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

• When mathematical models of infectious diseases are used to inform health policy, an important first step is often to calibrate a model to disease surveillance data for a specific setting (or multiple settings). It is increasingly common to also perform sensitivity analyses to demonstrate the robustness, or lack thereof, of the modeling results. Doing so requires the modeler to find multiple parameter sets for which the model produces behavior that is consistent with the surveillance data. While frequently overlooked, the calibration process is nontrivial at best and can be inefficient, poorly communicated and a major hurdle to the overall reproducibility of modeling results.

In this work, we describe a general approach to calibrating infectious disease models to surveillance data. The technique is able to match surveillance data to high accuracy in a very efficient manner as it is based on the Newton-Raphson method for solving nonlinear systems. To demonstrate its robustness, we use the calibration technique on multiple models for the interacting dynamics of HIV and HSV-2.

Mathematics Subject Classification: Primary: 92D30; Secondary: 92D25, 37N25, 37N30, 92C60.

 Citation:

• Figure 1.  Flow diagram of the four-compartment $SI$-type HIV/HSV-2 coinfection model. $N=S+H+A+C$

Figure 2.  Visual illustration of Newton's Method in 1-dimension for finding the solution of $f(x)=0$. As illustrated, the iteration $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ results from finding the root of the tangent line at the current iteration $(x_n, f(x_n))$

Figure 3.  HIV prevalence in South Africa (% of population ages 15-49 infected with HIV) from 1990-2013. Data from the World Bank, World Development Indicators [18]

Figure 4.  Flow diagram of the modified Granich et al. HIV model. $I= I_1+I_2+I_3+I_4$, $N = S+I_1+I_2+I_3+I_4$ and $P = \frac{I}{N}$

Figure 5.  Illustration of population's behavioral response to HIV prevalence in the Granich et al. model via the transmission term $e^{-\alpha P^n}$. Notably, if $n=1$ sexual risk behavior decreases exponentially as a function of HIV prevalence. As $n \to \infty$, the decline in sexual risk behavior approaches a step function

Figure 6.  Flow diagram of the HIV/HSV-2 coinfection model with behavioral response. The rate of transmission for HIV is given by $\Omega = \lambda e^{-\alpha P^n} \frac{A+r_2 C}{N}$ where $A=A_1+A_2+A_3+A_4$ is the number of people infected with HIV only, $C= C_1+C_2+C_3+C_4$ is the number of coinfected individuals, $N = S+H+A+C$ is the size of the total population and $P = \frac{A+C}{N}$ is the prevalence of HIV. The rate of transmission for HSV-2 is given by $\Psi = \kappa e^{-\alpha P^n} \frac{H+r_3 C}{N}$

Figure 7.  Dynamics of HIV prevalence and HIV incidence of the first 12 calibrations: horizontal dashed blue line is sampled HIV prevalence at equilibrium ($\widehat{A}$), horizontal dashed red line is sampled peak HIV incidence ($\widehat{I}$) and horizontal dashed gray line is sampled HIV prevalence at the time of peak incidence ($\widehat{P}$). Calibration is successful if HIV prevalence (solid blue curve) approaches $\widehat{A}$ (dashed blue line) as $t \to \infty$; HIV incidence (solid red curve) peaks at $\widehat{I}$ (dashed red line); and HIV prevalence at time of peak incidence is $\widehat{P}$ (solid blue curve, horizontal dashed gray line and the dashed vertical gray line indicating time of peak incidence all intersect at the same point)

Figure 8.  Behavioral response to HIV prevalence via the transmission term ${e^{-\alpha P^n}}$ for the calibrations with largest $\alpha$ values and the prevalence/incidence curves for those calibrations. Note that the prevalence/incidence curves of Calibrations #4 and #10 are shown in Figure 7

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

 Symbol Baseline Range References Model Parameters Average time in sexually active population (years) ${1}/{\mu}$ 35 30-40 Recruitment rate into the sexually active population (/yr) $\Lambda$ $\frac{1}{35}$ $\frac{1}{40}$-$\frac{1}{30}$ Diseased-induced mortality due to HIV infection (/yr) ${\mu_A}$ $\frac{1}{20}$$^\dagger$$, \frac{1}{100}$$^\ddagger$ $\frac{1}{100}$-$\frac{1}{20}$ Cofactor for increased HIV susceptibility due to HSV-2 infection $r_1$ 1.80 1.4-2.4 [9,21,13,1] Cofactor for increased HIV infectivity due to HSV-2 infection $r_2$ 1.45 1.2-1.7 [1,12,17,4] Cofactor for increased HSV-2 infectivity due to HIV infection $r_3$ 1.30 1.2-1.5 [1] Calibration Conditions HIV prevalence at endemic equilibrium (%) $\widehat{A}$ 17$^\dagger$, 0.3$^\ddagger$ 0.3-17 [18] HSV-2 prevalence at endemic equilibrium (%) $\widehat{H}$ 50$^\dagger$, 4$^\ddagger$ 4-50 [2,16,22,20] Calibration Parameters Transmission coefficient for HIV $\beta$ determined by calibration procedure Transmission coefficient for HSV-2 $\sigma$ determined by calibration procedure

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 "0th 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

