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- 1. Modelling T cell immunology Lecture 1 Carmen Molina-París Department of Applied Mathematics, School of Mathematics, University of Leeds, UK 30th of March 2009 CMP (Leeds) Theoretical immunology I2 M GS 1 / 76
- 2. Outline 1 Mathematics and physics Mathematical framework for physics 2 Biology Cellular level Molecular (surface) level 3 Modelling a population Deterministic birth process Stochastic birth process 4 Immunology Infections: viruses Vertebrates Adaptive immune system 5 General Theory of Continuous Time Markov Chains (CTMC) Transition probabilities Kolmogorov equations 6 Birth and death processes Continuous time birth and death process with absorbing states CMP (Leeds) Theoretical immunology I2 M GS 2 / 76
- 3. Mathematics is a special science! CMP (Leeds) Theoretical immunology I2 M GS 3 / 76
- 4. Classical mechanics Figure: Planetary motion. CMP (Leeds) Theoretical immunology I2 M GS 4 / 76
- 5. Quantum mechanics Figure: Hydrogen atom orbitals for electron. CMP (Leeds) Theoretical immunology I2 M GS 5 / 76
- 6. General relativity Figure: Structure of a black hole. CMP (Leeds) Theoretical immunology I2 M GS 6 / 76
- 7. Quantum gravity We do not yet have a complete mathematical framework. Figure: Early history of the universe. CMP (Leeds) Theoretical immunology I2 M GS 7 / 76
- 8. Biology still lacks a mathematical framework Figure: A cell. CMP (Leeds) Theoretical immunology I2 M GS 8 / 76
- 9. Complexity in biology I Figure: Viral infection. CMP (Leeds) Theoretical immunology I2 M GS 9 / 76
- 10. Complexity in biology II Figure: Different immune responses. CMP (Leeds) Theoretical immunology I2 M GS 10 / 76
- 11. Complexity also on the surface of cells Figure: Immune cell signalling: surface receptors. CMP (Leeds) Theoretical immunology I2 M GS 11 / 76
- 12. Details of T cell receptor (TCR) Figure: The T cell receptor (TCR). CMP (Leeds) Theoretical immunology I2 M GS 12 / 76
- 13. Lots of different receptors (with different structures) Figure: Different cellular receptors CMP (Leeds) Theoretical immunology I2 M GS 13 / 76
- 14. Model the time evolution of a population Figure: A lot of penguins! CMP (Leeds) Theoretical immunology I2 M GS 14 / 76
- 15. Deterministic modelling Example 1 Let N(t) be the population of those penguins at time t. 2 N(t) is the number of individuals in the population at time t. 3 The change in the number of penguins in a small time interval, ∆t, is given by N(t + ∆t) = N(t) + births − deaths + migration . 4 This equation is a conservation equation for the number of individuals of the population. 5 The form of the various terms on the right hand side requires modelling the situation that we are concerned with. CMP (Leeds) Theoretical immunology I2 M GS 15 / 76
- 16. Deterministic birth process I Example Let us assume that 1 There are no death events in the population. 2 There are only birth events in the population. 3 The birth rate (number of births per unit of time), b, is the same for all individuals of the population. 4 We have N(t + ∆t) = N(t) + births . CMP (Leeds) Theoretical immunology I2 M GS 16 / 76
- 17. Deterministic birth process II Example 1 Change in the population is due to birth events. 2 The change in the population size in the time interval ∆t due to a single individual is b × ∆t. 3 The change in the population size due to all individuals is N(t) × b × ∆t. N(t+∆t)−N(t) 4 N(t + ∆t) = N(t) + N(t) b ∆t ⇒ ∆t = b N(t) . 5 For a very small time interval, ∆t → 0+ N(t + ∆t) − N(t) def dN(t) lim + = = b N(t) . ∆t→0 ∆t dt 6 This equation can be easily solved by integration (Undergraduate Year 1). CMP (Leeds) Theoretical immunology I2 M GS 17 / 76
