Focusing on the epidemiology of HIV/AIDS in Nigeria, several studies have been published, (Tomori, 2004; Isiugo-Abanihe, 1994; Isiugo Abanihe, 1993; Nasidi & Harry, 2004; USAID, 2002; Canning et al, 2004a; Canning et al, 2004b), but virtually nothing has been done in the area of modeling the course of the infection overtime. Infectious disease data have two features that distinguished them from other data. They are highly dependent and the infection process is only partially observable (De Angelis, Day and Gill, 1998).

A consequence of these features is that the analysis of data is usually most effective when it is based on a model that describes aspects of the infection process Becker and Britton (1999). Again, an understanding of the magnitude and trajectory of the HIV/AIDS epidemic, as well as the uncertainty around the parameters is critically important both for planning and evaluating control strategies and for preparing for vaccine efficacy trials (Salomon, Gakidon & Murray, 2001). Mathematical models can become very useful tools in this area. Therefore modeling is an integral part of statistical work in HIV/AIDS research.

Apart from that, modeling exercise are aimed at making use of the available data (no matter how little) to provide information about the trend inherent in the course of the epidemic. Since in Nigeria, data on HIV/AIDS are scanty, a better insight can be provided if analysis of the data is based on estimate of statistical models whose assumption are realistic and with parameters defined to capture the situations peculiar to the locality. One of such statistical models is the back-calculation method that was first proposed by Brookmeyer and Gail (1986) for estimating infection distribution and for providing short-term projection of future AIDS case ( Tan, 2000).

Back-calculation is a method for estimating past infection rates from AIDS incidence data (Brookmeyer & Gail, 1994). The model has been used with some successes in several countries and situations (Brookmeyer & Damino, 1989; Brookmeyer & Liao, 1990, Brookmeyer 1991; Rosenberg, 1994 and Marion & Schecter, 1993). To apply the method to modeling work in sub-Saharan African countries, some modification has been introduced by Salomon and Murray (2001).

In this work, we proposed a generalized logistic model for the infection distribution of HIV/AIDS epidemic in the North Central Zone of Nigeria. We adopted the modifications made to the method of back-calculation as proposed by Salomon & Murray (2001). In section 2, we present details of the modifications and how we used it. Our results are presented in section 3 and in Section 4, we discussed our findings.

We made use of the data obtained from past HIV/AIDS Sentinel Surveillance Survey (HSSS) in the zone and published by the FMOH. The biannual HIV/AIDS Sentinel Surveillance Survey (HSSS) conducted by the Nigerian Federal Ministry of Health (FMOH) remains one of the most readily available strategies that provides information about the epidemic in the country as well as in the focused zone.

The Federal Ministry of Health through the department of public health, National AIDS/STI Control Programme, publishes biannual technical reports on the prevalence of HIV/AIDS in the various Sentinel sites (which are antenatal clinics – ANCs) in the six geopolitical zones of the country and data for each HSSS are made available in the Technical reports. It is believed that the data from the antenatal clinics most closely approximate prevalence levels in the adult population (Glys et al, 2005 and Salomon & Murray, 2001). In Nigeria, the Sentinel Surveillance Programme was based on the unlinked anonymous method, using the screening for Syphillis as entry point. All samples were stripped of identity, recorded by state, site, and age, properly stored and sent for HIV testing with Capillus and Genie II kits as specified in the protocol. All results and samples were documented and forwarded to the National Reference Laboratory (NLA) in Abuja. The samples were subjected to quality control in NLA (FMOH, 2003).

The survey which is conducted every two years started in 1991 with a total of 16 sites which could not be dived into urban and rural. The most recent survey was conducted in 2005 with a total of 160 sites (86 urban-IMT and 74 rural-OMT), 30 of which originate from the North Central Zone (FMOH 2005).Although population based prevalence surveys would be the most useful, they have not been undertaken in Nigeria, due to cost and logistics.

At the earlier phase of the HSSS, sites, rather than being identified as urban and rural, were identified as major town (MT) and outside major town (OMT) respectively (FMOH, 2003), it is only in the 2005 HSSS that the former were used (FMOH, 2005).

Salomon & Murray (2001) adapted a model for the incidence of HIV from the original back-calculation framework (equation 1) by focusing on HIV seroprevalence data, rather than AIDS notification.

AIDS diagnosis rate at time t =∫₀ ^t (HIV infection rate at time s) x Pr (incubation time =t-s)ds. Which is equivalent to

Figure 1

Where a(t) is the number of AIDS cases diagnosed at time t, i(s) the infection rate at time s and f(τ) is the probability density function of the time from HIV infection to AIDS Diagnosis (the incubation period distribution, which are estimates from cohort studies of HIV- infected persons) and is assumed that this follows a Weibull distribution. (Salomon & Murray, 2001; Salomon, Gakidou & Murray, 2001; Nishiura et al, 2004; and Srinivasa Rao, 2003).