 $i$ $\beta_i$ $\sigma_i$ $\widehat{A}_i$ $\widehat{H}_i$ precision$_i$ South Africa (HIV prevalence 15%, HSV-2 prevalence 50%) 0 0.0367143341 0.0705450591 0.0000000447 0.5949903301 $3.10\times 10^{-3}$ 1 0.0528598059 0.0770336930 0.1775025497 0.4988482844 $2.77\times 10^{-4}$ 2 0.0522429752 0.0764216142 0.1700166525 0.5000125340 $1.44\times 10^{-6}$ 3 0.0522423466 0.0764181115 0.1700000000 0.5000000000 $3.38\times 10^{-11}$ United Kingdom (HIV prevalence 0.3%, HSV-2 prevalence 4%) 0 0.0353517304 0.0297663650 0.0000221420 0.0401412229 $5.78\times 10^{-6}$ 1 0.0359324850 0.0297727618 0.0033368789 0.0399611984 $1.86 \times 10^{-8}$ 2 0.0359185167 0.0297727520 0.0030000000 0.0400000000 $1.10 \times 10^{-13}$

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 $m$

 Sampled parameter values and calibration conditions Calibration results $j$ $\Lambda$ $\mu$ $\mu_A$ $r_1$ $r_2$ $r_3$ $\widehat{A}$ $\widehat{H}$ $\beta^*$ $\sigma^*$ $\widehat{A}^*$ $\widehat{H}^*$ $m$ 1 0.026 0.028 0.037 1.98 1.65 1.24 0.088 0.276 0.043 0.045 0.088 0.276 4 2 0.033 0.027 0.015 2.39 1.23 1.22 0.146 0.164 0.039 0.035 0.146 0.164 4 3 0.028 0.027 0.022 1.55 1.25 1.23 0.007 0.340 0.037 0.042 0.007 0.340 3 4 0.025 0.032 0.042 2.39 1.52 1.50 0.025 0.275 0.043 0.046 0.025 0.275 3 5 0.025 0.028 0.016 2.24 1.33 1.36 0.145 0.291 0.034 0.042 0.145 0.291 4 6 0.032 0.027 0.021 2.01 1.51 1.28 0.123 0.294 0.034 0.042 0.123 0.294 4 7 0.031 0.033 0.033 1.89 1.57 1.41 0.065 0.290 0.043 0.049 0.065 0.290 4 8 0.031 0.030 0.020 1.67 1.67 1.48 0.110 0.330 0.034 0.046 0.110 0.330 4 9 0.027 0.028 0.030 2.19 1.29 1.45 0.016 0.409 0.033 0.048 0.016 0.409 3 10 0.029 0.028 0.029 1.84 1.67 1.47 0.170 0.134 0.054 0.038 0.170 0.134 4 11 0.025 0.030 0.035 1.66 1.40 1.47 0.147 0.477 0.046 0.066 0.147 0.477 4 12 0.032 0.028 0.022 1.56 1.24 1.37 0.143 0.176 0.050 0.037 0.143 0.176 4 13 0.026 0.028 0.040 1.59 1.54 1.38 0.026 0.419 0.041 0.050 0.026 0.419 3 14 0.027 0.028 0.048 2.14 1.41 1.33 0.025 0.152 0.059 0.036 0.025 0.152 3 15 0.031 0.027 0.044 1.69 1.47 1.43 0.121 0.312 0.055 0.048 0.121 0.312 4 16 0.029 0.033 0.045 1.43 1.42 1.44 0.162 0.354 0.067 0.061 0.162 0.354 4 17 0.028 0.029 0.033 1.54 1.31 1.48 0.118 0.234 0.056 0.042 0.118 0.234 4 18 0.028 0.029 0.012 1.59 1.58 1.32 0.027 0.377 0.026 0.047 0.027 0.377 3 19 0.026 0.029 0.029 1.87 1.24 1.37 0.074 0.315 0.044 0.045 0.074 0.315 4 20 0.033 0.031 0.021 2.39 1.41 1.49 0.145 0.127 0.047 0.037 0.145 0.127 4 21 0.030 0.031 0.017 1.60 1.38 1.39 0.024 0.054 0.046 0.033 0.024 0.054 3 22 0.025 0.032 0.010 1.46 1.46 1.30 0.098 0.412 0.031 0.055 0.098 0.412 4 23 0.029 0.028 0.046 1.64 1.59 1.30 0.150 0.256 0.060 0.049 0.150 0.256 4 24 0.031 0.026 0.044 2.16 1.57 1.50 0.040 0.360 0.038 0.044 0.040 0.360 4 25 0.028 0.030 0.022 2.05 1.54 1.24 0.132 0.093 0.051 0.038 0.132 0.093 4 26 0.033 0.027 0.019 1.64 1.54 1.35 0.140 0.384 0.033 0.048 0.140 0.384 4 27 0.033 0.028 0.025 2.12 1.23 1.43 0.125 0.488 0.034 0.058 0.125 0.488 4 28 0.026 0.030 0.024 2.06 1.20 1.46 0.050 0.497 0.033 0.062 0.050 0.497 3 29 0.028 0.032 0.021 2.35 1.65 1.26 0.117 0.419 0.027 0.059 0.117 0.419 4 30 0.032 0.027 0.014 2.15 1.45 1.40 0.007 0.336 0.023 0.041 0.007 0.336 3 31 0.029 0.028 0.021 2.30 1.32 1.24 0.007 0.416 0.026 0.048 0.007 0.416 3 32 0.031 0.032 0.028 1.98 1.46 1.34 0.143 0.166 0.054 0.044 0.143 0.166 4 33 0.025 0.030 0.012 2.31 1.67 1.36 0.160 0.044 0.045 0.032 0.160 0.044 4 34 0.027 0.025 0.018 1.66 1.25 1.24 0.122 0.099 0.045 0.031 0.122 0.099 3 35 0.029 0.032 0.024 2.00 1.36 1.38 0.141 0.197 0.049 0.044 0.141 0.197 4 36 0.029 0.031 0.012 2.01 1.20 1.45 0.006 0.367 0.027 0.049 0.006 0.367 3 37 0.026 0.028 0.047 2.07 1.60 1.37 0.155 0.070 0.076 0.041 0.155 0.070 4 38 0.032 0.027 0.027 1.84 1.60 1.42 0.161 0.397 0.036 0.051 0.161 0.397 4 39 0.033 0.032 0.043 1.88 1.32 1.38 0.021 0.412 0.047 0.055 0.021 0.412 3 40 0.026 0.033 0.037 1.83 1.66 1.34 0.022 0.390 0.038 0.055 0.022 0.390 3 41 0.033 0.032 0.035 1.86 1.24 1.26 0.073 0.049 0.068 0.038 0.073 0.049 3 42 0.027 0.026 0.031 2.14 1.37 1.33 0.113 0.069 0.057 0.033 0.113 0.069 3 43 0.033 0.027 0.022 1.95 1.41 1.28 0.126 0.217 0.041 0.039 0.126 0.217 4 44 0.030 0.026 0.014 2.34 1.39 1.22 0.119 0.317 0.027 0.041 0.119 0.317 4 45 0.029 0.028 0.021 2.06 1.38 1.22 0.048 0.480 0.026 0.055 0.048 0.480 4 46 0.026 0.027 0.031 1.77 1.67 1.25 0.143 0.439 0.036 0.057 0.143 0.439 4 47 0.029 0.032 0.031 2.09 1.23 1.40 0.030 0.337 0.042 0.050 0.030 0.337 3 48 0.028 0.029 0.025 1.67 1.22 1.35 0.160 0.303 0.049 0.047 0.160 0.303 4 49 0.033 0.033 0.040 1.83 1.62 1.47 0.041 0.375 0.042 0.055 0.041 0.375 4 50 0.029 0.030 0.045 2.23 1.22 1.39 0.044 0.101 0.067 0.037 0.044 0.101 3