- 18. Deterministic birth process III Example 1 If the population at time t = t0 is given by N0 , we have N(t) = N0 eb(t−t0 ) . 2 In a deterministic birth process the population size is predicted at time t with absolute certainty, once the initial size N0 and birth rate b are given. 3 The population size N(t) and time t are both continuous variables (both take real values) and not discrete (take integer values). CMP (Leeds) Theoretical immunology I2 M GS 18 / 76
- 19. Number of penguins does not live in the “real” line. Example 1 2 Let us assume there are n penguins at time t. 3 n 4 If there is a birth event in the time interval ∆t, there will be n + 1 penguins at time t + ∆t. 5 n n+1 CMP (Leeds) Theoretical immunology I2 M GS 19 / 76
- 20. Stochastic modelling Example 1 Let Xt be the discrete random variable that describes the number of individuals of the population at time t. 2 1 2 3 4 n−1 n n+1 3 The population size is a discrete variable (not continuous) but time is still a continuous variable. 4 The stochastic process that describes our population satisﬁes Xt ∈ {1, 2, · · · } and t ∈ [0, +∞). 5 Denote by pn (t) the probability that at time t the size of the population is n (or the probability that at time t there are n individuals in the population) pn (t) = P(Xt = n) . CMP (Leeds) Theoretical immunology I2 M GS 20 / 76
- 21. Stochastic birth process I Example 1 Consider a small time interval ∆t. How is Xt+∆t related to Xt ? 2 We have the following rules: 1 There are no death events in the population. 2 There are birth events in the population: the probability that a birth takes place is b∆t. 3 The probability of more than one birth in a time interval ∆t is negligible. 3 What can we say about pn (t + ∆t) in terms of pm (t)? CMP (Leeds) Theoretical immunology I2 M GS 21 / 76
- 22. Stochastic birth process II Example 1 The probability that a population of size n − 1 increases to n in the time interval (t, t + ∆t) is b × ∆t × (n − 1). 2 The probability that a population of size n increases to n + 1 in the time interval (t, t + ∆t) is b × ∆t × n. 3 If at time t the population has n individuals, the probability that no birth event takes place in the time interval (t, t + ∆t) is 1 − b × ∆t × n. 4 Evolution equation for pn (t): pn (t + ∆t) = b ∆t (n − 1) pn−1 (t) + (1 − b ∆t n) pn (t) . 5 The state at time t + ∆t only depends on the state at time t. 6 This is known as a Markov property. CMP (Leeds) Theoretical immunology I2 M GS 22 / 76
- 23. Cell division Figure: Cell division. Birth event 1 At time t there are n cells. 2 During the time interval ∆t there is a single birth event. 3 At time t + ∆t there are n + 1 cells. CMP (Leeds) Theoretical immunology I2 M GS 23 / 76
- 24. Cell death Figure: Cell death. Death event 1 At time t there are n cells. 2 During the time interval ∆t there is a single death event. 3 At time t + ∆t there are n − 1 cells. CMP (Leeds) Theoretical immunology I2 M GS 24 / 76
- 25. T cell Figure: T cell. CMP (Leeds) Theoretical immunology I2 M GS 25 / 76
- 26. Cell-cell interactions lead to “events” Figure: T cell-dendritic cell interaction. CMP (Leeds) Theoretical immunology I2 M GS 26 / 76
- 27. Cell-cell signalling: differentiation and proliferation Figure: T cell immune response. CMP (Leeds) Theoretical immunology I2 M GS 27 / 76
- 28. T cell and tumour cell Figure: T cell and tumour cell. CMP (Leeds) Theoretical immunology I2 M GS 28 / 76
- 29. Virus replication inside a cell CMP (Leeds) Theoretical immunology I2 M GS 29 / 76
- 30. There is a wide range of viral infections CMP (Leeds) Theoretical immunology I2 M GS 30 / 76
- 31. Viral particles can attach and gain entry into the cell CMP (Leeds) Theoretical immunology I2 M GS 31 / 76
- 32. Different ways for viral uncoating CMP (Leeds) Theoretical immunology I2 M GS 32 / 76
- 33. Immunology (vertebrates) I Arms of the immune system 1 Innate immune system: ﬁrst line of defense. 2 Adaptive immune system: sophisticated defenses against infection. CMP (Leeds) Theoretical immunology I2 M GS 33 / 76
