Hi all

I am trying to fit some epidemiological data to a compartmental model. However, when fitting with only 400 (chains=4)iterations (since this already takes 12h), it gives the following problems

- 13 divergent transitions
- over 50% of the iterations exceed treedepth
- the chains do not mix
- in a prior-predictive-check, the parameters all get sampled as they should, however, sometimes the ode_bdf-integrator gives the error “CVode failed with error flag -1.”, which apparently hints at not finding an accurate solution
- during the PPC, also sometimes it calculates negative cases numbers in comp_diffM, which obviously does not make sense
- for most parameters, the n_eff is only 2 or 3

Previously, some more simple model worked perfectly well. With some increased complexity however, and unable to find coding-errors, coming from a biology side and being new to stan, i do not really know how to interpret or solve this issue. Below is the code, Adapt_delta is 0.8 currently.

Below is the traceplot for some parameters, however it looks the same for all of them, like it does not get out of the initial conditions…

If anyone has seen a familiar problem, i would be delighted if you could help me.

```
functions {
//function for a new behaviour reproductive number
real modulate_Re(real t, real t1, real t2,real nu1, real nu2,real eta1, real eta2, real xi1, real xi2) {
real sigup;
real sigdown;
sigup=eta1+(1-eta1)/(1+exp(-xi1*(t-t1-nu1)));
sigdown=(1-eta2)/(1+exp(-xi2*(t-t2-nu2)));
return(sigup-sigdown);
}
real[] discretize(int max_days, real shape, real rate){ //function allows delay between symptoms and test
real disc[max_days];
real normconstant;
disc[1]=gamma_cdf(0.5,shape,rate); //first step has only half the interval to integrate
for(i in 2:max_days){
disc[i]=gamma_cdf(i+0.5,shape,rate)-gamma_cdf(i-0.5,shape,rate); //wed want to integrate from day x-0.5 to day x+0.5 but for first different
}
normconstant=sum(disc);
for(i in 1:max_days){
disc[i]=disc[i]/normconstant; //wed want to integrate from day x-0.5 to day x+0.5 but for first different
}
return (disc);
}
real[] SEIR(real t,
real[] y,
real[] theta,
real[] x_r,
int[] x_i
) {
int K = x_i[1];
//real tswitch = x_r[1]; // time of control measures
real dydt[(6*K)]; // SEPIAR (ignoring R) then C
real nI; // total infectious
real beta; // transmission rate
real eta1; // level of Re in August
real eta2; // level of Re in December
real t1; // time of increase
real t2; // ltime of decrease
real nu1;//delays in increase and decrease
real nu2;
real k1; // level of peak in Re
real k2; // level of peak subtracted from max
real xi1; // slope of rise of Re
real xi2; // slope of decrease of Re
real tau_1; // infection to preclinical
real tau_2; // preclinical to symptoms (tau_1+tau_2 = incubation)
real q_P; // contribution of presymptomatics to transmission
real gt; // generation time
real mu; // infectious duration for symptomatics
real psi; // probability of symptoms
real kappa; // reduced transmissibility of preclinical and asymptomatics
real p_tswitch; // switch function
real contact[K*K]; // contact matrix, first K values, corresponds to number of contact between age class 1 and other classes, etc
real f_inf[K]; // force of infection
real init[K*2]; // initial values
real age_dist[K]; // age distribution of the general population
real pi; // number of cases at t0
// Estimated parameters
beta = theta[1];
xi1 = theta[2];
xi2=theta[3];
pi = theta[4];
psi = theta[5];
eta1=theta[6];
eta2=theta[7];
nu1=theta[8];
nu2=theta[9];
// Fixed parameters
tau_1 = x_r[1];
tau_2 = x_r[2];
q_P = x_r[3];
gt = x_r[4];
t1=x_r[5];
t2=x_r[6];
// Composite parameters
mu = (1-q_P)/(gt-1/tau_1-1/tau_2);
kappa = (q_P*tau_2*psi)/((1-q_P)*mu-(1-psi)*q_P*tau_2);
// Contact matrix
contact = x_r[7:(6+K*K)];
// Initial conditions
for(k in 1:K){
age_dist[k] = x_r[6+K*K + k];
init[k] = age_dist[k] * (1-pi);
init[K+k] = age_dist[k] * pi;
}
// Total number of infectious people
p_tswitch = modulate_Re(t,t1,t2,nu1,nu2,eta1,eta2,xi1,xi2);
// Force of infection by age classes: beta * p_tswitch * sum((number of infected people by age + kappa*number of preclinical by age + kappa*number of asympto) / (total number of people by age) * (number of contact by age))
