Mathematical Insights in Evaluating State Dependent Effectiveness of HIV Prevention Interventions

Abstract

Mathematical models have been used to simulate HIV transmission and to study the use of preexposure prophylaxis (PrEP) for HIV prevention. Often a single intervention outcome over 10 years has been used to evaluate the effectiveness of PrEP interventions. However, different metrics express a wide variation over time and often disagree in their forecast on the success of the intervention. We develop a deterministic mathematical model of HIV transmission and use it to evaluate the public-health impact of oral PrEP interventions. We study PrEP effectiveness with respect to different evaluation methods and analyze its dynamics over time. We compare four traditional indicators, based on cumulative number or fractions of infections prevented, on reduction in HIV prevalence or incidence and propose two additional methods, which estimate the burden of the epidemic to the public-health system. We investigate the short and long term behavior of these indicators and the effects of key parameters on the expected benefits from PrEP use. Our findings suggest that public-health officials considering adopting PrEP in HIV prevention programs can make better informed decisions by employing a set of complementing quantitative metrics.

Appendix

1.1 A.1 Approximations of the Indicators at the Start of the Intervention

Based on the initial conditions for the two models, we can obtain the approximate behavior of the indicator at the start of the intervention. To facilitate this, we let I _T (0)=I(0)+I ^p(0) and S _T (0)=S(0)+S ^p(0). Recall that N(0)=S(0)+I(0) when there is no PrEP user and N(0)=S(0)+S ^p(0)+I(0)+I ^p(0) when there are PrEP users.

As examples, we will provide the approximation computation for the prevalence indicator, cumulative indicator, and fractional indicator. Recall that $P_{I}= 1-\frac{[\frac{I^{p}+I}{S^{p}+S+I^{p}+I}]_{\mathrm{DP}}}{[\frac{I}{S+I}]}$. We will first estimate $[\frac{I}{S+I}]$ and $[\frac {I^{p}+I}{S^{p}+S+I^{p}+I}]_{\mathrm{DP}}$ separately:

$$ \everymath{\displaystyle} \begin{array}{rcl} \biggl[\frac{I}{S+I}\biggr] &\thickapprox& \frac{I(0)+\frac {dI}{dt}(0)\,dt}{I(0)+S(0)+(\frac{dI}{dt}+ \frac{dS}{dt})(0)\,dt} \\ \noalign{\vspace*{6pt}} &=& \frac{I(0)+[\beta\frac{S(0)I(0)}{N(0)}-(\mu+d)I(0)]\,dt}{I(0)+S(0)+ [\varLambda-\mu S(0)-(\mu+d)I(0)]\,dt} \\ \noalign{\vspace*{6pt}} &=& \frac{PN(0)+[\beta P(1-P)N(0)-(\mu+d)PN(0)]\,dt}{N(0)+ [\varLambda-\mu(1-P)N(0)-(\mu+d)PN(0)]\,dt} \\ \noalign{\vspace*{6pt}} &=& \frac{P+[\beta P(1-P)- (\mu+d)P]\,dt}{1+(\frac{\varLambda}{N(0)}-\mu-dP)\,dt}; \\ \end{array} \nonumber $$

$$ \everymath{\displaystyle} \begin{array}{l} \biggl[\frac{I^p+I}{S^p+S+I^p+I}\biggr]_{\mathrm{DP}} \\ \noalign{\vspace*{6pt}} \quad {}\thickapprox \frac{I_T(0)+(\frac{dI^p}{dt}+\frac {dI}{dt})(0)\,dt}{N(0)+(\frac{dS^p}{dt}+\frac{dS}{dt}+\frac {dI^p}{dt}+\frac{dI}{dt})(0)\,dt} \\ \noalign{\vspace*{6pt}} \quad {}= \frac{I_T(0)+\{[I_T(0)-\alpha_iI^p(0)][S_T(0)-\alpha_sS^p(0)]\frac{\beta}{N(0)}- (\mu+d)I_T(0)\}\,dt}{N(0)+[\varLambda-\mu N(0)-dI_T(0)]\,dt} \\ \noalign{\vspace*{6pt}} \quad {}= PN(0) \biggl(1+\biggl\{(1-P)N(0) \bigl[1-(1-\theta)k_1+ (1-\alpha_i) (1- \theta)k_1\bigr] \bigl[1-k_1 \\ \noalign{\vspace*{6pt}} \qquad {}+ (1-\alpha_s)k_1 \bigr]\cdot\frac{\beta}{N(0)}-(\mu+d)\biggr\}\,dt\biggr)\cdot\frac {1}{ N(0)+(\varLambda-\mu N(0)-dPN(0))\,dt} \\ \noalign{\vspace*{6pt}} \quad {}= \frac{P+\{[1-(1-\theta)k_1\alpha_i](1-\alpha_sk_1)\beta P(1-P)-(\mu+d)P\}\,dt}{1+(\frac{\varLambda}{N(0)}-\mu-dP)\,dt}. \\ \end{array} \nonumber $$