- 34. Immunology (vertebrates) II Adaptive immune system 1 In vertebrates, innate responses call the adaptive immune responses into play, but both work together. 2 Innate immune responses: general defense reactions. Adaptive responses: highly speciﬁc to the particular pathogen that induced them and provide long-lasting protection. 3 Adaptive immune responses eliminate invading pathogens and any toxic molecules they produce. 4 These responses are destructive: should be directed only against foreign molecules and not against molecules of the host itself. 5 The adaptive immune system uses multiple mechanisms to avoid damaging responses against self molecules. 6 If these mechanisms fail and the system turns against the host: autoimmune diseases, which can be fatal. CMP (Leeds) Theoretical immunology I2 M GS 34 / 76
- 35. Immunology (vertebrates) III Self versus non-self or harmless versus dangerous 1 Many harmless foreign molecules enter the body. It would be pointless and potentially dangerous to mount adaptive immune responses against them. 2 Allergic conditions are examples of adaptive immune responses against apparently harmless foreign molecules. 3 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. 4 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. CMP (Leeds) Theoretical immunology I2 M GS 35 / 76
- 36. Immunology (vertebrates) IV Antigens and immunisation 1 Any substance capable of eliciting an adaptive immune response is referred to as an antigen (antibody generator). 2 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. 3 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. CMP (Leeds) Theoretical immunology I2 M GS 36 / 76
- 37. Immune cells 1 Adaptive immune responses are carried out by white blood cells called lymphocytes. 2 There are two broad classes of such responses: 1 antibody responses and 2 T cell-mediated immune responses. CMP (Leeds) Theoretical immunology I2 M GS 37 / 76
- 38. B cells 1 B cells are activated to secrete antibodies, which are proteins called immunoglobulins. 2 Antibodies circulate in the bloodstream and permeate the other body ﬂuids, where they bind speciﬁcally to the foreign antigen that stimulated their production. 3 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. 4 Antibody binding also marks invading pathogens for destruction, mainly by making it easier for phagocytic cells of the innate immune system to ingest them. CMP (Leeds) Theoretical immunology I2 M GS 38 / 76
- 39. T cells 1 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 (antigen-presenting cell). 2 T cells can detect microbes hiding inside host cells and (i) either kill the infected cells or (ii) help the infected cells or other cells eliminate the microbes. 3 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. 4 In other cases, the T cell produces signal molecules that (i) activate macrophages to destroy the microbes that they have phagocytosed or (ii) help activate B cells to make antibodies against the microbes. CMP (Leeds) Theoretical immunology I2 M GS 39 / 76
- 40. Macrophages and neutrophils I 1 The rapid innate immune responses to an infection depend largely on pattern recognition receptors made by cells of the innate immune system. 2 These receptors recognise microbe-associated molecules that are not present in the host organism. CMP (Leeds) Theoretical immunology I2 M GS 40 / 76
- 41. Macrophages and neutrophils II 1 Some of the pattern recognition receptors are present on the surface of professional phagocytic cells (phagocytes), such as macrophages and neutrophils. 2 These receptors mediate the uptake of pathogens, which are then delivered to lysosomes for destruction. 3 Other receptors 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. 4 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. CMP (Leeds) Theoretical immunology I2 M GS 41 / 76
