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
           Adaptive immune system
        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.
                Adaptive immune responses eliminate invading pathogens and any toxic
                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
                expressed by invading pathogens.
                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
      Adaptive immune system



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
      Adaptive immune system



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
      Adaptive immune system



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
      Adaptive immune system



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
      Adaptive immune system



Macrophages and neutrophils II
Theoretical immunology
   Immunology
      Adaptive immune system



Macrophages and neutrophils III
Theoretical immunology
   Immunology
      Adaptive immune system



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
      Adaptive immune system



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
      Adaptive immune system



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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