Estimating the Size of the HCV Infection Prevalence: A Modeling Approach Using the Incidence of Cases Reported to an Official Notification System

Abstract

In this paper we propose two methods to give a first rough estimate of the actual number of hepatitis C virus (HCV)-infected individuals (prevalence) taking into account the notification rate of newly diagnosed infections (incidence of notification) and the size of the liver transplantation waiting list (LTWL) of patients with liver failure due to chronic HCV infection. Both approaches, when applied to the Brazilian HCV situation converge to the same results, that is, the methods proposed reproduce both the prevalence of reported cases and the LTWL with reasonable accuracy. We use two methods to calculate the prevalence of HCV that, as a first, and very crude approximation, assumes that the actual prevalence of HCV in Brazil is proportional to the reported incidence to the official notification system with a constant denoted $\kappa$. In the paper we discuss the limitations and advantages of this assumption. With the two methods we calculated $\kappa$, which reproduces both the reported incidence and the size of the LTWL. With the value of $\kappa$ we calculated the prevalence I(a) (the integral of which resulted in 1.6 million people living with the infection in Brazil, most of whom unidentified). Other variables related to HCV infection (e.g., the distribution of the proportion of people aged a who got infected n years ago) can be easily calculated from this model. These new variables can then be measured and the model can be recursively updated, improving its accuracy.

Appendices

Appendix 1: Calculating the Age of Infection from the Incidence Data of an Official Notification Systems

In the main text we calculate the proportion of individuals aged a who acquired the infection between 0 and $\tau$, from Eq. (11). That equation is correct but, as we shall see in this appendix, it can be only an approximation of the notification data recorded in an official notification system. To see this let the time between 0 and $\tau$ be divided in 1-year intervals. Then, Eq. (11) reads:

$$\begin{aligned} P^{\prime }(a,\tau )= & {} \frac{\int _0^1 {\lambda (y)S(y)e^{\left[ {-\int \limits _y^a {(\mu (x)+\alpha (x))\hbox {d}x} } \right] }\hbox {d}y} +\int _1^2 {\lambda (y)S(y)e^{\left[ {-\int \limits _y^a {(\mu (x)+\alpha (x))\hbox {d}x} } \right] }\hbox {d}y+\cdots } +\int _{\tau -1}^\tau {\lambda (y)S(y)e^{\left[ {-\int \limits _y^a {(\mu (x)+\alpha (x))\hbox {d}x} } \right] }\hbox {d}y} }{I(a)}\nonumber \\ P^{\prime }(a,\tau )= & {} \frac{\int _0^\tau {\lambda (y)S(y)e^{\left[ {-\int \limits _y^a {(\mu (x)+\alpha (x))\hbox {d}x} } \right] }\hbox {d}y} }{I(a)} \end{aligned}$$

(16)

Therefore, from Eq. (3) of the main text the prevalence of individuals aged a, who acquired the infection between 0 and $\tau \ (\tau <a)$ is naively given by

$$\begin{aligned} P_{Not} (a,\tau )=\kappa P^{\prime }(a,\tau )=\frac{\kappa }{I(a)}\int _0^\tau {\lambda (y)S(y)e^{\left[ {-\int \limits _y^a {(\mu (x)+\alpha (x))\hbox {d}x} } \right] }\hbox {d}y} \end{aligned}$$

(17)

However, this is an overestimate because some individuals who acquired the infection between 0 and $\tau -1$ may have already been notified before $\tau -1$. To correct this, we have to exclude all individuals notified between 0 and $\tau -1$:

$$\begin{aligned} P_{Not} (a,\tau )= & {} \frac{\kappa }{I(a)}\left\{ \int _0^\tau \lambda (y)S(y)e^{\left[ {-\int \limits _y^a {(\mu (x)+\alpha (x))\hbox {d}x} } \right] }\hbox {d}y\right. \nonumber \\&\left. -\kappa \int _0^{\tau -1} \lambda (y)S(y)e^{\left[ {-\int \limits _y^a {(\mu (x)+\alpha (x))\hbox {d}x} } \right] }\hbox {d}y+O\left( {\kappa ^{2}} \right) \right\} \end{aligned}$$

(18)

(the calculations for orders greater than $\kappa ^{2}$ are too involved to be included here).

If $\kappa$ is small enough, then Eq. (11) of the main text is a good approximation.

Appendix 2: Generalizing the “Notification Rate” $\kappa$

In the main text we assumed $\hbox {SINAN}(a)=\kappa I(a)$, that is, all infected individuals that are reported are so with the same “notification rate” $\kappa$. In this appendix we relax this assumption and consider that there are two classes (as an example) of infected individuals that are reported with two different “notification rates” $\kappa _1$ and $\kappa _2$. Here we still keep the assumption that notification rates are age independent. In a future work we will relax this assumption.

Assuming that among the infected individuals there are two classes, reported differently, system (1) now reads:

$$\begin{aligned} \frac{\hbox {d}S(a)}{\hbox {d}a}= & {} -\lambda (a)S(a)-\mu (a)S(a) \nonumber \\ \frac{\hbox {d}I_1 (a)}{\hbox {d}a}= & {} p\lambda (a)S(a)-(\mu (a)+\gamma +\alpha _1 (a))I_1 (a) \nonumber \\ \frac{\hbox {d}I_2 (a)}{\hbox {d}a}= & {} (1-p)\lambda (a)S(a)-(\mu (a)+\gamma +\alpha _2 (a))I_2 (a) \nonumber \\ \frac{\hbox {d}R(a)}{\hbox {d}a}= & {} \gamma (I_1 (a)+I_2 (a))-\mu (a)R(a) \end{aligned}$$

(19)

Equation (3) of the main text becomes:

$$\begin{aligned} \hbox {SINAN}(a)=\kappa _1 pI_1 (a)+\kappa _2 (1-p)I_2 (a) \end{aligned}$$

(20)

This can be generalized for n classes as follows

$$\begin{aligned} \hbox {SINAN}(a)=\sum _{i=1}^n {q_i \kappa _i I_i (a)} ,\quad \sum _{i=1}^n {q_i =1} , \end{aligned}$$

(21)

Both approaches described in the main text can now be applied to system (19). This will be done in a future publication when more epidemiological data are available.