- 42. Macrophages and neutrophils III CMP (Leeds) Theoretical immunology I2 M GS 42 / 76
- 43. Macrophages and neutrophils IV CMP (Leeds) Theoretical immunology I2 M GS 43 / 76
- 44. Dendritic cells I 1 The cells of the vertebrate innate immune system that activate adaptive immune responses most efﬁciently are dendritic cells. 2 Present in most tissues, dendritic cells express high levels of TLRs and other pattern recognition receptors. 3 They function by presenting microbial antigens to T cells in peripheral lymphoid organs. 4 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. CMP (Leeds) Theoretical immunology I2 M GS 44 / 76
- 45. Dendritic cells II 1 In other cases, they pick up microbes or their products directly in a peripheral lymphoid organ, such as the spleen. 2 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. 3 Once activated, some of the T cells then migrate to the site of infection, where they help destroy the microbes. 4 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. CMP (Leeds) Theoretical immunology I2 M GS 45 / 76
- 46. Dendritic cells III CMP (Leeds) Theoretical immunology I2 M GS 46 / 76
- 47. Dendritic cells IV CMP (Leeds) Theoretical immunology I2 M GS 47 / 76
- 48. 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. CMP (Leeds) Theoretical immunology I2 M GS 48 / 76
- 49. 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, . . . . CMP (Leeds) Theoretical immunology I2 M GS 49 / 76
- 50. Transition probabilities II Deﬁnition We will say 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 +∞ pji (t) = 1 for t ≥ 0 , ∀i = 0, 1, 2, . . . . j=0 The matrix of transition probabilities, or transition matrix, P, is for all t ≥ 0 given by P = (pji (t)) . (2) CMP (Leeds) Theoretical immunology I2 M GS 50 / 76
- 51. 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. +∞ pji (t + ∆t) = pjk (∆t) pki (t) forward Kolmogorov equation , k =0 +∞ (4) = pjk (t) pki (∆t) backward Kolmogorov equation . k =0 CMP (Leeds) Theoretical immunology I2 M GS 51 / 76
- 52. Proof of the forward Kolmogorov equations +∞ 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 +∞ = 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 +∞ = 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 +∞ = pjk (∆t) pki (t) . k =0 +∞ 2 CMP (Leeds) Theoretical immunology I M GS 52 / 76
- 53. Backward Kolmogorov equations Example Homework: Derive the backward Kolmogorov equations: +∞ pji (t + ∆t) = pjk (t) pki (∆t) backward Kolmogorov equation . (5) k =0 CMP (Leeds) Theoretical immunology I2 M GS 53 / 76
- 54. Birth and death processes I 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 = X(t + ∆t) − X(t) . (6) The inﬁnitesimal transition probabilities of a general birth and death process are: pi+ji (∆t) = P{∆X(t) = j | X(t) = i} λi ∆t + o(∆t) j =1, µ ∆t + o(∆t) i j = −1 , (7) = 1 − (λ + µ )∆t + o(∆t) j =0, i i o(∆t) j = −1, 0, 1 . CMP (Leeds) Theoretical immunology I2 M GS 54 / 76
- 55. Birth and death processes II 1 We set λi = birth rate, when the population is of size i. 2 We set µi = death rate, when the population is of size i. 3 λi , µi ≥ 0 and o(∆t) is the Landau order symbol: o(∆t) lim =0. (8) ∆t→0 ∆t 4 The forward Kolmogorov differential equations for pji (t) can be derived directly from Equation (7). CMP (Leeds) Theoretical immunology I2 M GS 55 / 76
- 56. Forward Kolmogorov equations: birth and death I 1 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: pj i (t + ∆t) = pj−1 i (t)[λj−1 ∆t + o(∆t)] + pj+1 i (t)[µj+1 ∆t + o(∆t)] + 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 with the exception of the endpoints, j = 0 and j = N. CMP (Leeds) Theoretical immunology I2 M GS 56 / 76
