Branching within branching: a general model for host-parasite co-evolution

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1 Branching within branching: a general model for host-parasite co-evolution Gerold Alsmeyer (joint work with Sören Gröttrup) May 15, 2017 Gerold Alsmeyer Host-parasite co-evolution 1 of 26

2 1 Model 2 The associated branching process in random environment (ABPRE) 3 Extinction results 4 Limit theorems in the case of survival Gerold Alsmeyer Host-parasite co-evolution 2 of 26

3 Model description The basic ingredients a cell population (the hosts) a population of parasites colonizing the cells The basic assumptions cells form an ordinary Galton-Watson tree (GWT) parasites sitting in different cells multiply and share their progeny into daughter cells independently, but for parasites hosted by the same cell v, offspring numbers and sharing of progeny are conditionally independent given the number of daughter cells of v Gerold Alsmeyer Host-parasite co-evolution 3 of 26

4 Notational details: the cell population Ulam-Harris tree V = n 0 N n with N 0 = { } cell population: a GWT T = n N 0 T n V with T 0 = { } and T n := {v 1 v n V v 1 v n 1 T n 1 and 1 v n N v1 v n 1 }, where N v denotes the number of daughter cells of v the N v, v V are iid with common law (p k ) k 0, the offspring distribution of cells, having finite mean ν = k 1 kp k the number of cells process: T n = v Tn 1 N v (a GWP) Gerold Alsmeyer Host-parasite co-evolution 4 of 26

5 Details: the parasites the number of parasites in cell v are denoted by Z v the number of parasites process is then defined by Z n := v T n Z v, n N 0 the set of contaminated cells: T n = {v T n : Z v > 0} the number of contaminated cells: T n = #T n cell counts with a specific number of parasites: T n := ( T n,0,t n,1,t n,2, ), where T n,k gives the number of cells in generation n hosting k parasites Gerold Alsmeyer Host-parasite co-evolution 5 of 26

6 Z =1 Z 0 Z 1 =2 Z 2 =4 Z 3 =1 Z 1 Z 11 =3 Z 12 =1 Z 31 =0 Z 32 =5 Z 33 =2 Z 2 Figure: A typical realization of the first three generations of a BwBP Gerold Alsmeyer Host-parasite co-evolution 6 of 26

7 Reproduction of parasites To describe reproduction of parasites consider those hosted by a cell v T and suppose that Z v = z 1 and that v has k daughter cells, labeled v1,,vk, thus N v = k For i = 1,,z, let X (,k) i,v := ( X (1,k) i,v ),,X (k,k) i,v be iid copies of a random vector X (,k) := ( X (1,k),,X (k,k)) with arbitrary law on N k 0 and independent of any other occurring rv s Then, given N v = k, and for i = 1,,Z v = z, X (j,k) i,v equals the number of offspring of the i th parasite in v that is shared into daughter cell vj Gerold Alsmeyer Host-parasite co-evolution 7 of 26

8 Model parameters and basic assumptions µ j,k = EX (j,k) γ = EZ 1 = k 1 p k k j=1 µ j,k, the mean number of offspring per parasite, is supposed to be positive and finite, thus µ j,k < for all j k and P(N v = 0) < 1 We further assume p 1 = P(N v = 1) < 1, P(Z 1 = 1) < 1, and a positive chance for more than one parasite to be shared in the same daughter cell: p k P(X (j,k) 2) > 0 for at least one (j,k), 1 j k Gerold Alsmeyer Host-parasite co-evolution 8 of 26

9 A very short list of earlier contributions M Kimmel Quasistationarity in a branching model of division-within-division In Classical and modern branching processes (Minneapolis, MN, 1994), volume 84 of IMA Vol Math Appl, pages Springer, New York, 1997 V Bansaye Proliferating parasites in dividing cells: Kimmel s branching model revisited Ann Appl Probab, 18(3): , 2008 G Alsmeyer and S Gröttrup A host-parasite model for a two-type cell population Adv in Appl Probab, 45(3): , 2013 Gerold Alsmeyer Host-parasite co-evolution 9 of 26

10 The notorious questions Extinction-explosion dichotomy for the number of parasites process (Z n ) n 0 Extinction-explosion dichotomy for the number of contaminated cells process (T n ) n 0 {Z n } = {T n }? Necessary and sufficient conditions for almost sure extinction of contaminated cells Limit theorems for the relevant processes in the survival case (after normalization) Gerold Alsmeyer Host-parasite co-evolution 10 of 26

11 The ABPRE The associated branching process in random environment (ABPRE) is obtained by picking an infinite random cell-line (spine) in a size-biased version of T: Gerold Alsmeyer Host-parasite co-evolution 11 of 26

12 The construction (standard) Let (T n,c n ) n 0 be a sequence of iid random vectors independent of (N v ) v V and (X (,k) i,v ) k 1,i 1,v V The law of T n equals the size-biasing of the law of the N v, ie P(T n = k) = kp k ν for each n N 0 and k N, and P(C n = j T n = k) = 1 k for 1 j k, which means that C n has a uniform distribution on {1,,k} given T n = k Gerold Alsmeyer Host-parasite co-evolution 12 of 26

13 The ABPRE The random cell-line (spine) (V n ) n 0 is then recursively defined by V 0 = and V n := V n 1 C n 1 for n 1 Then =: V 0 V 1 V 2 V n provides us with a random cell line in V (not picked uniformly) as depicted in the following picture Gerold Alsmeyer Host-parasite co-evolution 13 of 26