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

 Symbol Baseline Range References Model Parameters Background mortality rate (/yr) $\mu$ 0.018 $\frac{1}{60} - \frac{1}{30}$ [11]$^\dagger$ Recruitment rate into the population (/yr) $\Lambda$ 0.018 $\frac{1}{60} - \frac{1}{30}$ $^\ddagger$ Location of transmission term $\alpha$ 0.055 0.05-50 [11]$^\dagger$ Shape of transmission term $n$ 0.996 0.90-30 [11]$^\dagger$ Rate of HIV progression (/yr) $\rho$ 0.303 0.1-0.4 [11]$^\dagger$ Calibration Conditions HIV prevalence at endemic equilibrium $\widehat{A}$ 0.17 0.3-17 [18] Calibration Parameters Initial value of transmission term (/yr) $\lambda$ determined by calibration procedure

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 $m$

 Sampled parameters and calibration condition Calibration results $j$ $\Lambda$ $\mu$ $\alpha$ $n$ $\widehat{A}$ $\lambda^*$ $\widehat{A}^*$ $m$ 1 0.018 0.021 34.3 29.1 0.115 0.104 0.115 2 2 0.028 0.024 31.7 14.6 0.019 0.065 0.019 2 3 0.022 0.025 5.8 4.9 0.126 0.066 0.126 2 4 0.021 0.019 30.2 6.1 0.142 0.044 0.142 2 5 0.028 0.017 35.9 5.7 0.006 0.097 0.006 2 6 0.026 0.028 8.7 22.0 0.015 0.049 0.015 2 7 0.031 0.018 6.0 27.3 0.061 0.087 0.061 2 8 0.030 0.019 29.9 17.2 0.014 0.077 0.014 2 9 0.024 0.026 32.4 6.3 0.073 0.078 0.073 2 10 0.022 0.022 47.6 18.5 0.125 0.098 0.125 2 11 0.022 0.030 25.0 19.3 0.160 0.124 0.160 2 12 0.023 0.030 49.1 10.6 0.114 0.059 0.114 2 13 0.032 0.017 17.4 14.4 0.061 0.041 0.061 2 14 0.025 0.030 9.0 3.0 0.140 0.063 0.140 2 15 0.020 0.022 8.3 14.9 0.093 0.126 0.093 2 16 0.024 0.028 24.0 23.0 0.060 0.077 0.060 2 17 0.029 0.032 8.0 24.5 0.023 0.108 0.023 2 18 0.025 0.025 25.3 15.1 0.119 0.109 0.119 2 19 0.018 0.031 18.8 29.1 0.126 0.084 0.126 2 20 0.031 0.026 41.9 24.7 0.075 0.107 0.075 2 21 0.027 0.018 22.6 18.3 0.131 0.102 0.131 2 22 0.027 0.019 23.9 12.6 0.106 0.094 0.106 2 23 0.031 0.026 28.6 10.0 0.086 0.095 0.086 2 24 0.019 0.027 45.3 6.9 0.067 0.075 0.067 2 25 0.025 0.033 42.0 7.9 0.040 0.102 0.040 2 26 0.020 0.027 36.7 21.5 0.112 0.085 0.112 2 27 0.020 0.020 24.4 7.2 0.088 0.049 0.088 2 28 0.021 0.026 49.6 25.3 0.154 0.123 0.154 2 29 0.030 0.026 9.8 12.5 0.100 0.072 0.100 2 30 0.021 0.019 2.4 22.5 0.058 0.106 0.058 2 31 0.022 0.026 4.6 23.2 0.098 0.101 0.098 2 32 0.026 0.022 46.7 17.7 0.108 0.090 0.108 2 33 0.029 0.024 5.8 9.6 0.120 0.125 0.120 2 34 0.023 0.030 9.5 17.1 0.027 0.087 0.027 2 35 0.025 0.024 0.7 23.5 0.111 0.117 0.111 2 36 0.021 0.030 16.0 21.0 0.023 0.101 0.023 2 37 0.027 0.018 29.4 6.3 0.165 0.123 0.165 2 38 0.025 0.033 48.2 6.1 0.071 0.096 0.071 2 39 0.027 0.030 12.0 22.4 0.057 0.071 0.057 2 40 0.025 0.033 43.6 4.3 0.017 0.059 0.017 2 41 0.031 0.031 1.7 13.0 0.151 0.075 0.151 2 42 0.031 0.031 26.1 11.1 0.041 0.077 0.041 2 43 0.022 0.024 39.9 16.9 0.100 0.100 0.100 2 44 0.029 0.022 17.7 13.2 0.103 0.061 0.103 2 45 0.023 0.024 6.7 21.4 0.158 0.117 0.158 2 46 0.028 0.023 4.2 10.5 0.082 0.051 0.082 2 47 0.023 0.031 13.1 18.4 0.022 0.084 0.022 2 48 0.031 0.022 46.3 28.3 0.058 0.118 0.058 2 49 0.020 0.025 42.7 27.7 0.077 0.112 0.077 2 50 0.024 0.030 46.1 2.2 0.082 0.085 0.082 2

