +∞.
i=2
Then, for large m, deaths are much greater than births, so that
1
zm → τm − τm+1 ≈ µm+1 as m → +∞, which is the mean time for a death
to occur when the population size is m + 1.
If we let m → +∞ so that
λ1 . . . λm        λ1 . . . λm
zm →               →0.
µ1 . . . µ m      µ1 . . . µm+1
Theoretical immunology
Continuous time birth and death process with absorbing states
Time to extinction

Expected times to extinction: proof IV
We then have
+∞
1    X λ1 . . . λi−1
τ1 =         +                  ,
µ1   i=2
µ1 . . . µ i
m                      +∞
µ1 . . . µm h X λ1 . . . λi−1   1    X λ1 . . . λi−1 i
zm =                                     −    −
λ1 . . . λm i=1 µ1 . . . µi     µ1   i=2
µ1 . . . µ i
+∞
µ1 . . . µm h X λ1 . . . λi−1 i
=−                                      ,
λ1 . . . λm i=m+1 µ1 . . . µi
m−1
X
τm − τ1 = −                  zs
s=1
m−1 h                 +∞
X        µ1 . . . µs X λ1 . . . λi−1 i
=                                          ,
s=1
λ1 . . . λs i=n+1 µ1 . . . µi
m−1 h                  +∞
X         µ1 . . . µs X λ1 . . . λi−1 i
τm = τ1 +                                                 .
s=1
λ1 . . . λs i=n+1 µ1 . . . µi

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