Then by the expression of prevalence indicator, we have

$$ \everymath{\displaystyle} \begin{array}{rcl} P_I&\thickapprox& 1- \frac{\frac{P+\{[1-(1-\theta)k_1\alpha_i](1-\alpha_sk_1)\beta P(1-P)- (\mu+d)P\}\,dt}{1+(\frac{\varLambda}{N(0)}-\mu-dP)\,dt}}{\frac{P+[\beta P(1-P)-(\mu+d)P]\,dt}{1+(\frac{\varLambda}{N(0)}-\mu-dP)\,dt}} \\ \noalign{\vspace*{6pt}} &=& 1-\frac{P+\{[1-(1-\theta)k_1\alpha_i](1-\alpha_sk_1)\beta P(1-P)-(\mu+d)P\}\,dt}{P+[\beta P(1-P)-(\mu+d)P]\,dt} \\ \noalign{\vspace*{6pt}} &=& 1- \frac{1+\{[1-(1-\theta)k_1\alpha_i](1-\alpha_sk_1)\beta (1-P)-(\mu+d)\}\,dt}{1+[\beta(1-P)-(\mu+d)]\,dt} \\ \noalign{\vspace*{6pt}} &\thickapprox& \bigl(-\bigl\{\bigl[1-(1- \theta)k_1\alpha_i\bigr](1-\alpha_sk_1) \beta (1-P)-(\mu+d)\bigr\}\,dt\bigr) \\ \noalign{\vspace*{6pt}} & & \bigl\{1-\bigl[\beta(1-P)-(\mu+d)\bigr]\,dt\bigr\} \\ \noalign{\vspace*{6pt}} &\thickapprox& -\bigl\{\bigl[1-(1- \theta)k_1\alpha_i\bigr](1-\alpha_sk_1) \beta (1-P)-(\mu+d)\bigr\}\,dt \\ \noalign{\vspace*{6pt}} & &{}+ \bigl[\beta(1-P)-(\mu+d)\bigr]\,dt \\ \noalign{\vspace*{6pt}} &=& \bigl[\alpha_s+(1-\theta) \alpha_i-(1-\theta)\alpha_s\alpha_ik_1 \bigr]k_1\beta(1-P)\,dt \\ \noalign{\vspace*{6pt}} &=& \bigl[\alpha_s+(1-\theta) \alpha_i(1-\alpha_s k_1)\bigr]k_1 \beta(1-P)\,dt. \\ \end{array} \nonumber $$

Thus, the slope at which the prevalence indicator increases at the beginning of the intervention can be approximated by

$$P_I'\thickapprox \bigl[\alpha_s+(1- \theta)\alpha_i(1-\alpha_s k_1) \bigr]k_1\beta(1-P). $$

By the expression of cumulative indicator, we have

Thus, the slope at which the cumulative indicator increases at the beginning of the intervention can be approximated by

$$C_I'\thickapprox \bigl[\alpha_s+(1- \theta)\alpha_i(1-\alpha_sk_1) \bigr]k_1\beta P(1-P)N(0). $$

By the expression of fractional indicator, we have

1.2 A.2 Asymptotic Approximations of the Indicators

To approximate the asymptotic behavior of indicators, we need to study the steady state of the two models. What is challenging is to study the steady state of the baseline model using PrEP (4). From

$$\frac{d(S^p+S+I^p+I)}{dt}=\varLambda-\mu \bigl(S^p+S+I^p+I \bigr)-d \bigl(I^p+I \bigr)=0 $$

we obtain that at steady state

$$N=\frac{\varLambda-d(I^p+I)}{\mu}. $$

From

$$\frac{d(S^p+I^p)}{dt}=k\varLambda-\mu \bigl(S^p+I^p \bigr)-dI^p=0 $$

we obtain that at steady state,

$$S^p=\frac{k\varLambda-(d+\mu)I^p}{\mu}. $$

Moreover, from

$$\frac{d(S+I)}{dt}=(1-k)\varLambda-\mu(S+I)-dI=0 $$

we obtain that at steady state,

$$S=\frac{(1-k)\varLambda-(d+\mu)I}{\mu}. $$

In the following, let m=μ+d and $\hat{I}=I^{p}+I$. Then at steady state,

Hence,

$$ \beta \bigl[I+(1-\alpha_i)I^p \bigr] \biggl[ \frac{(1-k)\varLambda-mI}{\mu}+(1-\alpha_s)\frac {k\varLambda- mI^p}{\mu} \biggr]= m\hat{I} \frac{\varLambda-d\hat{I}}{\mu}. \nonumber $$

