Probability of disease transmission to future generations

Question

Assume disease D is transmitted through a father (male) to his children. Let us assume that every family in a society have $C\geq1$ children. Let the father of a family have the disease D. What is the probability that disease D is transmitted to at least one person in the $N$-th generation after him $(N\geq1)$? $N=1$ represents the children of the father, $N=2$ represents the grandchildren, etc.

One can calculate the probability for a given $N$ and $C$, but I was wondering if it possible to find a general closed-form solution.

P.S.: The problem is equivalent to the problem of passing the family-name. In a society where the family-name is passed from father to children, what is the probability of having the family-name of a father in the $N$-th generation after him.

Answer 1

What you're referring to is sometimes called a Branching Markov Chain, or simply Branching Process.

To solve we'll introduce some notation.

Let $X_n$ be the number of diseased people at time(generation) $n$. $p_{d}$ be the probability of having $d$ offspring from a person with disease $D$. $u_n = Pr(X_n = 0)$. Our goal is to find $Pr(X_n > 0) = 1 - u_n$.

Using the law of total probability we can recursively find:

$u_n = \sum\limits_{d=0}^{C^n}p_d(u_{n-1})^d$.

Or more generally

$u_n = \sum\limits_{d=0}^\infty p_d(u_{n-1})^d$.

Allowing us to find the extinction probability at time $n$. From there we simply take it's complement to find the probability of $D$ existing at time $n$

Reference: An Introduction to Stochastic Modeling, Pinsky and Karlin

EDIT:

I've made an error and edited above. $p_d$ is the probability of having $d$ children who could inherit the disease. At $N = 1$ this is simply $C$, but at greater $N$ this some multiple of $C$ depending on previous generations.

$u_n$ is found recursively at each generation using the previous generations $u$ and the total law of probability.

Answer 2

I don't think it's possible.

I assume $C > 0$. I also suppose that repartition of male/female is balanced.

Let $q_N$ be the probability that the disease is not transferred to any person of the $N$-th generation. (This is the complementary of the probability you are looking for.)

Then:

• $q_1 = 0$
• $q_{N+1} = (0.5 + 0.5 \times q_N)^C$

This sequence converges (because it's bounded and monotonic) and the limit $q_{\infty}$ satisfies:

$$q_{\infty} = (0.5 + 0.5 \times q_{\infty})^C$$

I'm quite sure the solution of this equation has no closed-form in general.

That's not a complete answer of your question and I hope someone will be able to help you more on this.

Answer 3

As it has already been pointed out by @darthline and @b-maréchal, there is no closed-form solution to this problem and the answer should be found recursively.

For this specific example, one can find $q_N$ as the probability that the disease is not transferred to any person of the $N$-th generation. The probability of having the disease is then $p_N = 1-q_N$.

To find $q_N$ we have:

• $q_1 = 0$
• $q_2 = (\frac{1}{2})^C$
• $q_N = \sum_{i=0}^{C} {C \choose i} (\frac{1}{2})^C (q_{N-1})^i, \quad N\geq3$

($i$ represents the number of males among the children of a father)