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

 Symbol Range References Model Parameters Background mortality rate (/yr) $\mu$ $\frac{1}{60} - \frac{1}{30}$ [11] Recruitment rate into the population (/yr) $\Lambda$ $\frac{1}{60} - \frac{1}{30}$ $^\ddagger$ Location of HIV transmission term $\alpha$ 0.05-50 [11] Shape of HIV transmission term $n$ 0.90-30 [11] Rate of HIV progression (/yr) $\rho$ 0.1-0.4 [11] Calibration Conditions HIV prevalence at endemic equilibrium (/yr) $\widehat{A}$ 0.3-17 [18] HSV-2 prevalence at endemic equilibrium (/yr) $\widehat{H}$ 4-50 [2,16,22,20] Calibration Parameters Initial value of HIV transmission term (/yr) $\lambda$ determined by calibration procedure Initial value of HSV-2 transmission term (/yr) $\kappa$ determined by calibration procedure

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 $m$

 Sampled parameters and calibration condition Calibration results $j$ $\Lambda$ $\mu$ $\rho$ $\alpha$ $n$ $\widehat{A}$ $\widehat{H}$ $\lambda^*$ $\kappa^*$ $\widehat{A}^*$ $\widehat{H}^*$ $m$ 1 0.033 0.030 0.136 25.388 7.775 0.100 0.429 0.040 0.057 0.100 0.429 4 2 0.026 0.025 0.294 25.452 10.004 0.061 0.433 0.057 0.053 0.061 0.433 4 3 0.022 0.032 0.268 9.053 18.416 0.065 0.333 0.066 0.055 0.065 0.333 4 4 0.027 0.022 0.159 30.943 15.777 0.107 0.286 0.042 0.038 0.107 0.286 4 5 0.020 0.032 0.184 35.273 19.612 0.025 0.484 0.033 0.063 0.025 0.484 4 6 0.019 0.017 0.363 46.871 29.933 0.081 0.050 0.105 0.029 0.081 0.050 4 7 0.021 0.026 0.164 35.976 10.444 0.050 0.437 0.034 0.050 0.050 0.437 4 8 0.017 0.023 0.149 45.299 27.869 0.013 0.349 0.031 0.035 0.013 0.349 4 9 0.018 0.022 0.122 2.960 2.103 0.161 0.067 0.050 0.031 0.161 0.067 4 10 0.020 0.025 0.210 47.955 15.737 0.099 0.044 0.072 0.034 0.099 0.044 4 11 0.018 0.021 0.120 22.432 23.321 0.114 0.424 0.028 0.042 0.114 0.424 4 12 0.032 0.023 0.155 27.625 4.781 0.107 0.234 0.044 0.037 0.107 0.234 4 13 0.029 0.027 0.324 36.150 6.654 0.087 0.303 0.063 0.051 0.087 0.303 4 14 0.018 0.020 0.144 29.338 26.770 0.008 0.357 0.027 0.032 0.008 0.357 4 15 0.019 0.032 0.377 42.820 16.166 0.168 0.403 0.084 0.086 0.168 0.403 4 16 0.028 0.017 0.126 15.874 24.707 0.160 0.237 0.041 0.030 0.160 0.237 4 17 0.021 0.024 0.275 14.494 1.718 0.097 0.111 0.111 0.047 0.097 0.111 4 18 0.032 0.026 0.254 14.086 11.617 0.085 0.231 0.053 0.044 0.085 0.231 4 19 0.020 0.030 0.300 18.883 1.811 0.022 0.358 0.054 0.051 0.022 0.358 4 20 0.024 0.024 0.274 5.479 9.004 0.161 0.068 0.091 0.043 0.161 0.068 4 21 0.033 0.024 0.149 26.541 10.778 0.003 0.325 0.032 0.035 0.003 0.325 4 22 0.024 0.031 0.173 0.848 2.736 0.019 0.206 0.050 0.040 0.019 0.206 4 23 0.017 0.020 0.147 35.029 14.683 0.102 0.117 0.046 0.028 0.102 0.117 4 24 0.026 0.030 0.121 