Let $p=[\frac{I^{p}}{I^{p}+I}]_{\mathrm{DP}}=[\frac{I^{p}}{\hat{I}}]_{\mathrm{DP}}$, and assume that $\hat{I}=I^{p}+I\neq0$ at steady state. Then

$$ \everymath{\displaystyle} \begin{array}{rcl} & & \beta(\hat{I}-\alpha_ip\hat{I}) \biggl[\frac{(1-k)\varLambda-m(1-p)\hat {I}}{\mu}+ (1-\alpha_s)\frac{k\varLambda-mp\hat{I}}{\mu} \biggr]-m \hat{I}\frac{\varLambda-d\hat {I}}{\mu}=0, \\ \noalign{\vspace*{6pt}} & & \beta\hat{I}(1-\alpha_ip) \biggl[ \frac{(1-k)\varLambda-m(1-p)\hat{I}}{\mu }+(1-\alpha_s) \frac{k\varLambda-mp\hat{I}}{\mu} \biggr]=m\hat{I} \frac{\varLambda-d\hat{I}}{\mu}, \\ \noalign{\vspace*{6pt}} & & \beta(1-\alpha_ip) \biggl[ \frac{(1-k)\varLambda+(1-\alpha_s)k\varLambda }{m}-(1-p)\hat{I}- (1-\alpha_s)p\hat{I} \biggr]= \varLambda-d\hat{I}, \\ \noalign{\vspace*{6pt}} & & \frac{\beta(1-\alpha_ip)(1-\alpha_sk)}{\mu+d}\varLambda-\beta (1- \alpha_ip) (1-\alpha_sp)\hat{I} =\varLambda-d\hat{I}, \\ \noalign{\vspace*{6pt}} & & \hat{I}=\frac{\frac{\beta(1-\alpha_ip)(1-\alpha_sk)}{\mu +d}-1}{\beta(1-\alpha_ip)(1-\alpha_sp)-d}\varLambda= \frac{R_0(1-\alpha_ip)(1-\alpha_sk)-1}{\beta(1-\alpha_ip)(1-\alpha_sp)-d}\varLambda. \\ \end{array} \nonumber $$

Notice that in the model with no intervention (6), the basic reproduction number is $R_{0}=\frac{\beta}{\mu+d}$, and the endemic steady state is $(\frac{1}{\beta-d}\varLambda,\frac{R_{0}-1}{\beta-d}\varLambda )$. By the definitions of the indicators, now we can find the expressions to approximate the asymptotic behavior of $\hat{C}_{I}$ and $\hat{F}_{I}$:

$$\hat{C}_I= \biggl[\frac{R_0-1}{\beta-d}- \frac{R_0(1-\alpha_ip)(1-\alpha_sk)-1}{\beta(1-\alpha_ip)(1-\alpha_sp)-d} \biggr] \varLambda; $$

$$\hat{F}_I=1-\frac{R_0(1-\alpha_ip)(1-\alpha_sk)-1}{ R_0-1}\frac{\beta-d}{\beta(1-\alpha_ip)(1-\alpha_sp)-d}. $$

Now consider the prevalence indicator, we want to find $[\frac{I^{p}+I}{S^{p}+S+I^{p}+I}]_{\mathrm{DP}}$ at the steady state.

$$ \everymath{\displaystyle} \begin{array}{l} \frac{dI^p}{dt}+(1-\alpha_s) \frac{dI}{dt} \\ \noalign{\vspace*{6pt}} \quad {}= (1-\alpha_s)\beta \frac{(S^p+S)I}{N}+(1-\alpha_s) (1-\alpha_i)\beta \frac{(S^p+S)I^p}{N} \\ \noalign{\vspace*{6pt}} \qquad {}- (\mu+d)\bigl[I^p+(1- \alpha_s)I\bigr] \\ \noalign{\vspace*{6pt}} \quad {}= (1-\alpha_s)\beta\bigl[I+(1- \alpha_i)I^p\bigr]\frac{S^p+S}{N}-(\mu +d) \bigl[I^p+(1-\alpha_s)I\bigr]=0. \\ \end{array} \nonumber $$

