Age-dependent branching processes with incubation

Transcription

1 Age-dependent branching processes with incubation I. RAHIMOV Department of Mathematical Sciences, KFUPM, Box. 1339, Dhahran, 3161, Saudi Arabia We study a modification of the branching stochastic process which takes into account the incubation period of individual s life time. Consider a population of individuals of the same type who colonize a region. Assume that at time zero we have a single individual (ancestor) of age zero labelled by I. This individual lives a random time L I. At the end of the time interval L I the individual dies or leaves the region (emigrate) after laying a random number of ν I eggs (seeds). Each egg E, after a random incubation period τ E, independently of others generates ξ E individuals of age zero, with P {ξ E = 1} = 1 P {ξ E = 0} = p. This means that each egg generates one individual with probability p and will be destroyed with probability q, 0 q < 1, p + q = 1. These new individuals, independently each of other, behave in the same manner as the initial ancestor, i.e. live a random period of time and lay a random number of eggs, before they die or emigrate, and so on. We assume that pairs (L I, ν I ) and (τ E, ξ E ) are independent and for distinct I and E are independent copies of some pairs (L, ν) and (τ, ξ) respectively. It is known that susceptible-infectious-removed (SIR) epidemic model can be approximated by branching processes, when the initial number of susceptible individuals is large, (see Andersson and Britton [1], p.). In the framework of the epidemic models L and ν may be understood as the infectious period and the number of contacts during the infectious period of a single infective individual. Naturally, the variable τ is the incubation period and q may be considered as immune rate or as the rate of vaccination. 1

2 We denote G 1 (t) = P {L t}, G (t) = P {τ t}, t 0, P k = P {ν = k}, k 0. Assume that p > 0 to exclude the trivial case, when the process will extinct in the first generation. Realizations of the process are given by vector X(t) = (X 1 (t), X (t)) with X 1 (t) being the number of individuals (the number of infectious individuals) and X (t) being the number of eggs (the number of individuals who had a contact with an infectious individual). The process X(t) can be considered as a multi-type age-dependent process with types of individuals T 1 and T. Individuals of type T 1 generate only individuals of type T and vice versa i. e. evolution of the process has the form of transformations T 1 T and T T 1. The components of the vector X(t) are, naturally, the numbers of individuals of types T 1 and T at time t. 1. Subexponential case In the case when the Malthusian parameter exists asymptotic properties of process X(t) can be derived using results from the theory of multi type processes. However in mortal processes, which is the case in most epidemic models, the Malthusian parameter may do not exist. In this case the study of the process requires more delicate analysis and needs additional restrictions on the life time distributions. These restrictions define a class of so called subexponential distributions, which have tails that decay at a slower than exponential rate [3]. We note that this class includes most important for applications distributions, such as, Weibull, with decreasing hazard function, Log-normal, Log-logistic, Pareto and some of other heavy tailed distributions. Possibilities of using the heavy tailed distributions in modelling of the incubation period of infectious diseases, including HIV or AIDS, discussed in Chapter of [5]. This situation, in particular, explains the interest in study of the family of subexponential distributions by a number of authors, who have investigated various aspects of the family. We note [4] (and references therein), where an example, demonstrating that the exponential family is not closed under convolution. In this note we obtain certain properties of subexponential distributions related to the finite number of convolutions. In particular we describe a subclass of the subexponential family, which is closed with respect to convolution. Based on these results we study iting behavior of the process X(t) as t, when the Malthusian parameter does not exist.

3 Let A(t), t [0, ) be cumulative distribution of a positive random variable. Following [3] we define family R n, n, of distributions such that 1 A n (t) t 1 A(t) = n. (1) We note that, as it was proved in [3], R R n for any n. Distributions belonging to the class R are called subexponential, describing the property of their tail which decay at a slower than exponential rate (see [], p.147, for example). We obtain certain properties of subexponential distributions. Let τ 1 and τ be independent nonnegative random variables and A i (t) = P {τ t}, i = 1,. We denote C(t) := (1 A 1 (t))/(1 A (t)). Assume that A i (t), i = 1, be such that there exist it C(t) = C [0, ]. () t Let R be a subclass of R such that for each pair A i (t) R, i = 1, the condition () is satisfied. The following result is important in the proof of main theorems. We denote A(t) = A 1 A (t). Lemma 1. If A i (t) R, i = 1,, then for each i 1 1 A A i (t) t 1 A(t) = C + i. (3) As it was mentioned earlier, R is not closed under the convolution. The following result shows that the subclass R is closed with respect to the convolution. Lemma. a) If A i (t) R, i = 1, and () is satisfied, then A(t) R. b) The subclass of subexponential distributions R is closed with respect to the convolution. Now we consider the following moments of the process A i j(t) = E[X j (t) X(0) = ε i ], B i jk(t) = E[X j (t)(x k (t) δ jk ) X(0) = ε i ], where i, j, k = 1,. The matrix M α = (a ij, i, j = 1, ), where a ii = 0, i = 1, and a 1 = m 0 e αu dg 1 (u), a 1 = p 3 0 e αu dg (u),

