We present a spatial model for the mean and correlation of highly dispersed count data, and apply it to individual-level counts of the nematode Wuchereria bancrofti, a parasite of humans which causes the disease lymphatic filariasis. Our model uses the negative binomial distribution, whose shape parameter is a convenient index of over-dispersion. Spatial association is quantified in terms of a characteristic length, which has an intuitive interpretation as the distance over which correlation decreases by half. Demographic surveillance and mapping enable us to include individual-level covariates such as age and sex. We discuss the distinctive features of our model and interpret the results in terms of the epidemiology of lymphatic filariasis and possible implications for control programmes.