The model used in the analyses of this report extends the 'static' mathematical expression (equation 1, equation 2 and equation 3) by including time-dependent parameter estimates for all demographic parameters and simulating the dynamic model-based prevalence of HIV and HCV in the population. Various assumptions about the role NSPs and syringe distribution among heterogeneous groups of IDUs were also considered.

Furthermore, an extensive natural history model of HIV and HCV monoinfection or coinfection was developed to dynamically track the number of people in each HIV and HCV health state. A schematic diagram of compartments of the HIV and HCV transmission model for IDUs in Australia is presented in Figure A.4. The change in the number of people in each compartment was tracked mathematically by formulating a system of 473 ordinary differential equations, one for each compartment. One compartment represents IDUs who are not infected with HIV or HCV. Fifteen compartments represent IDUs who are monoinfected with HCV: in acute stage, fibrosis stages F0, F1, F2, F3, F4, and for each of these, whether they are untreated or receiving treatment. People infected with HCV who have advanced fibrosis can progress to clinical outcomes of liver failure, hepatocellular carcinoma, or may receive a liver transplant. It is assumed that individuals that progress to these three clinical outcomes no longer inject drugs.

Sixteen compartments represent IDUs who are monoinfected with HIV: individuals who become HIV-infected are initially untreated and are assumed to have a CD4+ T cell count above 500 cells per ěl, then will progress in their disease through categories according to CD4+ T cell levels (350-500 cells per ěl, 200-350 cells per ěl, and <200 cells per ěl); HIV-infected individuals may initiate antiretroviral therapy (for each CD4+ T cell category the number of individuals on effective first-line treatment, treatment failure, or effective secondline treatment are also tracked). This model also tracks potential co-infection of HIV and HCV, including all possible combinations of HIV and HCV disease states; however, it is assumed that HIV-infected individuals with CD4 counts less than 350 cells per ěl and on antiretroviral therapy will not also receive treatment for their HCV infection at the same time.

Thus, 205 ordinary differential equations are used to describe the co-infection of HIV and HCV among IDUs. This model also tracks the disease progression of individuals who have stopped injecting drugs but are infected with HIV and/or HCV. The number of equations is then doubled, plus the equation for uninfected but susceptible IDUs, leading to a total of 473 ordinary differential equations, one for each model compartment, to describe the number of people in each health state. The flows in the number of people between these compartments are due to biological, behavioural, clinical, or epidemiological parameters (specified in detail in Appendix B).

Below is the mathematical description of the model. The equations describe the change in the number of people in each health state for HIV and HCV monoinfection. The complete mathematical model also includes equations for each possible HIV and HCV coinfection combination, with the terms of the ordinary differential equations amalgamating the appropriate HIV and HCV terms.Top of page

Figure A.4: Schematic diagram of compartments of the HIV and HCV transmission model for IDUs in Australia

Text equivalent below for Figure A.4: Schematic diagram of compartments of the HIV and HCV transmission model for IDUs in Australia
Large image of Figure A.4 (GIF 123 KB)

Text version of Figure A.4

Top of page
Uninfected IDUs are initially infected with either HCV or HIV.

IDUs infected with acute HCV are either:
  • spontaneously cleared and returned to uninfected status
  • treated for acute HCV and returned to uninfected status
  • not treated for acute HCV and develop fibrosis (stages F0 to F4).
The IDUs with fibrosis can:
  • be treated for fibrosis at each stage, cleared of the virus and returned to uninfected status, or
  • develop liver failure and/or hepatocellular carcinoma, which can lead to a liver transplant.
IDUs infected with HIV are either:
  • not on ART and progress through four stages (CD4 more than 500, CD4 350-500, CD4 200-350, CD4 less than 200)
  • subsequently on ART which either:
    • maintains CD4 at more than 500
    • results in increased CD4 (effective first line therapy or effective second line therapy)
    • does not prevent CD4 from decreasing (treatment failure)
IDUs infected with HIV or HCV may be coinfected with both HIV and HCV upon which progression for each disease follows as described above, except IDUs spontaneously cleared or treated for HCV do return to unifected status, but are considered monoinfected with HIV.Top of page

Mathematical models

The mathematical description of the model is below:

Uninfected IDUs


Mathematical equation for change in uninfecteds. Text equivalent not available.
Large image of uninfected IDUs mathematical model (GIF 39 KB)

HCV-infected individuals


Mathematical equation for change in acute infecteds. Text equivalent not available.
Large image of change in acute infecteds equation (GIF 22 KB)Top of page
Mathematical equation for change in F0 infecteds. Text equivalent not available.
Large image of change in F0 infecteds equation (GIF 22 KB)
Mathematical equation for change in F1 infecteds. Text equivalent not available.