- 57. Forward Kolmogorov equations: birth and death II 1 If j = 0 and assuming that µ0 = 0, then p0i (t + ∆t) = p1i (t)µ1 ∆t + p0i (t)(1 − λ0 ∆t) + o(∆t) . (10) 2 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. 3 We can now derive the forward Kolmogorov differential equations using the transition probabilities of the previous slides. CMP (Leeds) Theoretical immunology I2 M GS 57 / 76
- 58. Forward Kolmogorov equations: birth and death III 1 We obtain from equation (9) pji (t + ∆t) − pji (t) = pj−1 i (t)λj−1 + pj+1 i (t)µj+1 − pji (t)(λi + µj ) ∆t o(∆t) + . ∆t o(∆t) 2 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. CMP (Leeds) Theoretical immunology I2 M GS 58 / 76
- 59. Forward Kolmogorov equations: birth and death IV 1 Using equation (10) we obtain: dp0i (t) = p1i (t)µ1 − λ0 p0i (t) , for i ≥ 0 . (13) dt 2 Using equation (11) we obtain: dpNi (t) = pN−1i (t)λN−1 − µN pNi (t) , for i ≥ 0 . (14) dt CMP (Leeds) Theoretical immunology I2 M GS 59 / 76
- 60. 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: +∞ P(t)π = π , πi = 1 , and πi ≥ 0 , i=0 for t ≥ 0 and i = 0, 1, 2, . . . CMP (Leeds) Theoretical immunology I2 M GS 60 / 76
- 61. 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 +∞ λ0 λ1 . . . λi−1 < +∞ . (15) µ1 µ2 . . . µi i=1 The stationary probability distribution is given by λ0 λ1 . . . λi−1 πi = π0 , for i = 1, 2, . . . (16) µ1 µ2 . . . µi 1 π0 = +∞ λ0 λ1 ...λi−1 . (17) 1+ i=1 µ1 µ2 ...µi CMP (Leeds) Theoretical immunology I2 M GS 61 / 76
- 62. 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. CMP (Leeds) Theoretical immunology I2 M GS 62 / 76
- 63. Continuous time birth and death process with absorbing states 1 Let us assume that a continuous time birth and death process has λ0 = 0 = µ0 . 2 We have just shown that a positive stationary probability distribution does not exist. 3 Furthermore, the zero state is absorbing and eventually the total population will become extinct as t → +∞. lim p0 (t) = 1 . t→+∞ 4 This implies that extinction is certain with probability one. CMP (Leeds) Theoretical immunology I2 M GS 63 / 76
- 64. 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 +∞ µ1 µ2 . . . µi = +∞ , (18) λ1 λ2 . . . λ i i=1 we have limt→+∞ p0 (t) = 1 , and if +∞ µ1 µ2 . . . µi < +∞ , (19) λ1 λ2 . . . λ i i=1 then we have +∞ µ1 µ2 ...µi i=m λ1 λ2 ...λi lim p0 (t) = +∞ µ1 µ2 ...µi . (20) t→+∞ 1+ i=1 λ1 λ2 ...λi CMP (Leeds) Theoretical immunology I2 M GS 64 / 76
- 65. Conditional probability distribution I 1 In birth and death models, when state zero is absorbing, there is no stationary probability distribution. 2 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. 3 This approximate stationary distribution is known as the conditional probability distribution. 4 We denote the conditional probability distribution associated with X(t) as: qi (t) = P(X(t) = i | non-extinction) pi (t) (21) = , for i = 1, 2, . . . . 1 − p0 (t) CMP (Leeds) Theoretical immunology I2 M GS 65 / 76
- 66. Conditional probability distribution II 1 The conditional 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 = λi−1 qi−1 − (λi + µi )qi + µi+1 qi+1 + qi (µ1 q1 ) . for i ≥ 1 (22) 2 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 (23) = q2 µ2 − q1 (λ1 + µ1 ) + q1 (q1 µ1 ) . 3 If the state space is ﬁnite, we have dqN 1 dpN pN 1 dp0 = + dt 1 − p0 dt 1 − p0 1 − p0 dt (24) CMP (Leeds) Theoretical − qN µ = qN−1 λN−1immunology N + qN (q1 µ1 ) . 2 I M GS 66 / 76
- 67. Approximation to the conditional distribution I 1 The conditional probability distribution can be approximated by making the assumption µ1 = 0. 2 We thus have dq = Qq, with dt ˜ −λ1 µ2 0 ... λ1 −λ2 − µ2 µ3 ... ˜ = Q 0 λ2 −λ3 − µ3 ... . . . . . . .. . . . . . 3 This new continuous time birth and death process will have a unique positive stationary probability distribution given by π = (˜1 , π2 , . . .)T . ˜ π ˜ CMP (Leeds) Theoretical immunology I2 M GS 67 / 76