Therefore, if both the AIDS diagnosis rate and the incubation time distribution were known exactly, the underlying infection process could be reconstructed. The estimated infection process can then be used together with the incubation time distribution to predict future AIDS cases.

The basic idea of the original back-calculation is to use AIDS incidence data together with an estimate of the incubation period distribution to reconstruct the numbers of individuals who must have been previously infected in order to give rise to the observed pattern of AIDS incidence (Brookmeyer & Gail, 1994). But AIDS Incidence data are practically difficult to come by in Nigeria. Consequently, the foundation of our model was the relationship between prevalence, incidence, and survivorship over time for infected individuals. The discrete form of equation (1) is given by:

Defining t = 0 as the first year of the epidemic, the number of HIV-infected individuals at time t is equal to the total number of individuals who were infected before time t and are still alive at time t:

Figure 2

where Ncprev(t) is the prevalence of infected people at time t expressed as an absolute number, NCInc(s) is the number of new infections occurring between time s and (s+1), andf(τ) is the probability that an individual will survive at least τ years after being infected.

As noted in Salomom & Murray (2001), half a year is subtracted from the progression to AIDS' function, under the assumption that the average moment of infection within a given time period is the midpoint of that period, e. g. prevalence at year 10 in the epidemic would include those individuals infected during the ninth year who have endured an average of 0.5 year's mortality risk, plus those individuals infected during the eighth year who have endured an average of 1.5 years of mortality risk, and so on.

Expressing equation (2) in proportion rather than in absolute value we have:

Figure 3

where Pop(t) is the size of the population at time t, Pop(s) is the size of the population at time s, assuming that the population size is constant over time. Equation (2) may be further simplified as follows:

Figure 4

where NCPrev_r(t) is the proportion of the population who have prevalent infection at time t and NCInc_r(s) is the incidence of new infection between time s and (s+1), expressed as a proportion of the population at time s. If the population size changes over time, then

Figure 5

We considered a point process of new HIV infections and assume that the distribution of the incubation periods (from infection to AIDS diagnosis) is independently and identically distributed random variables with probability density function PDF (t). Let t be the period between infection and AIDS diagonis and f(t) be the PDF of this incubation period t. We always describe the PDF of the incubation period of HIV infection with the use of weibull disbribution (Anderson et al, 1991; Nishiura et al, 2004; Brookmeyer & Gail, 1994 and Wai-Yaum, 2000). This could be expressed as follows;

Figure 6

where α and β are the shape and scale parameters respectively.

The logistic model has an age long application in epidemic modeling of infections disease with particular reference to HIV/AIDS. For details see Haberman (1990), Fuhrer, (1998), Hendriks et al (1992, Taylor (1989), Broo kmeyer and Damiano (1989), Arzrouni (2004), Rosenberg, Gail and Pee (1991), Daley and Gani (2005). The generalized logistics model is given by:

Figure 7

where µ is the location parameter and s > 0 is the scale parameter. Based on the above backgrounds we propose a four-parameter logistic model, equation (8), for the incidence of HIV/AIDS in the North Central Zone of Nigeria.

Figure 8

where NCInc_R(s) is the incidence of new infection between time s and (s+1) and α_NC is the rate /level of intervention activities (a small value of α_NC caused the incidence to increase rapidly and vice versa).β_NC is the initial proportion of adults population that is at risk of infection (it determines the peak of the epidemic (UNAIDS, 2003)), µ_NC is the incidence rate/pattern adjustment parameter (it determines the pattern of infection overtime, higher value of which implies faster pattern of infection and vice versa and π_NC > 0 is a parameter incorporated to cater for the behaviour of the entire curve in response to the intervention program (large value of π_NC stretches out the curve and hence implies low incidence, which further implies high level of intervention activities and vice versa).

Our model overestimated the incidence rate, particularly, prior to the peak incidence a further modification of equation (8) was necessary, which yields the parametric model for the incidence curve of the HIV/AIDS in the North Central Zone given by the four-parameter model:

Figure 9

where all the parameters are as defined earlier, in equation (8) above.

Stage Modeling (combining models prior to and after a specified time point e.g. treatment initiation, awareness campaign etc. is a usual practice in HIV/AIDS modeling see Solomon & Wilson (1990), Salomon & Murray (2001) and Brookmeyer and Liao(1990) for details.

The authors are grateful to the Management of the Redeemer's University, Nigeria for providing an enabling environment and research resources that have encouraged and enhanced the output of this work. We are also grateful to the Federal Ministry of Health, Nigeria (FMOH) and the National Action Committee on AIDS (NACA) for providing the data used for this study. We appreciate the contribution(s) of all the Academic Staff of the Department of Statistics, University of Ilorin, Nigeria, where this work was first presented as an unpublished seminar paper.