4 plays an important role in the study of asymptotic behavior of the process. The Malthusian parameter α of the process is defined from condition ρ α = 1, where ρ α is the Perron eigenvalue of M α. Since ρ α = (a 1 a 1 ) 1/ and the random variables L and τ are independent, the Malthusian parameter is the root of equation mpee α(l+τ) = 1. (4) Note that, if mp = 1, then α = 0 and, if mp > 1, then α > 0. When mp < 1, then α may not exist. But, if it does exist, then α < 0. The right and left eigenvectors of M α corresponding to the Perron eigenvalue are U α = ( ρα a 1 + ρ α, a 1 a 1 + ρ α ) T, V α = ( a1 + ρ α ρ α, ) a 1 + ρ α. a 1 Note that U T α1 = 1, V α U α = 1, where 1 T = (1, 1). Assume that there exists c [0, ] such that 1 G (t) = c. (5) t 1 G 1 (t) We also denote G = G 1 G, a i = δ 1i + δ i c and b i = δ 1i cm + δ i p. We put by definition c(1 + c) 1 = 1, when c =. Theorem 1. Let mp < 1, G i (t) R, i = 1, and (5) is satisfied. Then for i, j, k = 1, a) t A i j(t) 1 G(t) = δ ija i + (1 δ ij )b i (1 + c)(1 mp). b) If in addition σ (0, ), then for i, j, k = 1, t B i jk (t) 1 G(t) = 0. Now we provide a it theorem which gives asymptotic behavior of the non-extinction probability and the iting distribution of the process conditioned on non-extinction. We denote for vectors s = (s 1, s ) and x = (x 1, x ) the quantity s x = s x 1 1 s x. 4

5 Theorem. If mp < 1, σ (0, ), G i (t) R, i = 1, and (5) is satisfied, then a) b) t Q i (t) 1 G(t) = a i + b i (1 + c)(1 mp) ; t E[sX(t) X(t) 0, X(0) = ε i ] = a is i + b i s j. a i + b i Remark. It follows from Theorem that, if the process does not become extinct, in the long run with probability one a single individual is alive. It is the individual of the initial type with probability a i (a i + b i ) 1 and of the opposite type with probability b i (a i + b i ) 1. Recall that a i = δ 1i + δ i c, b i = δ 1i cm + δ i p. In terminology of epidemics the results illustrate the following situation. By preventive measures, such as isolating of infectives or increasing of the immunization rate one can ensure that mp < 1, which leads to extinction of the epidemic with probability one. However, if an epidemic initiated by a single infective does not cease, then in the long run one infective may exist with probability (1 + cm) 1. With probability cm(1 + cm) 1 an individual who had a contact with an invective may exist in the long run.. The extinction time Now we consider an important variable related to survival of the process, namely the time to extinction. It is defined as T i 0 = min{t : X(t) = 0 X(0) = ε i }, i = 1,. The time to the extinction measured by the number of generations can similarly be defined as N i 0 = min{n : X n = 0 X 0 = ε i }, i = 1,. If N0 i = n, then there is at least one individual α in n 1th generation. Therefore the survival time T0 i of the process equals to the sum of the life time of α and the life times of all parents of α. Thus we obtain the following relationship between T0 i and N0: i T0 1 = N 1 0 / i=1 (L i + τ i ), if n is even, (N 1 0 1)/ i=1 (L i + τ i ) + L 0, if n is odd, 5

6 where L i, τ i, i 0 are independent random variables such that L d i = L, τ d i = τ and d means equality of distributions. Similarly we find N 0 / T0 i=1 (L i + τ i ), if n is even, = (N 0 1)/ i=1 (L i + τ i ) + τ 0, if n is odd. Since the life times and offspring numbers of the individuals are independent, when EN i 0, EL and Eτ are finite, we obtain: ET 1 0 = EL + Eτ EN EL Eτ P {N 1 0 is odd}, (6) ET0 EL + Eτ = EN Eτ EL 0 + P {N0 is odd}. (7) Now we focus our attention on the distribution of N0. i We obtain exact formulas for the distribution of N0, i i = 1,, in the case, when the offspring generating function has the form of a linear fractional transformation: Φ(s) = α + βs 1 δs, (8) where 0 δ < 1, and Φ(s) is the probability generating function of P k, k 0. Proposition. If Φ(s) has the form of (8) and mp 1, then a) P {N 1 0 k} = 1 (pδ α qβ)(pδ (α + qβ) k 0) 1 ; b) P {N 1 0 k + 1} = 1 (pδ α qβ)(pδ + 1 k 0) 1 ; c) P {N 0 k} = 1 (δ q pα)(δ (q + pα) k 0) 1 ; d) P {N 0 k + 1} = 1 (δ q pα)(δ + k 0) 1, where 0 = p 1 (α + β)/(1 α), 1 = (1 p 0 ) 1 (pp 0 δ α qβ), = p 1 (qδ q pα) and k = 0, 1,,... Similar formulas are also obtained in the case of mp = 1. Now we consider some particular cases of the offspring distribution. 6

