Ejemplo de consumo del servicio

One of the earliest household models developed was due to Longini and Koop-man (1982). The model by Longini and KoopKoop-man (1982) considers individuals in a population partitioned into independent households. Individuals in a given household can be infected by their family members as well as by other members of the population. The model also assumes that the transmission processes within a given household does not depend on the transmission dynamics of the entire community. Addy et al. (1991) studied generalized stochastic models involving a population partitioned into households which was applied on serologic data from two influenza epidemics. Becker and Dietz (1995) consider a household model for

highly infectious diseases, such as smallpox. They assume that once a member of a given household contracts infection, then every member of the household gets infected. The models were used to evaluate various vaccination strategies taking the household structures into account. Findings from Becker and Dietz (1995) sug-gested the use of different vaccination coverage for different household structures.

For example, it was observed that it is better to immunize randomly selected in-dividuals when the households are of equal size, while it is better to immunize all members of large households when the sizes of the households are unequal. Ex-amples of other works on the development of household epidemic models are due to House and Keeling (2008) and Goldstein et al. (2009).

Many of the results obtained for household disease models are asymptotic as n → ∞.

1.7.1 Household-based epidemics with two-levels mixing

Ball et al. (1997) introduced two levels mixing in household-based epidemic model and since then household models for the spread of infectious diseases have re-ceived a considerable attention, see, for example Ball and Neal (2004), Ball and Lyne (2001), Neal (2006), Britton and Neal (2010) and Longini et al. (2005).

Suppose we have n mutually exclusive households each of size h so that that the population size is N = nh. The two levels mixing household-based epidemic model of Ball et al. (1997) assumes that an infectious individual makes global contact and local contacts. A global contact is made with an individual chosen uniformly at random from the N (= nh) population at rate λ > 0. Similarly, a local contact is

made with an individual chosen uniformly at random from the n individuals in the infective’s household at a typically larger rate hβ > 0. Therefore, the individual to individual global and local infection rates are λ/N and β. Contacts are made at the points of mutually independent Poisson processes. The infectious period of different infectives are independent and identically distributed according to a ran-dom variable I with an arbitrary, but specified distribution. This model can also be generalized for various household structures including when the household sizes are unequal. Let h = 1, 2, . . . , denote the possible sizes of the households in the population. Let n_h denote the number of households of size h so that n =P

n=1h_n and N = P∞

n=1nh_n are the total number of households and the population size respectively. The models we analyse in this thesis are based on the concept of two levels mixing epidemics.

1.7.2 Need for Household based epidemic models

There are a number of reasons why models are developed. Developing models which capture the basic household structures of a population is needed to effec-tively analyse infectious disease data emanating from household level. According to Ball et al. (2015), household is the most crucial aspect of human society that can affect disease transmission. Contacts among individuals of a given household are longer and more frequent than with members of another household. Individu-als become ill after being infected by the members of their household or by other members of the community often stay at home making regular contact with their household members. Several control strategies are implemented and monitored on

household levels. A plausible household-based epidemic model will potentially pro-vide appropriate answers to certain public health questions, such as, ’who infects whom?’. Household models take into account heterogeneity in population behav-ior, which is a key determinant among the factors that determine the occurrence of major epidemic outbreak, how fast it spreads if it does occur and the number of individuals ultimately infected during the course of the epidemic. Moreover, household-based infectious disease models can suggest applicable control strate-gies given the household structures, for example in the administration of vaccines (Becker and Dietz, 1995). Other measures such as contact tracing and isolation of infected individuals (if necessary) are readily applicable through households.

Therefore, there is need to develop epidemic model which accurately captures the key transmission mechanisms of infectious diseases at household levels. In this thesis, we develop Bayesian inference methods on such models which capture the inherent structure in both human and animal populations, where in this case a household could be childcare facilities, workplaces, dwelling places for humans or animal holdings (farms). Our main focus is the so-called two levels mixing model of Ball et al. (1997).

1.7.3 Inference on household models

Despite the advances so far recorded by household based endemic models, only a few works are channeled on inference. Drawing inference from household based model is usually very complicated due to the high computational complexity in-volved. Household epidemic data are often very highly dependent especially for

temporal data where information is obtained from the same group of individuals over time. Also, as with most epidemic data, some processes are unobserved, for example, actual infection time, thereby giving rise to outbreak data that are only partially observed. Although the problem of high dependence among household outbreak data can be minimized by using simplified assumptions. For example assuming that the households are independent households (Addy et al., 1991) makes it possible for the likelihood function π(x|θ) to be expressed as the product of likelihood function of all the n households. That is, the dependence between households is broken by independence households assumption then the likelihood function of the data given the parameters is

π(x|θ) = π(x1, x2, . . . , xn|θ)

= π(x₁|θ) × π(x₂|θ) × . . . × π(x_n|θ) (independence)

=

n

Y

i=1

π(x_i|θ)

On the other hand, the problem of incomplete data is minimized by designing appropriate data imputation strategies, see,for example, O’Neill (2009) and Neal and Kypraios (2015). Most available literature on household based epidemic use the Markov chain Monte Carlo (MCMC) algorithms to sample from the target dis-tribution, example, Britton and O’Neill (2002), Cauchemez et al. (2004), O’Neill et al. (2000) and O’Neill (2009). In most practical situations, the likelihood func-tion in (1.7.1) is very complicated that the posterior distribufunc-tion can never be available in a closed form no matter what the choice of the prior distribution may

be. Other likelihood-free methods of inference on household epidemics have been developed, see, for example, Neal (2012).

In this thesis, we shall focus on developing inference methods on household-based SIS epidemics with respect to two different data forms which we shall introduce in Chapter 2.