23.521 10.510 0.114 0.340 0.036 0.049 0.114 0.340 4 25 0.029 0.030 0.169 35.260 8.151 0.122 0.496 0.035 0.070 0.122 0.496 4 26 0.028 0.031 0.357 19.775 5.288 0.086 0.101 0.094 0.049 0.086 0.101 4 27 0.019 0.030 0.312 8.664 27.919 0.126 0.219 0.080 0.055 0.126 0.219 4 28 0.024 0.021 0.357 4.805 29.623 0.009 0.469 0.056 0.041 0.009 0.469 4 29 0.030 0.020 0.259 32.938 28.889 0.099 0.436 0.052 0.049 0.099 0.436 4 30 0.018 0.017 0.182 4.482 2.045 0.027 0.288 0.041 0.026 0.027 0.288 4 31 0.028 0.026 0.164 26.525 25.949 0.117 0.084 0.058 0.035 0.117 0.084 4 32 0.023 0.020 0.372 39.113 5.546 0.136 0.443 0.075 0.064 0.136 0.443 4 33 0.032 0.022 0.353 1.944 19.318 0.082 0.353 0.084 0.047 0.082 0.353 4 34 0.029 0.031 0.110 2.429 4.162 0.059 0.432 0.026 0.056 0.059 0.432 4 35 0.027 0.024 0.165 47.290 21.058 0.041 0.089 0.051 0.029 0.041 0.089 4 36 0.025 0.022 0.348 17.164 26.863 0.063 0.455 0.076 0.052 0.063 0.455 4 37 0.017 0.024 0.361 9.553 7.454 0.027 0.054 0.103 0.029 0.027 0.054 3 38 0.027 0.022 0.222 46.392 21.362 0.167 0.320 0.051 0.051 0.167 0.320 4 39 0.028 0.018 0.247 21.006 7.505 0.115 0.465 0.046 0.048 0.115 0.465 4 40 0.018 0.026 0.143 34.607 19.392 0.022 0.103 0.046 0.030 0.022 0.103 4 41 0.021 0.033 0.203 46.859 24.756 0.139 0.184 0.068 0.049 0.139 0.184 4 42 0.028 0.021 0.251 22.415 11.566 0.026 0.043 0.072 0.025 0.026 0.043 4 43 0.021 0.020 0.390 14.424 26.982 0.078 0.408 0.079 0.049 0.078 0.408 4 44 0.018 0.017 0.113 4.843 4.594 0.037 0.277 0.028 0.026 0.037 0.277 4 45 0.020 0.028 0.376 2.375 17.141 0.054 0.302 0.083 0.048 0.054 0.302 4 46 0.029 0.023 0.125 2.299 25.985 0.052 0.456 0.024 0.044 0.052 0.456 4 47 0.028 0.033 0.353 34.218 7.959 0.010 0.207 0.081 0.042 0.010 0.207 4 48 0.018 0.033 0.222 38.289 27.674 0.134 0.430 0.046 0.072 0.134 0.430 4 49 0.030 0.022 0.272 25.403 9.279 0.021 0.315 0.054 0.035 0.021 0.315 4 50 0.023 0.030 0.119 2.163 27.532 0.135 0.126 0.051 0.037 0.135 0.126 4

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

 Symbol Range References Model Parameters Background mortality rate (/yr) $\mu$ $\frac{1}{60} - \frac{1}{30}$ [11] Recruitment rate into the population (/yr) $\Lambda$ $\frac{1}{60} - \frac{1}{30}$ $^\ddagger$ Rate of HIV progression (/yr) $\rho$ 0.1-0.4 [11] Calibration Conditions HIV prevalence at endemic equilibrium $\widehat{A}$ 0.3-17 [18] Peak HIV incidence $\widehat{I}$ 10%-50% of $\widehat{A}$ HIV prevalence at time of peak HIV incidence $\widehat{P}$ 5%-95% of $\widehat{A}$ Calibration Parameters Initial value of transmission term (/yr) $\lambda$ determined by calibration procedure Location of transmission term $\alpha$ determined by calibration procedure Shape of transmission term $n$ determined by calibration procedure