Hence,

$$ \everymath{\displaystyle} \begin{array}{rcl} \frac{S^p+S}{N}&=&\frac{\mu+d}{(1-\alpha_s)\beta} \frac{1-\alpha_s\frac {I}{I^p+I}}{1-\alpha_i\frac{I^p}{I^p+I}}=R_0^{-1}\frac{1}{1-\alpha_s}\frac{1-\alpha_s(1-p)}{1-\alpha_ip}; \\ \noalign{\vspace*{6pt}} \frac{I^p+I}{N}&=&1-\frac{\mu+d}{(1-\alpha_s)\beta} \frac{1-\alpha_s\frac{I}{I^p+I}}{1-\alpha_i\frac{I^p}{I^p+I}}=1-R_0^{-1}\frac {1}{1-\alpha_s} \frac{1-\alpha_s(1-p)}{1-\alpha_ip}. \\ \end{array} \nonumber $$

By the definition of prevalence indicator, the asymptotical behavior of the prevalence indicator can be determined by

$$P_I=1-\frac{1-R_0^{-1}\frac{1}{1-\alpha_s}\frac{1-\alpha_s(1-p)}{1-\alpha_ip}}{1-R_0^{-1}}=1-\frac{R_0-\frac{1-\alpha_s(1-p)}{(1-\alpha_s)(1-\alpha_ip)}}{R_0-1}. $$

For the cumulative indicator at the steady state, we have

$$ \everymath{\displaystyle} \begin{array}{rcl} \frac{d}{dt}C_I&=&\biggl[ \frac{d}{dt}I_{\mathrm{New}}\biggr]-\biggl[\frac {d}{dt} \bigl(I^p_{\mathrm{New}}+I_{\mathrm{New}}\bigr)\biggr]_{\mathrm{DP}}, \\ \noalign{\vspace*{6pt}} \frac{d}{dt}C_I&=&\bigl[(\mu+d)I \bigr]-\bigl[(\mu+d) \bigl(I^p+I\bigr)\bigr]_{\mathrm{DP}} \\ \noalign{\vspace*{6pt}} &=& \biggl(\frac{\beta-(\mu+d)}{\beta-d}-\frac{\beta(1-\alpha_i p)(1-\alpha_s k)-(\mu+d)}{ \beta(1-\alpha_i p)(1-\alpha_s p)-d}\biggr) \varLambda \\ \noalign{\vspace*{6pt}} &=& (\mu+d) \biggl(\frac{R_0-1}{\beta-d}- \frac{R_0(1-\alpha_i p)(1-\alpha_s k)-1}{ \beta(1-\alpha_i p)(1-\alpha_s p)-d}\biggr)\varLambda. \\ \end{array} \nonumber $$

We see that the cumulative indicator keeps increasing, and eventually the rate approaches a constant.

Finally, we look at the annual incidence indicator

$$ \everymath{\displaystyle} \begin{array}{rcl} aI_I(n)&=&1-\frac{[\frac {I^p_{\mathrm{New}}(n+1)+I_{\mathrm{New}}(n+1)-(I^p_{\mathrm{New}}(n)+I_{\mathrm{New}}(n))}{S^p(n)+S(n)}]_{\mathrm{DP}}}{ [\frac{I_{\mathrm{New}}(n+1)-I_{\mathrm{New}}(n)}{S(n)}]} \\ \noalign{\vspace*{6pt}} &=& 1-\frac{[\frac{\frac{d}{dt}(I^p_{\mathrm{New}}(n+q)+ I_{\mathrm{New}}(n+s))}{S^p(n)+S(n)}]_{\mathrm{DP}}}{[\frac{\frac{d}{dt}I_{\mathrm{New}} (n+r)}{S(n)}]} \quad \mbox{with some}\ q,s,r\in[0,1], \\ \end{array} \nonumber $$

then at steady state, we have

$$ \everymath{\displaystyle} \begin{array}{rcl} aI_I&=&1-\frac{[\frac{\frac{d}{dt}(I^p_{\mathrm{New}}+I_{\mathrm{New}})}{S^p+S}]_{\mathrm{DP}}}{ [\frac{\frac{d}{dt}I_{\mathrm{New}}}{S}]}= 1- \frac{[\frac{(\mu+d)(I^p+I)}{S^p+S}]_{\mathrm{DP}}}{[\frac{(\mu+d)I}{S}]}= 1-\frac{[\frac {I^p+I}{S^p+S}]_{\mathrm{DP}}}{[\frac{I}{S}]} \\ \noalign{\vspace*{6pt}} &=& 1-\frac{R_0\frac{(1-\alpha_s)(1-\alpha_i p)}{1-\alpha_s(1-p)}-1}{R_0-1}. \\ \end{array} \nonumber $$