14 The ABPRE V 0 BP 0 V 1 BP 1 V 2 BP 2 V 3 Gerold Alsmeyer Host-parasite co-evolution 14 of 26

15 The number of parasites along the spine Regarding the structure of the number of parasites process along the spine (Z Vn ) n 0, the following lemma is fundamental Lemma Let (Z n) n 0 be a BPRE with Z ancestors and iid environmental sequence Λ := (Λ n ) n 0 taking values in {L (X (j,k) ) 1 j k < } and such that ( ) P Λ 0 = L (X (j,k) ) = p k ν = 1 k kp k ν = P(C 0 = j,t 0 = k) for all 1 j k < Then (Z Vn ) n 0 and (Z n) n 0 are equal in law Gerold Alsmeyer Host-parasite co-evolution 15 of 26

16 Definition of the ABPRE The BPRE (Z n) n 0 is now called the associated branching process in random environment (ABPRE) It is one of the major tools used in the study of the BwBP, and the following lemma provides a key relation between this process and its associated ABPRE Gerold Alsmeyer Host-parasite co-evolution 16 of 26

17 A key result Lemma For all n,k,z N 0, and P z ( Z n = k ) = ν n E z T n,k P z ( Z n > 0 ) = ν n E z T n Gerold Alsmeyer Host-parasite co-evolution 17 of 26

18 Generating functions For n N and s [0,1] E(s Z n Λ) = g Λ0 g Λn 1 (s) is the quenched generating function of Z n with iid g Λn and g λ defined by g λ (s) := E(s Z 1 Λ0 = λ) = λ n s n n 0 for any distribution λ = (λ n ) n 0 on N 0 Moreover, Eg Λ 0 (1) = EZ 1 = where γ = EZ 1 p k ν EX (j,k) = EZ 1 ν 1 j k = γ ν <, (1) Gerold Alsmeyer Host-parasite co-evolution 18 of 26

19 Regimes of the ABPRE It is well-known that (Z n) n 0 survives with positive probability iff Elogg Λ 0 (1) > 0 and Elog (1 g Λ0 (0)) < Recall that γ < is assumed and that there exists 1 j k < such that p k > 0 and P(X (j,k) 1) > 0, which ensures that Λ 0 δ 1 with positive probability The ABPRE is called supercritical, critical or subcritical if Elogg Λ 0 (1) > 0, = 0 or < 0, respectively Gerold Alsmeyer Host-parasite co-evolution 19 of 26

20 Regimes of the ABPRE The subcritical case further divides into the three subregimes when Eg Λ 0 (1)logg Λ 0 (1) < 0,= 0, or > 0, respectively, called strongly, intermediate and weakly subcritical case The quite different behavior of the process in each of the three subregimes is shown by the limit results derived in J Geiger, G Kersting, and V A Vatutin Limit theorems for subcritical branching processes in random environment Ann Inst H Poincaré Probab Statist, 39(4): , 2003 Gerold Alsmeyer Host-parasite co-evolution 20 of 26

21 Notation For s = (s 0,s 1,) N = { (x i ) i 0 N 0 x i > 0 finitely often } and z N 0, we use P s for probabilities conditioned upon the event that the initial generation consists of s k cells hosting exactly k parasites for k = 0,1,, ie T 0,k = s k for k = 0,1, P z for probabilities given that initially there is one cell which contains z parasites, ie N = 1 and Z = z P = P 1 Gerold Alsmeyer Host-parasite co-evolution 21 of 26

22 The number of parasites process Theorem (Extinction-explosion principle) The parasite population of a BwBP either dies out or explodes, ie for all s N = { (x i ) i 1 N 0 x i > 0 finitely often } P s (Z n 0) + P s (Z n ) = 1 Gerold Alsmeyer Host-parasite co-evolution 22 of 26

23 The number of contaminated cells process Theorem Let P(Surv) > 0 and P z := P z ( Surv) (a) If P 2 (T1 2) > 0, then P z(tn ) = 1 and thus Ext = {sup n 0 Tn < } P z -as for all z N (b) If P 2 (T1 2) = 0, then P z(tn = 1 fa n 0) = 1 for all z N Gerold Alsmeyer Host-parasite co-evolution 23 of 26

24 Almost sure extinction of contaminated cells Theorem (a) If P 2 (T 1 2) = 0, then P(Ext) = 1 if, and only if, ElogE(Z 1 N ) 0 or Elog P(Z 1 > 0 N ) = (b) If P 2 (T1 2) > 0, then the following statements are equivalent: (i) P(Ext) = 1 (ii) ETn 1 for all n N 0 (iii) sup n N0 ETn < (iv) ν 1, or ν > 1, Elogg Λ 0 (1) < 0 and inf 0 θ 1 Eg Λ 0 (1) θ 1 ν Gerold Alsmeyer Host-parasite co-evolution 24 of 26

25 A second size-biasing V 0 BP 0 V 1 BP 1 V 2 BP 2 V 3 Gerold Alsmeyer Host-parasite co-evolution 25 of 26

26 Want to know more? G Alsmeyer and S Gröttrup Branching within branching: a model for host-parasite co-evolution Stochastic Process Appl, 126(6): , 2016 Gerold Alsmeyer Host-parasite co-evolution 26 of 26