Large image of change in F1 infecteds equation (GIF 21 KB)
Mathematical equation for change in F2 infecteds. Text equivalent not available.
Large image of change in F2 infecteds equation (GIF 21 KB)Top of page
Mathematical equation for change in F3 infecteds. Text equivalent not available.
Large image of change in F3 infecteds equation (GIF 22 kB)
Mathematical equation for change in F4 infecteds. Text equivalent not available.
Large image of change in F4 infecteds equation (GIF 24 KB)
Mathematical equation for change in acute infecteds on treatment. Text equivalent not available.
Large image of change in acute infecteds on treatment equation (GIF 24 KB)Top of page
Mathematical equation for change in F0 infecteds on treatment. Text equivalent not available.
Large image of change in F0 infecteds on treatment equation (GIF 27 KB)
Mathematical equation for change in F1 infecteds on treatment. Text equivalent not available.
Large image of change in F1 infecteds on treatment equation (GIF 27 KB)
Mathematical equation for change in F2 infecteds on treatment. Text equivalent not available.
Large image of change in F2 infecteds on treatment equation (GIF 27 KB)Top of page
Mathematical equation for change in F3 infecteds on treatment. Text equivalent not available.
Large image of change in F3 infecteds on treatment equation (GIF 27 KB)
Mathematical equation for change in F4 infecteds on treatment. Text equivalent not available.
Large image of change in F4 infecteds on treatment equation (GIF 25 KB)
Mathematical equation for change in liver failure infecteds. Text equivalent not available.
Large image of change in liver failure infecteds equation (GIF 19 KB)Top of page
Mathematical equation for change in HCC infecteds. Text equivalent not available.
Large image of change in HCC infecteds equation (GIF 18 KB)
Mathematical equation for change in liver transplants. Text equivalent not available.
Large image of change in liver transplants equation (GIF 19 KB)

HIV-infected individuals


Mathematical equation for change in infecteds (CD4 greater than 500). Text equivalent not available.
Large image of change in infecteds (CD4 greater than 500) equation (GIF 32 KB)
Mathematical equation for change in infecteds (CD4 is more than 350 but less than 500). Text equivalent not available.
Large image of change in infecteds (CD4 is more than 350 but less than 500) equation (GIF 37 KB)Top of page
Mathematical equation for change in infecteds (CD4 is more than 200 but less than 350). Text equivalent not available.
Large image of change in infecteds (CD4 is more than 200 but less than 350) equation (GIF 38 KB)
Mathematical equation for change in infecteds (CD4 is less than 200). Text equivalent not available.
Large image of change in infecteds (CD4 is less than 200) equation (GIF 33 KB)
Mathematical equation for change in infecteds (CD4 is more than 500) during first treatment. Text equivalent not available.
Large image of change in infecteds (CD4 is more than 500) during first treatment equation (GIF 34 KB)Top of page
Mathematical equation for change in infecteds (CD4 is more than 350 but less than 500) during first treatment. Text equivalent not available.
Large image of change in infecteds (CD4 is more than 350 but less than 500) during first treatment equation (GIF 37 KB)
Mathematical equation for change in infecteds (CD4 is more than 200 but less than 350) during first treatment. Text equivalent not available.
Large image of change in infecteds (CD4 is more than 200 but less than 350) during first treatment equation (GIF 36 KB)
Mathematical equation for change in infecteds (CD4 is less than 200) during first treatment. Text equivalent not available.
Large image of change in infecteds (CD4 is less than 200) during first treatment equation (GIF 29 KB)Top of page
Mathematical equation for change in treatment failure infecteds (CD4 is more than 500). Text equivalent not available.
Large image of change in treatment failure infecteds (CD4 is more than 500) equation (GIF 37 KB)
Mathematical equation for change in treatment failure infecteds (CD4 is more than 350 but less than 500). Text equivalent not available.
Large image of change in treatment failure infecteds (CD4 is more than 350 but less than 500) equation (GIF 45 KB)
Mathematical equation for change in treatment failure infecteds (CD4 is more than 200 but less than 350). Text equivalent not available.
Large image of change in treatment failure infecteds (CD4 is more than 200 but less than 350) equation (GIF 46 KB)Top of page
Mathematical equation for change in treatment failure infecteds (CD4 is less than 200). Text equivalent not available.
Large image of change in treatment failure infecteds (CD4 is less than 200) equation (GIF 40 KB)
Mathematical equation for change in infecteds (CD4 is more than 500) on 2nd treatment. Text equivalent not available.
Large image of change in infecteds (CD4 is more than 500) on 2nd treatment equation (GIF 32 KB)
Mathematical equation for change in infecteds (CD4 is more than 350 but less than 500) on 2nd treatment. Text equivalent not available.
Large image of change in infecteds (CD4 is more than 350 but less than 500) on 2nd treatment equation (GIF 37 KB)Top of page
Mathematical equation for change in infecteds (CD4 is more than 200 but less than 350) on 2nd treatment. Text equivalent not available.
Large image of change in infecteds (CD4 is more than 200 but less than 350) on 2nd treatment equation (GIF 36 KB)
Mathematical equation for change in infecteds (CD4 is less than 200) on 2nd treatment. Text equivalent not available.
Large image of change in infecteds (CD4 is less than 200) on 2nd treatment equation (GIF 28 KB)Top of page