The average age at which IDUs in Australia acquire HCV infection is their early to mid 20s^{53, 67, 68} and it is likely that IDUs who acquire HIV infection would do so a few years thereafter on average. Approximately *p*_{MSM}=65% of IDUs with HIV are men who have sex with men, *p*_{F}=7% are females and the remaining *p*_{M}=28% are heterosexual males^{54}. For IDUs with HCV infection, approximately *p*_{MSM}=5% of IDUs with HIV are men who have sex with men, *p*_{F}=32% are females and the remaining *p*_{M}=63% are heterosexual males^{54}. It is assumed that the average number of long-term sexual partners each IDU has after acquiring infection is *c*_{reg}=3-5. In addition, it is assumed that each heterosexual IDU would have an average of *c*_{cas}=5-10 casual short-term sexual partners (each with one penetrative act) after acquiring infection and male homosexual IDUs would have an average of *c*_{cas}=20-30 casual partners after acquiring infection. Most IDUs who share syringes tend to do so with sexual partners or close friends.^{69, 70} Therefore, some of the potential partners are not susceptible to transmission because they are already infected (and probably the primary source for the case in question); this complexity is not considered here. It is assumed that HIV transmission rates are *β*_{HL}=0.01 for heterosexual transmission for long-term partnerships, *β*_{ML}=0.1 for male homosexuals for long-term partnerships, and per-act probabilities of HIV transmission during casual partnerships are *β*_{FM}=0.0005 for female-to-male transmission, *β*_{MF}=0.001 for male-to-female transmission, and *β*_{MM}=0.01 for male-to-male transmission during unprotected sex.^{ 71-77} It is assumed that condom usage is *p*=80%,^{78} with efficacy of 95%.^{79-83} HCV transmission per sexual contact is assumed to be *β*_{HCV-a}=0.1% per act and *β*_{HCV-p}=2% per long-term partnership.^{84-86}

The average fertility rate in Australia is *f*=1.93 babies per woman over her lifetime^{87} and the median age of all mothers of births is approximately 31 years.^{88} Based on the average age at infection and the relatively similar infection ages for HIV and HCV, it is assumed that 75% of a woman's births occur after she acquires infection.^{88} The probability of mother-to-child transmission is *β*_{HIV-MTCT}=2% for HIV (with the use of antiretrovirals and Caesarean section)^{89-92} and *β*_{HCV-MTCT}=5% for HCV.^{93-95}

Therefore, the average number of secondary infections through sexual transmission or mother-to-child transmission per HIV infection is:

*S _{HIV}* =

*p*(

_{MSM}*c*+

_{reg}β_{ML}*c*) +

_{cas}β_{MM}*p*(

_{M}*c*+

_{reg}β_{HL}*c*) +

_{cas}β_{MF}*p*(

_{F}*c*+

_{reg}β_{HL}*c*+

_{cas}β_{FM}*0.75 fβ*)

_{HIV-MTCT}*(see below for the plain english version of the HIV equation)*

and the average number of secondary HCV transmissions expected per HCV infection is:

*S*=

_{HCV}*p*(

_{MSM}*c*+

_{reg}β_{HCV-p}*c*) +

_{cas}β_{HCV-a}*p*(

_{M}*c*+

_{reg}β_{HCV-p}*c*) +

_{cas}β_{HCV-a}*p*(

_{F}*c*+

_{reg}β_{HCV-p}*c*+

_{cas}β_{HCV-a}*0.75 fβ*)

_{HCV-MTCT}*(see below for the plain english version of the HCV equation)*

Substituting parameter estimates leads to 0.44 and 0.11 secondary HIV and HCV cases, respectively, for each primary infection. Top of page

## Plain english version of equation for secondary HIV transmission

The average number of secondary infections through sexual transmission or mother-to-child transmission per**HIV infection**is:

- the average number of long-term sexual partners each IDU has after acquiring infection times the transmission rate of unprotected male-to-male sex
- plus the average number of short-term sexual partners each IDU has after acquiring infection times the transmission rate of unprotected male-to-male sex
- times the number of IDUs with HIV who are men who have sex with men

**plus**

- the average number of long-term sexual partners each IDU has after acquiring infection times the transmission rate of unprotected heterosexual sex in long-term relationships
- plus the average number of short-term sexual partners each IDU has after acquiring infection times the transmission rate of unprotected male-to-female sex
- times the number of IDUs with HIV who are men who have sex with women

**plus**

- the average number of long-term sexual partners each IDU has after acquiring infection times the transmission rate of unprotected heterosexual sex in long-term relationships
- plus the average number of short-term sexual partners each IDU has after acquiring infection times the transmission rate of unprotected female-to-male sex
- plus the probability of mother-to-child transmission times 0.75 of the average fertility rate
- times the number of IDUs with HIV who are women who have sex with men.

## Plain english version of equation for secondary HCV transmission

The average number of secondary infections through sexual transmission or mother-to-child transmission per**HCV infection**is:

- the average number of long-term sexual partners each IDU has after acquiring infection times the transmission rate of HCV per long-term partnership
- plus the average number of short-term sexual partners each IDU has after acquiring infection times the transmission rate of HCV per sexual act
- times the number of IDUs with HCV who are men who have sex with men

**plus**

- the average number of long-term sexual partners each IDU has after acquiring infection times the transmission rate of HCV per long-term partnership
- plus the average number of short-term sexual partners each IDU has after acquiring infection times the transmission rate of HCV per sexual act
- times the number of IDUs with HCV who are men who have sex with women

**plus**

- the average number of long-term sexual partners each IDU has after acquiring infection times the transmission rate of HCV per long-term partnership
- plus the average number of short-term sexual partners each IDU has after acquiring infection times the transmission rate of HCV per sexual act
- plus the probability of mother-to-child transmission times 0.75 of the average fertility rate
- times the number of IDUs with HCV who are women who have sex with men