^{1}

^{*}

^{1}

^{*}

^{1}

^{*}

Discrete epidemic models are applied to describe the physical phenomena of spreading infectious diseases in a household. In this paper, an attempt has been made to develop a modified epidemic chain model by assuming a beta distribution of third kind for the probability of being infected by contact with a given infective from the same household with closed population. This paper emphasizes mainly on developing the probabilities of all possible epidemic chains with one introductory case for three, four and five member household. The key phenomenon towards developing this paper is to provide an alternative model of chain binomial model.

The chain binomial models (Bailey, 1975) [

A more detailed comparison of the fits provided by the two models namely, Reed-Frost chain binomial model and the stochastic version of the Kermack-McKendrick epidemic model, is not attempted by Becker for any epidemic chain model developed by assuming any other kind of Beta distribution for the probability of being infected by contacting with a given infective from the same household. In order to make a more exhaustive comparison, we formulate a modified epidemic chain model by assuming a beta distribution of third kind for the probability of being infected by contacting with a given infective from the same household.

An epidemic chain model was developed by Becker in 1980 [

Consider a disease say, influenza, which is able to spread from person in a household. Let the time at which the disease is introduced to the household as the time origin and suppose that the outbreak within the household is over by time t^{*}. Assume that during the time interval (0, t^{*}) the chance of infection from outside the household is negligible compared with the chance of infection from within the household. Following a latent period of random duration, an infected person becomes infectious and remains so until his removal by isolation, death or recovery, with immunity for the duration of the outbreak. The probability that a given infected person A, say, transmits the disease to any given susceptible during the time increment (t, t + h) is assumed to be

So, ^{*}) into n small time increment ^{*}) is

which tends in the limit as

In particular case when A assumes the constant value

It is not always possible to determine which infective is responsible for a certain infection. It is easier by making use of the gaps between cases, to partition the cases of a household into generations: the susceptible infected by direct contact with the introductory cases are said to make up the first generation of cases; the susceptibles infected by direct contact with first generation cases are said to make up the second generation and so forth. By an epidemic chain we mean the enumeration of the number of cases in each generation.

Thus, we should use 1-2-1-0 to denote the chain consisting of one introductory case, two first generation cases, one second generation case and no cases in later generation.

1-2-1-0

1: Introductory case

2: First Generation case

1: Second generation case

0: Third Generation case

Corresponding to a given infective A, the conditional probability that r out of k susceptibles of the household escape infection by A is

given the infection potential I of infective A.

Corresponding to a given infective A, unconditional probability that r out of k susceptibles of the household escape infection A is given by

Becker (1980) has considered

and

Now, let us consider

Then

of being infected by contact with a given infective from the same household. So the higher herms of

neglected.

For the practical application the term

The above term is resulted after applying the test for convergence of the infinite series

test and succeed when the Ratio test fails. For the test of convergence of the infinite beta series, the Raabe’s test is applied when the test fails for the Ratio test.

Further,

where

Then expression (i) using equation (ii) is given by

To illustrate the computation of the probabilities associated with the different possible epidemic chains we consider the chain 1-1-2-0 in a household of size five including one introductory case. The probability of this chain, conditional on the probabilities

The unconditional probability is obtained by taking the expectation of this conditional probability and using the fact that

Since,

The probabilities of the possible type of chains for household of size three with one introductory case for the probability assuming beta distribution of third kind is given in

The paper aims to develop a probability model of infectious diseases which is an alternative to the epidemic chain binomial model of Becker (1980) [

Type of chain | Probability assuming beta type I (Becker, 1980) [ | Probability assuming beta type III |
---|---|---|

1-0 | ||

1-1-0 | ||

1-2 | ||

1-1-1 |

Type of chain | Probability assuming beta type I (Becker, 1980) [ | Probability assuming beta type III |
---|---|---|

1-0 | ||

1-1-0 | ||

1-2-0 | ||

1-1-1-0 | ||

1-3 | ||

1-2-1 | ||

1-1-2 | ||

1-1-1-1 |

Type of chain | Probability assuming beta type I (Becker, 1980) [ | Probability assuming beta type III |
---|---|---|

1-0 | ||

1-1-0 | ||

1-2-0 | ||

1-1-1-0 | ||

1-3-0 | ||

1-1-2-0 | ||

1-2-1-0 | ||

1-1-1-1-0 |

1-4 | ||
---|---|---|

1-3-1 | ||

1-1-3 | ||

1-2-2 | ||

1-2-1-1 | ||

1-1-2-1 | ||

1-1-1-2 | ||

1-1-1-1-1 |

model than that of the epidemic chain binomial model of Becker, so the estimation procedure for the proposed model is also complicated as compared to the existing epidemic chain binomial model. However, we are in the process of illustrating the application of this method to the data on common cold for three, four, five member household with closed population in our next communication.

This work is financially supported by University Grants Commission, New Delhi, under UGC-BSR one time grant (No. F.19-145/2015(BSR)) and provided to the first author.

None.

Dilip C.Nath,Kishore K.Das,TandrimaChakraborty, (2016) A Modified Epidemic Chain Binomial Model. Open Journal of Statistics,06,1-6. doi: 10.4236/ojs.2016.61001