# Force of infection and analysis of 'static' incidence

Page last updated: 2009

Based on these factors, we formulated a mathematical expression for the 'force of infection', which refers to the dynamic rate at which susceptible individuals become infected. The force of infection used in this analysis was developed by first considering a static and homogeneous population of N IDUs and was then adapted to include heterogeneous and dynamic features. In a homogenous population, if each IDU injects an average of n times per year, a proportion, s, of IDUs share their syringes with others in a proportion, q, of their injections, and sharing occurs in groups of m people, then the total number of 'sharing events' in the population per year is (Nnsq)/m . The total number of expected transmissions will be this number multiplied by the average number of transmissions per 'sharing event'. If the prevalence in the population is P, then the probability of r infected people in a sharing group of size m is (rm) Pr (1 - P)m-r, using standard binomial theory. If the group members inject using the shared syringe in random order, then an average of (m - r)/(r - 1) uninfected people will inject before the first infected person (and between each infected person). Therefore, in each sharing event an average of m - [(m - r)/(r + 1)] - r = [(rm - r2)/(r + 1)] uninfected people will use a syringe after an infected person has used it. If a shared syringe is used δs times before disposal then ms syringes are used in each 'sharing event' and the average number of uninfected people in the group to use the same syringe after an infected person becomes [(rm - r2)/(r + 1)][(δs)/m] . If the probability of infection from a contaminated syringe per use is β, but transmission is reduced by an effectiveness of εc through syringe cleaning and cleaning occurs before a proportion, pc, of shared injections, then each susceptible person could acquire infection with probability (1 - pcεc)β if using a contaminated syringe. Therefore, the expected number of transmissions in a given sharing group (or probability of a transmission occurring) is Then the total number of transmissions expected each year, or incidence (I), is (equation 1)

The reader is referred to reference 63 for details of thorough analyses of this static expression, applied to Australian IDUs. Below are summary results from these analyses.

The expected reproduction ratio, R, per IDU was calculated for HIV and HCV as a function of the average duration of injecting post-seroconversion (Figure A.1): if each IDU injects for an average of D years after seroconversion, then the average number of secondary cases per IDU is (equation 2)

An epidemic is sustained if R is greater than one,64 implying that each infected person is associated with at least one secondary transmission on average. It was found that the threshold duration of injecting post-seroconversion required to sustain an epidemic is 11.6 (7.0-22.4, IQR) years for HIV and 2.3 (1.8-3.2, IQR) years for HCV (Figure A.1). Based on behavioural data54, 65 it is reasonable to assume that the average duration of injecting post-HCV seroconversion is approximately ten years. This is considerably greater than the threshold of 2.3 years required to control HCV incidence. In contrast, the duration of injecting for HIV-infected IDUs, post-seroconversion, is assumed to be much less than for HCV (less than ten years) and thus less than the critical 11.6 years required to control HIV incidence.
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To identify factors that could provide effective targets for intervention a sensitivity analysis was conducted, by means of calculating partial rank correlation coefficients40 between incidence and the sampled model parameters (results not shown). It was determined that the number of times each syringe is used before disposal is the most sensitive behavioural factor in determining the incidence of both HIV and HCV infection, followed by the percentage of injections that are shared. Therefore, the expected change in incidence for HIV and HCV was investigated in relation to the frequency of shared injections and the average number of times each syringe is used (Figure A.2).

The number of times each syringe is used may be decreased by greater dissemination of sterile syringes through NSPs. The number of syringes distributed through NSPs has remained relatively constant over the last decade (see Table B.1), suggesting that saturation levels have been reached. However, there is also reason to believe that there are opportunities for public sector NSP services to increase client reach. It is difficult to estimate the proportion of all IDUs that access NSPs, however, the recent National Drug Strategy Household Survey revealed that only 51% of those who had injected in the last 12 months usually obtained their injecting equipment from public sector NSPs66. Structural and policy factors may limit access to current NSP services. With the exception of pharmacy-based services, few NSPs operate into the evening or are open on weekends. Whilst syringe dispensing machines operate 24 hours a day, these not are operational throughout Australia. There are also limits on the quantity and range of syringes freely available at some NSP services. Secondary exchange of sterile needles and syringes (from one IDU to another) is prohibited in most states and territories, and there are some locations where there is demand for NSP, but where services are not well developed. These factors suggest that syringe distribution in Australia is limited by supply rather than demand, and that increased coverage is possible.