for(k in 1:K) {
f_inf[k] = beta * p_tswitch * sum((to_vector(y[(3*K+1):(4*K)])+kappa*to_vector(y[(2*K+1):(3*K)])+kappa*to_vector(y[(4*K+1):(5*K)]))./ to_vector(age_dist) .* to_vector(contact[(K*(k-1)+1):(k*K)]));
}
// Compartments
for (k in 1:K) {
// S: susceptible
dydt[k] = - f_inf[k] * (y[k]+init[k]);
// E: incubating (not yet infectious)
dydt[K+k] = f_inf[k] * (y[k]+init[k]) - tau_1 * (y[K+k]+init[K+k]);
// P: presymptomatic (incubating and infectious)
dydt[2*K+k] = tau_1 * (y[K+k]+init[K+k]) - tau_2 * y[2*K+k];
// I: symptomatic
dydt[3*K+k] = psi * tau_2 * y[2*K+k] - mu * y[3*K+k];
// A: asymptomatic
dydt[4*K+k] = (1-psi) * tau_2 * y[2*K+k] - mu * y[4*K+k];
// C: cumulative number of infections by date of disease onset
dydt[5*K+k] = psi * tau_2 * y[2*K+k];
}
return(dydt);
}
}
data {
// Structure
int K; // number of age classes
vector[K] age_dist; // age distribution of the population
int pop_t; // total population
//real tswitch; // time of introduction of control measures
// Controls
real t0; //starting time
int t_data; //time of first data
int S;
real ts[S]; // time bins
int inference; // 0: simulating from priors; 1: fit to data
int doprint;
// Data to fit
int D; // number of days with reported incidence
int incidence_cases[D]; // overal incidence for W weeks
int incidence_deaths[D]; // overal incidence for W weeks
int agedistr_cases[K]; // number of cases at tmax for the K age classes
int agedistr_deaths[K]; // mortality at tmax for the K age classes
int tests_age_groups[D,K];
// Priors
real p_beta;
real p_eta[2];
real p_pi[2];
real p_epsilon[2];
//real p_rho[2]; replaced by lambda
real p_phi;
real p_xi;
real p_nu;
real p_psi[2];
real p_lambda;//prior for initial slope of saturation function
// Fixed parameters
real contact[K*K]; // contact matrix
real p_q_P; // proportion of transmission that is caused by presymptomatics
real p_incubation; // incubation period
real p_preclinical; // preclinical period (part of the incubation with possible transmission)
real p_generation_time;
real p_children_trans; // relative transmissibility in children 1-10
// Fixed corrections
real p_report_80plus; // fixed ascertainment proportion for ages 80+
// Fixed delays
int G;
int D_max;
real p_gamma[G]; // from onset to death
real mean_shape;//mean for prior for shape of gamma which captures testing delay
real mean_rate;//mean for prior for rate of gamma which captures testing delay
real p_nu1nu2[2];//prior parameters for t1 (2 of them) and t2
real p_eta1eta2[4];//prior parameters for the end and start levels of the sigmoids
int t1;//estimation of increase time onset
int t2;// estimation of start of decrease
}
transformed data {
real tau_1 = 1.0 / (p_incubation - p_preclinical);
real tau_2 = 1.0 / p_preclinical;
real q_P = p_q_P;
real gt = p_generation_time;
real x_r[6+K*K+K]; // 6 parameters + K*K contact matrix parameters + K age_dist parameters
int x_i[1] = {K};
real init[K*6] = rep_array(0.0, K*6); // initial values
real contact2[K*K] = contact;
for(i in 1:(2*K)) contact2[i] = contact[i] * p_children_trans; // apply lower transmissibility in children
x_r[1] = tau_1;
x_r[2] = tau_2;
x_r[3] = q_P;
x_r[4] = gt;
x_r[5] = t1;//time of increase in Re
x_r[6] = t2;//time of decrease in Re
x_r[7:(6+K*K)] = contact2;
for(k in 1:K) {
x_r[6+K*K+k] = age_dist[k];
}
}
parameters{
real<lower=0,upper=1> beta; // base transmission rate
real<lower=0,upper=1> eta1; // reduction in transmission rate after incresed relaxations
real<lower=0,upper=1> eta2; // reduction in transmission rate after quarantine measures
real<lower=0,upper=30> nu1; //delays from a manually chosen timepoint which is start increase and decrease
real<lower=0,upper=30> nu2; //delays from a manually chosen timepoint which is start increase and decrease
vector<lower=0,upper=1> [K] epsilon; // age-dependent mortality probability
real<lower=0, upper=1> pi; // number of cases at t0
real<lower=0> phi[2]; // variance parameters
real<lower=0,upper=1> xi1; // slope of Re_increase
real<lower=0,upper=1> xi2; // slope of Re_decrease