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 $m$

 Sampled parameters and calibration condition Calibration results $j$ $\Lambda$ $\mu$ $\rho$ $\widehat{A}$ $\widehat{I}$ $\widehat{P}$ $\lambda^*$ $\alpha^*$ $n^*$ $\widehat{A}^*$ $\widehat{I}^*$ $\widehat{P}^*$ ${m}$ 1 0.031 0.028 0.306 0.095 0.016 0.011 224.91 1.22212E+01 0.2 0.095 0.016 0.011 2 2 0.017 0.027 0.115 0.054 0.017 0.036 0.54 2.80696E+09 7.2 0.054 0.017 0.036 2 3 0.031 0.022 0.171 0.015 0.007 0.006 1.82 2.22785E+04 2.1 0.015 0.007 0.006 2 4 0.024 0.026 0.372 0.023 0.009 0.020 0.46 1.09674E+42 25.5 0.023 0.009 0.020 3 5 0.032 0.032 0.192 0.168 0.066 0.151 0.52 1.91294E+33 42.7 0.168 0.066 0.152 3 6 0.022 0.023 0.102 0.120 0.044 0.105 0.48 1.30078E+32 34.4 0.120 0.044 0.104 3 7 0.031 0.018 0.307 0.093 0.013 0.021 15.58 1.04219E+01 0.3 0.093 0.013 0.021 2 8 0.024 0.017 0.224 0.143 0.057 0.037 3.86 3.28571E+01 1.1 0.143 0.057 0.039 2 9 0.027 0.033 0.124 0.160 0.042 0.013 62.63 1.27961E+01 0.3 0.160 0.042 0.014 2 10 0.032 0.023 0.215 0.105 0.029 0.097 0.34 1.15023E+57 58.1 0.105 0.029 0.096 3 11 0.023 0.020 0.138 0.110 0.049 0.064 0.97 2.69922E+05 5.2 0.110 0.049 0.066 2 12 0.032 0.024 0.393 0.102 0.044 0.068 0.80 3.07050E+06 6.3 0.102 0.044 0.070 2 13 0.031 0.018 0.199 0.074 0.013 0.055 0.27 5.60380E+09 8.5 0.074 0.013 0.055 2 14 0.026 0.023 0.386 0.049 0.019 0.018 2.04 2.41472E+02 1.5 0.049 0.019 0.018 2 15 0.033 0.032 0.364 0.138 0.067 0.073 1.32 1.62810E+03 3.3 0.138 0.067 0.071 2 16 0.018 0.032 0.378 0.055 0.010 0.024 1.11 4.61171E+01 1.0 0.055 0.010 0.024 2 17 0.027 0.019 0.254 0.170 0.064 0.075 1.36 1.83155E+02 2.4 0.170 0.064 0.075 2 18 0.018 0.025 0.223 0.062 0.028 0.051 0.62 5.34456E+22 18.6 0.062 0.028 0.050 3 19 0.025 0.017 0.177 0.147 0.058 0.051 2.07 9.90795E+01 1.7 0.147 0.058 0.049 2 20 0.017 0.032 0.337 0.013 0.005 0.012 0.43 2.57848E+149 79.2 0.013 0.005 0.012 3 21 0.031 0.028 0.367 0.083 0.016 0.050 0.48 1.12025E+03 2.7 0.083 0.016 0.050 2 22 0.017 0.023 0.213 0.157 0.035 0.030 6.57 1.27404E+01 0.6 0.157 0.035 0.028 2 23 0.023 0.022 0.278 0.124 0.051 0.010 104.17 1.40918E+01 0.3 0.124 0.051 0.009 2 24 0.020 0.018 0.280 0.044 0.013 0.025 0.69 2.06649E+05 3.7 0.044 0.013 0.025 2 25 0.027 0.023 0.234 0.169 0.018 0.084 0.54 1.16327E+01 1.0 0.169 0.018 0.084 2 26 0.025 0.017 0.112 0.041 0.008 0.018 0.69 5.50369E+03 2.4 0.041 0.008 0.018 2 27 0.021 0.030 0.135 0.146 0.067 0.094 0.90 9.15850E+05 6.6 0.146 0.067 0.093 2 28 0.026 0.022 0.237 0.076 0.014 0.021 2.34 2.52938E+01 0.8 0.076 0.014 0.021 2 29 0.021 0.025 0.245 0.105 0.038 0.092 0.47 2.27510E+28 28.8 0.105 0.038 0.091 3 30 0.021 0.030 0.277 0.119 0.057 0.106 0.62 2.40327E+33 35.8 0.119 0.057 0.107 3 31 0.024 0.017 0.382 0.092 0.017 0.013 137.54 1.16973E+01 0.2 0.092 0.017 0.013 2 32 0.017 0.033 0.111 0.097 0.020 0.034 1.18 9.80157E+01 1.5 0.097 0.020 0.034 2 33 0.023 0.019 0.150 0.041 0.013 0.038 0.36 3.52296E+107 77.2 0.041 0.013 0.039 3 34 0.026 0.019 0.115 0.065 0.020 0.058 0.38 9.88557E+45 38.4 0.065 0.020 0.058 3 35 0.018 0.029 0.133 0.123 0.039 0.046 1.45 1.60470E+02 1.9 0.123 0.039 0.047 2 36 0.018 0.023 0.212 0.128 0.048 0.049 1.72 1.44867E+02 1.9 0.128 0.048 0.050 2 37 0.029 0.022 0.194 0.131 0.036 0.021 9.28 1.59178E+01 0.6 0.131 0.036 0.021 2 38 0.023 0.029 0.113 0.147 0.030 0.120 0.30 2.46362E+14 17.0 0.147 0.030 0.120 2 39 0.030 0.030 0.132 0.093 0.041 0.071 0.66 5.17855E+13 12.9 0.093 0.041 0.071 2 40 0.028 0.032 0.370 0.034 0.007 0.031 0.23 6.39613E+38 26.6 0.034 0.007 0.031 2 41 0.026 0.028 0.264 0.012 0.002 0.010 0.26 7.91021E+25 13.5 0.012 0.002 0.010 2 42 0.022 0.021 0.139 0.038 0.018 0.003 60.79 2.99343E+01 0.4 0.038 0.018 0.003 2 43 0.029 0.025 0.175 0.131 0.044 0.017 16.96 1.61755E+01 0.5 0.131 0.044 0.015 2 44 0.031 0.019 0.163 0.061 0.013 0.042 0.38 6.60195E+08 7.0 0.061 0.013 0.041 2 45 0.021 0.017 0.238 0.141 0.053 0.106 0.61 7.72114E+09 11.3 0.141 0.053 0.104 2 46 0.023 0.027 0.104 0.124 0.052 0.103 0.59 1.88292E+20 22.0 0.124 0.052 0.105 3 47 0.024 0.029 0.295 0.076 0.025 0.052 0.57 9.65496E+07 6.9 0.076 0.024 0.053 2 48 0.027 0.029 0.378 0.106 0.012 0.091 0.21 2.25508E+02 2.7 0.106 0.012 0.091 2 49 0.030 0.033 0.202 0.024 0.003 0.016 0.24 6.01293E+05 3.5 0.024 0.003 0.016 2 50 0.029 0.020 0.116 0.028 0.008 0.007 2.62 2.59513E+02 1.2 0.028 0.008 0.007 2

