An exact approach to calibrating infectious disease models to surveillance data: The case of HIV and HSV-2

Table 1.  Model parameters and calibration conditions for the four-compartment $SI$-type HIV/HSV-2 coinfection model. $^\dagger$ Values used in the high HIV burden example; South Africa. $^\ddagger$ Values used in the low HIV burden example; United Kingdom

Table 2.  Calibration of the four-compartment $SI$-type HIV/HSV-2 coinfection model for a high prevalence and low prevalence setting; South Africa and the United Kingdom, respectively. Note that $i=0$ the "0^th iteration" refers to our initial approximation to the solution of the calibration system (4)-(8). HIV and HSV-2 prevalence at the endemic equilibrium that results from using the $i$th approximations for $\beta$ and $\sigma$ are denoted $\widehat{A}_i$ and $\widehat{H}_i$, respectively. Precision is measured by average absolute value of the calibration system (4)-(8) evaluated at the current approximation (i.e. $\frac{||f(h_i, a_i, c_i, \sigma_i, \beta_i)||_1}{5})$. A stopping criteria of $\max\{|\beta_i-\beta_{i-1}|, |\sigma_i-\sigma_{i-1}|\} < 10^{-9}$ is used for Newton's Method

Table 3.  Robustness of calibrating the four-compartment $SI$-type HIV/HSV-2 coinfection model to HIV and HSV-2 prevalence at the endemic equilibrium. For each sample $j=1, 2, ..., 50$, the resulting values of the calibration parameters ($\beta^*, \sigma^*$) and the resulting disease prevalences at equilibrium ($\widehat{A}^*, \widehat{H}^*$) are given. The number of iterations of Newton's Method required to achieve a stopping criterion of $\max\{|\beta_i-\beta_{i-1}|, |\sigma_i-\sigma_{i-1}|\} < 10^{-9}$ is denoted by $m$

Table 4a.  Calibration of Granich et al. model to HIV prevalence. (a) Model parameters and calibration conditions for parameterizing the Granich et al. model with behavioral response to HIV prevalence at endemic equilibrium. $^\dagger$Baseline value taken from [11]; range is modeling assumption of this work. $^\ddagger$Recruitment rate range is same as background mortality to give a total population of 1 in the absence of HIV, without loss of generality

Table 4b.  Calibration of Granich et al. model to HIV prevalence. (b) Results of calibrating the Granich et al. model to 50 independent sets of parameters and calibration conditions (randomly sampled from ranges above). For each sample $j=1, 2, ..., 50$, the resulting values of the calibration parameter ($\lambda^*$) and the resulting HIV prevalence at equilibrium ($\widehat{A}^*$) are given. The number of iterations of Newton's Method required to achieve a stopping criterion of $|\lambda_i-\lambda_{i-1}| < 10^{-9}$ is denoted by $m$

Table 5a.  Calibration of HIV/HSV-2 coinfection model with behavioral response to HIV and HSV-2 prevalence. (a) Model parameters and calibration conditions for parameterizing the HIV/HSV-2 coinfection model with behavioral response to HIV and HSV-2 prevalence at endemic equilibrium. $^\ddagger$Recruitment rate range is same as background mortality to give a total population of 1 in the absence of HIV, without loss of generality

Table 5b.  Calibration of HIV/HSV-2 coinfection model with behavioral response to HIV and HSV-2 prevalence.(b) Results of calibrating the HIV/HSV-2 confection model with behavioral response to 50 independent sets of parameters and calibration conditions (randomly sampled from ranges above).For each sample $j=1, 2, ..., 50$, the resulting values of the calibration parameters ($\lambda^*, \kappa^*$) and the resulting diseases prevalences at equilibrium ($\widehat{A}^*, \widehat{H}^*$) are given. The number of iterations of Newton's Method required to achieve a stopping criterion of $\max\{|\lambda_i-\lambda_{i-1}|, |\kappa_i-\kappa_{i-1}|\} < 10^{-9}$ is denoted by $m$

Table 6a.  Calibration of Granich et al. model to HIV prevalence, peak HIV incidence and timing of peak HIV incidence. (a) Model parameters and calibration conditions for parameterizing the Granich et al. model to HIV prevalence, peak HIV incidence and timing of peak HIV incidence. $^\ddagger$Recruitment rate range is same as background mortality to give a total population of 1 in the absence of HIV, without loss of generality

Table 6b.  Calibration of Granich et al. model to HIV prevalence, peak HIV incidence and timing of peak HIV incidence. (b) Results of calibrating the Granich et al. model to 50 independent sets of parameters and calibration conditions (randomly sampled from ranges above). For each sample, the resulting values of the calibration parameters ($\lambda^*, \alpha^*, n^*$) and the resulting calibration values ($\widehat{A}^*, \widehat{I}^*, \widehat{P}^*$) are given. The number of iterations of Newton's Method required to achieve a stopping criterion of $\max\{|\lambda_i-\lambda_{i-1}|, |\alpha_i-\alpha_{i-1}|, |n_i-n_{i-1}|\} < 10^{-9}$ is denoted by $m$

Table 7a.  Calibration of HIV/HSV-2 confection model with behavioral response to HIV prevalence, HSV-2 prevalence, peak HIV incidence and timing of peak HIV incidence. (a) Model parameters and calibration conditions for parameterizing the HIV/HSV-2 confection model with behavioral response to HIV and HSV-2 prevalence, peak HIV incidence and timing of peak HIV incidence. $^\ddagger$Recruitment rate range is same as background mortality to give a total population of 1 in the absence of HIV

Table 7b.  Calibration of HIV/HSV-2 confection model with behavioral response to HIV prevalence, HSV-2 prevalence, peak HIV incidence and timing of peak HIV incidence. (b) Calibration of HIV/HSV-2 confection model with behavioral response to 50 independent sets of parameters and calibration conditions. For each sample $j=1, 2, ..., 50$, the resulting values of the calibration parameters ($\lambda^*, \kappa^*, \alpha^*, n^*$) and the resulting calibration values ($\widehat{A}^*, \widehat{H}^*, \widehat{I}^*, \widehat{P}^*$) are given. Number of iterations required to satisfy the stopping criteria of Steps 1 and 2 (i.e. $\max\{|\lambda_i-\lambda_{i-1}|, |\alpha_i-\alpha_{i-1}|, |n_i-n_{i-1}|\} < 10^{-9}$ and $\max\{|\lambda_i-\lambda_{i-1}|, |\kappa_i-\kappa_{i-1}|\} < 10^{-9}$) denoted ${m}_1$ and ${m}_2$, respectively