real<lower=0,upper=1> psi; // proportion of symptomatics
real<lower=0,upper=10> shape; //
real<lower=0,upper=1> rate; //
real<lower=0,upper=1> lambda; //initial slope of the saturation function
}
transformed parameters {
// change of format for integrate_ode_rk45
real theta[9]; // vector of parameters
real y[S,K*6]; // raw ODE output
vector[K] comp_C[S+G];
vector[K] comp_diffC[S+G];
//vector[K] comp_T[S+D_max];//JE
vector[K] comp_diffT[S+D_max];//JE
vector[K] comp_M[S+G];
vector[K] comp_diffM[S+G];
vector[D_max] p_delay;
// outcomes
vector[K] output_incidence_cases_age[D]; // overall case incidence by day and age group
vector[D] output_incidence_cases; // overall case incidence by day
vector[K] output_cum_cases_age[D]; //cumulative number of cases by age
vector[D] output_incidence_deaths; // overal mortality incidence by day
simplex[K] output_agedistr_cases; // final age distribution of cases
simplex[K] output_agedistr_deaths; // final age distribution of deaths
// change of format for integrate_ode_rk45
theta[1:9] = {beta, xi1,xi2,pi,psi,eta1,eta2,nu1,nu2};
// run ODE solver
y = integrate_ode_bdf(
SEIR, // ODE function
init, // initial states
t0, // t0
ts, // evaluation dates (ts)
theta, // parameters
x_r, // real data
x_i, // integer data
1.0E-10, 1.0E-10, 1.0E3); // tolerances and maximum steps
// extract and format ODE results (1.0E-9 correction to avoid negative values due to unprecise estimates of zeros as tolerance is 1.0E-10)
for(i in 1:S) {
comp_C[i] = (to_vector(y[i,(5*K+1):(6*K)]) + 1.0E-9) * pop_t;
comp_diffC[i] = i==1 ? comp_C[i,] : 1.0E-9*pop_t + comp_C[i,] - comp_C[i-1,]; // lagged difference of cumulative incidence of symptomatics
}
for(i in 1:D_max){
p_delay[i]=discretize(D_max,shape,rate)[i]; // calculate the discretized fit
}
// Incidence and cumulative incidence after S
for(g in 1:G) {
comp_C[S+g] = comp_C[S];
comp_diffC[S+g] = rep_vector(1.0E-9,K);
}
for(i in 1:(S+D_max)){
comp_diffT[i] = rep_vector(1.0E-9,K);
}
for(i in 1:S){
for(d in 0:(D_max-1)){
comp_diffT[i+d] += comp_diffC[i] * p_delay[d+1]; //People are tested at onset plus
}
}
// Mortality
for(i in 1:(S+G)){
comp_diffM[i] = rep_vector(1.0E-9,K);
}
for(i in 1:S){
for(g in 1:G){
comp_diffM[i+g] += comp_diffC[i] .* epsilon * p_gamma[g]; //new deaths at i+g are new cases at i*mortality*delay_prob for g
}
}
for(i in 1:(S+G)){
for(k in 1:K){
comp_M[i,k] = sum(comp_diffM[1:i,k]);// Compute outcomes
}
}
for(i in t_data:S){
output_incidence_cases_age[i-t_data+1] = to_vector(tests_age_groups[i-t_data+1]) ./ ( to_vector(tests_age_groups[i-t_data+1]) ./ to_vector(comp_diffT[i]) + 1/lambda );
output_incidence_cases[i-t_data+1] = sum(output_incidence_cases_age[i-t_data+1]);
for(k in 1:K){
output_cum_cases_age[i-t_data+1,k] = sum(output_incidence_cases_age[1:(i-t_data+1),k]);// Compute outcomes
}
output_incidence_deaths[i-t_data+1] = sum(comp_diffM[i]);
}
output_agedistr_cases = output_cum_cases_age[D,] ./ sum(output_cum_cases_age[D,]);
output_agedistr_deaths = (comp_M[D,]) ./ sum(comp_M[D,]);
}
model {
// priors
beta ~ beta(p_beta,p_beta);
eta1 ~ beta(p_eta1eta2[1],p_eta1eta2[2]);// draw the start and end levels of the sigmoids p_Rleves has prior parameters for all of those
eta2 ~ beta(p_eta1eta2[3],p_eta1eta2[4]);// draw the start and end levels of the sigmoids
for(k in 1:K){
epsilon[k] ~ beta(p_epsilon[1],p_epsilon[2]);
}
pi ~ beta(p_pi[1],p_pi[2]); // p_pi=C(1,999)
phi ~ exponential(p_phi);
xi1 ~ beta(1,1); //draw for slopes xi1 and xi2 of the modulate_Re function
xi2 ~ beta(1,1);
nu1 ~ exponential(p_nu1nu2[1]); // pnu1nu2=c(1/20,1/30)
nu2 ~ exponential(p_nu1nu2[2]);
lambda ~ beta(1,1);
psi ~ beta(p_psi[1],p_psi[2]);
shape ~ exponential(1/mean_shape);
rate ~ exponential(1/mean_rate);
// likelihood
if (inference!=0) {
for(i in 1:D) {
target += neg_binomial_2_lpmf( incidence_cases[i] | output_incidence_cases[i], output_incidence_cases[i]/phi[1]);
target += neg_binomial_2_lpmf( incidence_deaths[i] | output_incidence_deaths[i], output_incidence_deaths[i]/phi[2]);
}
target += multinomial_lpmf(agedistr_cases | output_agedistr_cases);
target += multinomial_lpmf(agedistr_deaths | output_agedistr_deaths);
}
}
```