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

 Symbol Range References Model Parameters Background mortality rate (/yr) $\mu$ $\frac{1}{60} - \frac{1}{30}$ [11] Recruitment rate into the population (/yr) $\Lambda$ $\frac{1}{60} - \frac{1}{30}$ $^\ddagger$ Rate of HIV progression (/yr) $\rho$ 0.1-0.4 [11] Calibration Conditions HIV prevalence at endemic equilibrium $\widehat{A}$ 0.3-17 [18] HSV-2 prevalence at endemic equilibrium $\widehat{H}$ 4-50 [2,16,22,20] Peak HIV incidence $\widehat{I}$ 10%-50% of $\widehat{A}$ HIV prevalence at time of peak HIV incidence $\widehat{P}$ 5%-95% of $\widehat{A}$ Calibration Parameters Initial value of HIV transmission term (/yr) $\lambda$ determined by calibration procedure Initial value of HSV-2 transmission term (/yr) $\kappa$ determined by calibration procedure Location of transmission term $\alpha$ determined by calibration procedure Shape of transmission term $n$ determined by calibration procedure

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

 Sampled parameters and calibration condition Calibration results $j$ $\Lambda$ $\mu$ $\rho$ $\widehat{A}$ $\widehat{H}$ $\widehat{I}$ $\widehat{P}$ $\lambda^*$ $\kappa^*$ $\alpha^*$ $n^*$ $\widehat{A}^*$ $\widehat{H}^*$ $\widehat{I}^*$ $\widehat{P}^*$ ${m}_1$ ${m}_2$ 1 0.019 0.030 0.245 0.066 0.278 0.017 0.007 27.363 24.30 1.57188E+01 0.3 0.066 0.278 0.023 0.010 2 4 2 0.033 0.019 0.222 0.073 0.499 0.026 0.032 0.623 0.82 1.33110E+03 2.4 0.073 0.499 0.037 0.033 2 4 3 0.018 0.021 0.119 0.105 0.422 0.012 0.028 0.980 1.32 2.00261E+01 0.8 0.105 0.422 0.017 0.030 2 4 4 0.027 0.027 0.183 0.149 0.314 0.025 0.022 9.194 9.80 1.09897E+01 0.4 0.149 0.314 0.036 0.028 2 4 5 0.027 0.021 0.271 0.087 0.125 0.026 0.059 0.462 0.20 1.78937E+07 6.6 0.087 0.125 0.033 0.059 2 4 6 0.027 0.020 0.366 0.013 0.383 0.004 0.010 0.267 0.14 1.04947E+22 11.5 0.013 0.383 0.005 0.010 2 4 7 0.027 0.017 0.101 0.035 0.475 0.011 0.014 0.726 1.09 5.92861E+03 2.2 0.035 0.475 0.014 0.014 2 4 8 0.027 0.025 0.127 0.102 0.408 0.042 0.072 0.407 0.59 1.09128E+10 9.7 0.102 0.408 0.063 0.073 2 4 9 0.027 0.033 0.171 0.042 0.456 0.013 0.009 2.812 4.86 5.02452E+01 0.8 0.042 0.456 0.017 0.010 2 4 10 0.022 0.027 0.188 0.088 0.167 0.030 0.032 1.258 0.95 1.90229E+02 1.7 0.088 0.167 0.045 0.035 2 4 11 0.017 0.024 0.393 0.068 0.354 0.012 0.031 0.593 0.43 4.13872E+01 1.1 0.068 0.354 0.016 0.031 2 4 12 0.022 0.021 0.132 0.108 0.446 0.022 0.086 0.145 0.25 1.00813E+14 14.3 0.108 0.446 0.032 0.085 2 4 13 0.022 0.018 0.274 0.090 0.325 0.027 0.041 0.664 0.44 5.42991E+02 2.2 0.090 0.325 0.037 0.042 2 4 14 0.030 0.029 0.158 0.154 0.447 0.053 0.024 5.053 8.46 1.69047E+01 0.7 0.154 0.447 0.078 0.026 2 4 15 0.020 0.021 0.315 0.099 0.064 0.032 0.087 0.349 0.15 2.83629E+28 28.2 0.099 0.064 0.042 0.088 2 4 16 0.029 0.032 0.213 0.013 0.174 0.004 0.010 0.319 0.25 3.11897E+23 12.3 0.013 0.174 0.007 0.010 2 4 17 0.019 0.026 0.225 0.087 0.313 0.019 0.031 0.941 0.77 6.22601E+01 1.3 0.087 0.313 0.026 0.031 2 4 18 0.017 0.021 0.251 0.092 0.265 0.013 0.027 1.891 1.09 1.44818E+01 0.6 0.092 0.265 0.016 0.029 2 4 19 0.026 0.030 0.167 0.098 0.242 0.016 0.041 0.531 0.50 8.84472E+01 1.5 0.098 0.242 0.024 0.043 