- 68. Approximation to the conditional distribution II 1 The stationary probability distribution must satisfy +∞ λ1 λ2 . . . λi−1 πi = ˜ π1 , and ˜ πi = 1 . ˜ (25) µ2 µ3 . . . µi i=1 2 Therefore, a unique positive stationary probability distribution exists for the system dq = Qq, if: dt ˜ +∞ λ1 λ2 . . . λi−1 < +∞ . (26) µ2 µ3 . . . µi i=1 3 The probability distribution π is an approximation to the exact ˜ limiting conditional probability distribution. CMP (Leeds) Theoretical immunology I2 M GS 68 / 76
- 69. Approximation to the conditional distribution III 1 The stationary probability distribution π satisﬁes Q˜ = 0, which ˜ π can be written as 0 = λi−1 πi−1 − (λi + µi )˜i + µi+1 πi+1 , i = 2, 3, . . . , (27) ˜ π ˜ 0 = −λ1 π1 + µ2 π2 , ˜ ˜ (28) +∞ and = 1. i=1 πi ˜ 2 We now solve these equations and ﬁnd π2 : ˜ λ1 π2 = ˜ π1 . ˜ µ2 3 For π3 we have ˜ (λ2 + µ2 ) µ3 π3 = (λ2 + µ2 )˜2 − λ1 π1 = ˜ π ˜ λ 1 − λ 1 π1 ˜ µ2 λ1 λ2 π3 = ˜ π1 . ˜ µ2 µ3 CMP (Leeds) Theoretical immunology I2 M GS 69 / 76
- 70. Approximation to the conditional distribution IV 1 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 . . . µi 2 We can now write for πi+1 : ˜ µi+1 πi+1 = (λi + µi )˜i − λi−1 πi−1 ˜ π ˜ λ1 λ2 . . . λi−1 (λi + µi ) λ1 λ2 . . . λi−1 = − π1 ˜ µ2 µ3 . . . µi µ2 µ3 . . . µi−1 λ1 λ2 . . . λi−1 λi πi+1 = ˜ π1 . ˜ µ2 µ3 . . . µi+1 CMP (Leeds) Theoretical immunology I2 M GS 70 / 76
- 71. Approximation to the conditional distribution V 1 We have solved by induction in terms of π1 . ˜ 2 We now must solve for π1 . ˜ 3 We make use of the condition +∞ πi = 1 . ˜ i=1 4 We ﬁnd +∞ +∞ λ1 λ2 . . . λi−1 π i = 1 − π1 = ˜ ˜ π1 ˜ µ2 µ3 . . . µi i=2 i=2 +∞ λ1 λ2 . . . λi−1 ⇒1= 1+ π1 ˜ µ2 µ3 . . . µi i=2 1 π1 = ˜ +∞ λ1 λ2 ...λi−1 . 1+ i=2 µ2 µ3 ...µi CMP (Leeds) Theoretical immunology I2 M GS 71 / 76
- 72. Expected times to extinction 1 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, . . .. 2 Furthermore, we assume that limt→∞ p0 (t) = 1. 3 The expected time to extinction τm = E(τ0,m ) satisﬁes: 1 + ∞ λ1 λ2 ...λi−1 , m = 1 , µ1 i=2 µ1 ...µi τm = λ1 ...λi−1 τ1 + m−1 µ1 ...µs ∞ s=1 λ1 ...λs i=s+1 µ1 ...µi , m = 2, 3, . . . . (29) CMP (Leeds) Theoretical immunology I2 M GS 72 / 76
- 73. Expected times to extinction: proof I 1 If we let zi = τi − τi+1 ≤ 0, we have 1 λi µi τi = + τi+1 + τi−1 . λi + µi λi + µi λi + µi 2 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 ) , (30) λi λi 1 µi zi = + zi−1 . λi λi CMP (Leeds) Theoretical immunology I2 M GS 73 / 76
- 74. Expected times to extinction: proof II 1 We continue by induction to obtain 1 µm µ2 . . . µm µ1 . . . µm zm = + + ... + + z0 λm λm λm−1 λ1 . . . λ m λ1 . . . λ m µ1 . . . µm λ1 . . . λm−1 1 = + ... + + z0 λ1 . . . λm µ1 . . . µm µ1 m µ1 . . . µm λ1 . . . λi−1 = − τ1 , as z0 = −τ1 λ1 . . . λ m µ1 . . . µi i=1 m λ1 . . . λ m 1 λ1 . . . λi−1 ⇒ zm = + − τ1 . µ1 . . . µm µ1 µ1 . . . µi i=2 (31) CMP (Leeds) Theoretical immunology I2 M GS 74 / 76
- 75. Expected times to extinction: proof III ∞ λ1 ...λi−1 Suppose i=2 µ1 ...µi = +∞ ⇒ τ1 = +∞ . 1 2 But since {τi }∞ is a non-decreasing sequence τm i=1 = +∞. ∞ λ1 ...λi−1 Suppose i=2 µ1 ...µi < +∞ ⇒ τ1 < +∞. 3 4 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. 5 If we let m → +∞ so that λ1 . . . λ m λ1 . . . λ m zm → →0. µ1 . . . µm µ1 . . . µm+1 CMP (Leeds) Theoretical immunology I2 M GS 75 / 76
- 76. Expected times to extinction: proof IV 1 We then have +∞ 1 λ1 . . . λi−1 τ1 = + , µ1 µ1 . . . µi i=2 m +∞ µ1 . . . µm λ1 . . . λi−1 1 λ1 . . . λi−1 zm = − − λ1 . . . λ m µ1 . . . µi µ1 µ1 . . . µi i=1 i=2 +∞ µ1 . . . µm λ1 . . . λi−1 =− , λ1 . . . λ m µ1 . . . µi i=m+1 m−1 m−1 +∞ µ1 . . . µs λ1 . . . λi−1 τm − τ1 = − zs = , λ1 . . . λ s µ1 . . . µi s=1 s=1 i=n+1 m−1 +∞ µ1 . . . µs λ1 . . . λi−1 τm = τ1 + . λ1 . . . λ s µ1 . . . µi s=1 i=n+1 CMP (Leeds) Theoretical immunology I2 M GS 76 / 76

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