7 Example 1. Let us consider the Bernoulli offspring distribution, i.e. Φ(s) = p 0 +p 1 s. Then from Proposition we obtain that P {N0 i > k} = (pp 1 ) k, P {N0 1 > k + 1} = p 1 (pp 1 ) k and P {N0 > k + 1} = p(pp 1 ) k. Therefore we find, when pp 1 < 1, EN 1 0 = n=0 P {N 1 0 > n} = 1 + p 1 1 pp 1. (9) Since P {N 1 0 = k + 1} = p 0 (pp 1 ) k, we get P {N 1 0 is odd} = p 0 (1 pp 1 ) 1. Hence we conclude from this and relation (6) that ET 1 0 = (EL + Eτ)(1 + p 1) (1 pp 1 ) By similar arguments we obtain from (7) + p 0(EL Eτ) (1 pp 1 ) ET 0 = (EL + Eτ)(1 + p) (1 pp 1 ) + q(eτ EL) (1 pp 1 ). Example. Let now the offspring distribution be geometric i.e. p k = d k (1 d), 0 < d < 1, k = 0, 1,,... Then Φ(s) = 1 d 1 ds, m = d 1 d. In this case we obtain from Proposition 4 that, if mp 1, then P {N 1 0 k} = 1 P {N 1 0 k + 1} = 1 1 (mp) 1, (10) 1 (mp) k 1 1 (mp) 1, (11) 1 + m 1 (1 (pd) 1 )(mp) k Using the Proposition one can derive similar formulas for N 0. Example 3. The rate of vaccination (proportion of vaccinated individuals in the population) is an important parameter in the preventive medicine. The formulas (10) and (11) allow to compute desired rate of vaccination to have the epidemic ceased before a given generation with a given probability for a given mean number of contacts. For a numerical example, if the mean number of contacts is 4, what should be the vaccination rate to the epidemic 7

8 cease before third generation with probability say 0.95? We denote by N the extinction generation number of the population of infective individuals. Since in our model the infective individuals correspond to generations labelled by even numbers, we obtain from equation and formula (10) the following: P {N } = P {N 1 0 4} = (4p) 1 = 0.05, 1 (4p) 3 which is equivalent to ((4p) ) = From this we find p = Consequently we conclude that the vaccination rate should be q = , i. e. almost 93.5 per cent of individuals must be vaccinated. For illustration we provide the values of the vaccination rate for different mean numbers of the contacts in Table 1. m q Table 1. Vaccination rate for different mean numbers of the contacts. Example 4. Now we consider an example of spread of infections such as measles and mumps in vaccinated school populations presented in papers Nkowane et al. (1987) and Gustafson et al.(1987). In these papers the authors identified four generations of spread, for highly vaccinated populations. Using our results we can determine the probability that a single defective generates an outbreak of more than four generations depending on the rate of vaccination. Let the mean number of contacts during the infectious period be 4. Let again N be the extinction generation number of the population of infective individuals. In this case we have P {N > 4} = P {N 1 0 > 8} = Table gives some numerical examples. 1 (4(1 q)) 1 1 (4(1 q)) 5. 8

9 q P{N > 4} Table. Change of the probability with the vaccination rate. Table shows, if the vaccination rate is less than 0.6 outbreaks of more than four generations are more likely. Around 1 percent of importations will lead to such outbreaks, if the vaccination rate is 0.8. The geometric distribution of the number of contacts is appropriate if the duration of the infectious period is an exponential random variable and contacts occur at fixed intervals. In some applications modified geometric distribution, in which zero has not necessarily the probability 1 d may be appropriate. It can be given as P k = b(1 d)d k 1, k = 1,,... and P 0 = 1 b. Note that its generating function has also form of (8) and the the Proposition is applicable for this distribution as well. Acknowledgment. These results are part of the project No FT-006/03 funded by King Fahd University of Petroleum and Minerals. The author is indebted to KFUPM for excellent research facilities. REFERENCES 1. Andersson H., Britton T Stochastic Epidemic Models and their Statistical Analysis, Springer, Ser. LN in Statistics 151, New York.. Athreya K., Ney P Branching Processes. Springer, New York. 3. Chistyakov V.P A theorem on sums of independent positive random variables and its applications to branching random processes. Theory of Probab. Appl. V.9, Leslie J. R On the non-closure under convolution of the subexponential family. J. Appl. Probab. 6, Mode Ch., J. Sleeman C., K Stochastic Processes in Epidemiology, World Scientific, Singapore-New Jersey-London-Hong Kong. 9