2 4 20 0.032 0.018 0.162 0.004 0.253 0.001 0.003 0.315 0.21 1.69847E+39 16.4 0.004 0.253 0.002 0.004 2 4 21 0.023 0.018 0.317 0.114 0.205 0.050 0.024 5.369 2.61 2.49145E+01 0.8 0.114 0.205 0.063 0.025 2 4 22 0.022 0.018 0.349 0.097 0.156 0.013 0.043 1.036 0.37 1.20336E+01 0.7 0.097 0.156 0.015 0.047 2 4 23 0.020 0.021 0.121 0.112 0.244 0.015 0.038 0.671 0.62 4.06640E+01 1.2 0.112 0.244 0.023 0.041 2 4 24 0.021 0.022 0.275 0.131 0.207 0.044 0.109 0.321 0.22 9.01245E+16 19.0 0.131 0.207 0.068 0.111 2 4 25 0.033 0.027 0.128 0.048 0.134 0.021 0.040 0.412 0.35 1.47565E+32 24.1 0.048 0.134 0.038 0.040 2 4 26 0.031 0.032 0.232 0.017 0.195 0.004 0.007 1.048 0.71 1.01842E+03 1.5 0.017 0.195 0.006 0.007 2 4 27 0.032 0.032 0.183 0.163 0.278 0.022 0.141 0.132 0.12 4.93217E+16 21.3 0.163 0.278 0.031 0.142 2 4 28 0.030 0.027 0.363 0.080 0.128 0.031 0.041 0.975 0.39 2.89175E+03 2.8 0.080 0.128 0.036 0.042 2 4 29 0.032 0.026 0.199 0.033 0.041 0.010 0.006 6.908 3.13 4.36917E+01 0.7 0.033 0.041 0.011 0.006 2 4 30 0.017 0.026 0.211 0.018 0.071 0.004 0.006 1.130 0.53 4.52802E+02 1.2 0.018 0.071 0.004 0.006 2 4 31 0.027 0.017 0.339 0.008 0.434 0.001 0.003 0.583 0.36 2.40135E+02 1.0 0.008 0.434 0.002 0.004 2 4 32 0.033 0.029 0.151 0.103 0.352 0.019 0.089 0.158 0.19 1.69030E+26 26.4 0.103 0.352 0.028 0.090 2 4 33 0.026 0.027 0.150 0.111 0.363 0.051 0.074 0.471 0.63 5.48670E+07 7.7 0.111 0.363 0.080 0.076 2 4 34 0.029 0.028 0.327 0.118 0.083 0.041 0.025 6.144 2.99 1.75725E+01 0.7 0.118 0.083 0.051 0.028 2 4 35 0.026 0.019 0.298 0.063 0.433 0.012 0.056 0.107 0.11 8.21837E+39 33.2 0.063 0.433 0.017 0.057 2 4 36 0.019 0.028 0.222 0.102 0.162 0.022 0.018 6.609 4.04 1.50967E+01 0.5 0.102 0.162 0.029 0.021 2 4 37 0.025 0.017 0.321 0.153 0.449 0.023 0.029 13.872 14.87 9.29399E+00 0.3 0.153 0.449 0.033 0.035 3 4 38 0.017 0.026 0.306 0.152 0.272 0.028 0.089 0.310 0.24 2.90089E+02 2.8 0.152 0.272 0.039 0.090 2 4 39 0.021 0.028 0.220 0.027 0.130 0.009 0.015 0.748 0.41 4.01353E+05 3.3 0.027 0.130 0.012 0.015 2 4 40 0.021 0.018 0.102 0.105 0.395 0.049 0.027 2.254 3.06 9.83595E+01 1.4 0.105 0.395 0.071 0.031 2 4 41 0.026 0.021 0.354 0.080 0.480 0.018 0.037 0.437 0.45 1.73014E+02 1.8 0.080 0.480 0.025 0.039 2 4 42 0.017 0.029 0.170 0.124 0.043 0.031 0.107 0.315 0.17 1.61848E+24 26.5 0.124 0.043 0.032 0.108 2 4 43 0.031 0.031 0.370 0.064 0.332 0.012 0.011 31.695 19.29 1.22280E+01 0.3 0.064 0.332 0.014 0.014 2 4 44 0.031 0.026 0.143 0.010 0.375 0.004 0.001 54.142 84.59 3.92541E+01 0.4 0.010 0.375 0.006 0.001 2 4 45 0.021 0.024 0.244 0.098 0.287 0.014 0.019 7.851 6.15 1.13403E+01 0.4 0.098 0.287 0.019 0.022 3 4 46 0.023 0.021 0.385 0.120 0.139 0.052 0.021 11.064 4.83 1.65212E+01 0.6 0.120 0.139 0.063 0.022 2 4 47 0.020 0.032 0.166 0.122 0.420 0.028 0.010 36.862 65.08 1.30911E+01 0.3 0.122 0.420 0.040 0.012 2 4 48 0.024 0.025 0.323 0.152 0.235 0.029 0.101 0.260 0.18 4.59014E+03 4.4 0.152 0.235 0.043 0.102 2 4 49 0.025 0.026 0.384 0.031 0.171 0.011 0.006 6.999 3.11 3.49139E+01 0.6 0.031 0.171 0.014 0.007 2 4 50 0.025 0.018 0.313 0.052 0.479 0.012 0.004 165.669 157.22 1.52477E+01 0.2 0.052 0.479 0.014 0.006 2 4
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