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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