If K syringes are distributed each year and a proportion ω of all syringes are not used, then the number of syringes distributed that are used is P(1 - ω). The number of syringes used for individual injecting episodes among non-sharing IDUs is [nN(1 - s)]/(δp). Similarly, the total number of syringes used for individual injecting among all sharing IDUs is [n(1 - q)sN]/(δp). and the total number of syringes used in sharing events is (nqsN)/(δS). Therefore, (equation 3)

defines a relationship between the total number of syringes distributed and the use of syringes in this mathematical model (equation 2). Changes in the number of syringes distributed are likely to change any, or all, of the following factors in a way that is consistent with equation 3: the proportion of syringes that remain unused (ω) , the proportion of injections that are shared (q) , or the average number of times each syringe is used (in shared (δS) or individual (non-shared) injections (δp)). Changes to ω and δp will not influence transmission levels but changes to q and δS could potentially result in large reductions in incidence. It could be speculated that increased syringe coverage is most likely to influence a decrease in the number of injections per syringe (for both personal and shared syringes). Therefore, equation 3 was used to estimate the change in the average number of injections per syringe used in both individual and shared injections, assuming the same percentage increase or decrease for both, according to a change in the total number of syringes distributed. The new values for the usage per syringe (δp and δS) were then used in equation 1, and all other parameters were sampled independently from their original distributions as defined in Table B.1. This was used to estimate the expected incidence of HIV and HCV based on changes in syringe distribution (Figure A.3). It should be noted that very large increases in syringe distribution are likely to be infeasible and unrealistic. It is also important to acknowledge that other relationships between incidence and syringe distribution could be expected if syringe distribution affected other factors in equation 3. However, Figure A.3 does demonstrate that it greater NSP distribution of syringes may lead to reductions in incident cases of HIV and HCV and that if there was a decline in syringe distribution through NSPs then significant increases in incidence could be expected. It is likely that the provision of NSP services has contained the HIV epidemic among IDUs. Top of page

## Figure A.1: The average number of secondary cases of HIV (orange) and HCV (blue) transmission per IDU versus the duration of injecting post-seroconversion

The solid lines refer to median simulations and the dashed line refers to one secondary infection. ### Text version of Figure A.1

Figures in this description are approximate as they have been read from the graph.

For HIV:
• the average (median) number of secondary cases per IDU increases gradually as the years of injecting post-seroconversion pass, reaching 1 at 11.6 years and 2.4 at 25 years.
• the average number of secondary cases per IDU is 1 after 11.6 years of injecting post-seroconversion.
For HCV:
• the average (median) number of secondary cases of increases rapidly as the years of injecting post-seroconversion pass, increasing from 1 at 2.3 years to 11 at 25 years
• the average number of secondary cases per IDU is 1 after 2.3 years.

## Figure A.2: The simulated number of annual (a) HIV and (b) HCV transmissions among IDUs in Australia versus the percentage of injections that are shared and the average number of times each syringe is used before disposal

The dashed lines refer to current levels of sharing and syringe use. Top of page  ### Text version of Figure A.2

Figures in this description are approximate as they have been read from the graph.

Figure A.2 shows the simulated number of transmissions among IDUs versus the percentage of injections that are shared and the average number of times each syringe is used before disposal.

The greater the percentage of injections that are shared and the greater number of times each shared syringe is used before disposal, the larger the simulated number of transmissions.

For HIV, the current level of injections that are shared (15%) and the current number of times each shared syringe is used (2.6) results in 30-40 transmissions among IDUs in Australia annually.

For HCV, the current level of injections that are shared (15%) and number of times each shared syringe is used (2.6) results in 10,000 HCV transmissions among IDUs in Australia annually. Top of page

## Figure A.3: Scatter plots of the simulated number of annual (a) HIV and (b) HCV transmissions among IDUs in Australia versus the number of sterile syringes distributed in Australia

This is assuming that syringe distribution changes the average number of times each syringe is used before disposal. The blue dots are results from 1000 simulations, the red curves represent the median parameter values, and the black dashed lines refer to current levels of syringe distribution.  ### Text version of Figure A.3

Figures in this description are approximate as they have been read from the graph.

Figure A.3 consists of two graphs showing the simulated number of annual transmissions among IDUs in Australia versus the number of sterile syringes distributed in Australia.
For HIV, the simulated average (median) number of annual transmission among IDUs in Australia decreases from 120 incidences with 10 million syringes distributed a year to 18 incidences with 60 million syringes distributerd a year. The current level of syringe distribution (30 million per year) results in 35 incidences of HIV transmission.
For HVC, the simulated average (median) number of annual transmission among IDUs in Australia decreases from 40 incidences with 10 million syringes distributed a year to 5 incidences with 60 million syringes distributerd a year. The current level of syringe distribution (30 million/year) results in 10 incidences of HCV transmission.