diff --git a/NEWS.md b/NEWS.md
index 9ea7dc325..816f91704 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -53,6 +53,7 @@
 
 - Delay distribution discretisation now properly accounts for primary event censoring during model fitting, matching the correction already applied on the R side since v1.8.0. This improves accuracy for short delays where the observation window is large relative to the delay.
 - Left truncation of delay distributions (e.g. excluding generation times of zero) is now handled analytically rather than by zeroing and renormalising, giving more accurate PMFs near the truncation point.
+- Refactored the Gaussian process, convolution, and observation model Stan functions for efficiency and readability, including a shared `matern_indices()` helper for the Matern spectral densities, outer-product basis function construction in `PHI()` and `PHI_periodic()`, and a shared `reporting_phi()` helper for the negative binomial overdispersion.
 
 ## Package changes
 
diff --git a/inst/stan/functions/convolve.stan b/inst/stan/functions/convolve.stan
index 2a84df929..76e20cbdc 100644
--- a/inst/stan/functions/convolve.stan
+++ b/inst/stan/functions/convolve.stan
@@ -40,7 +40,7 @@ array[] int calc_conv_indices_len(int s, int xlen, int ylen) {
   int s_minus_ylen = s - ylen;
   int start_x = max(1, s_minus_ylen + 1);
   int end_x = xlen;
-  int start_y = max(1, 1 - s_minus_ylen);;
+  int start_y = max(1, 1 - s_minus_ylen);
   int end_y = ylen + xlen - s;
   return {start_x, end_x, start_y, end_y};
 }
@@ -62,7 +62,6 @@ array[] int calc_conv_indices_len(int s, int xlen, int ylen) {
 vector convolve_with_rev_pmf(vector x, vector y, int len) {
   int xlen = num_elements(x);
   int ylen = num_elements(y);
-  vector[len] z;
 
   if (xlen + ylen - 1 < len) {
     reject("convolve_with_rev_pmf: len is longer than x and y convolved");
@@ -72,16 +71,17 @@ vector convolve_with_rev_pmf(vector x, vector y, int len) {
     reject("convolve_with_rev_pmf: len is shorter than x");
   }
 
+  vector[len] z;
+
   for (s in 1:xlen) {
     array[4] int indices = calc_conv_indices_xlen(s, xlen, ylen);
     z[s] = dot_product(x[indices[1]:indices[2]], y[indices[3]:indices[4]]);
   }
 
-  if (len > xlen) {
-    for (s in (xlen + 1):len) {
-      array[4] int indices = calc_conv_indices_len(s, xlen, ylen);
-      z[s] = dot_product(x[indices[1]:indices[2]], y[indices[3]:indices[4]]);
-    }
+  // runs zero times unless len > xlen
+  for (s in (xlen + 1):len) {
+    array[4] int indices = calc_conv_indices_len(s, xlen, ylen);
+    z[s] = dot_product(x[indices[1]:indices[2]], y[indices[3]:indices[4]]);
   }
 
   return z;
diff --git a/inst/stan/functions/gaussian_process.stan b/inst/stan/functions/gaussian_process.stan
index 2f067b4bd..ca1f80a21 100644
--- a/inst/stan/functions/gaussian_process.stan
+++ b/inst/stan/functions/gaussian_process.stan
@@ -23,6 +23,26 @@ vector diagSPD_EQ(real alpha, real rho, real L, int M) {
   return factor * exp(exponent * square(indices));
 }
 
+/**
+  * Squared spectral indices shared by the Matern kernels
+  *
+  * Returns `square(pi() / (2 * L) * linspaced_vector(M, 1, M))`, the term that
+  * appears in the denominator of every Matern spectral density. The
+  * `linspaced_vector()` call uses data-only bounds and is scaled afterwards so
+  * the function compiles under the data-only argument constraint in older Stan
+  * versions (e.g. the one shipped with rstan).
+  *
+  * @param M Number of basis functions
+  * @param L Length of the interval
+  * @return A vector of squared spectral indices
+  *
+  * @ingroup estimates_smoothing
+  */
+vector matern_indices(int M, real L) {
+  vector[M] indices = linspaced_vector(M, 1, M);
+  return square(pi() / (2 * L) * indices);
+}
+
 /**
   * Spectral density for 1/2 Matern (Ornstein-Uhlenbeck) kernel
   *
@@ -35,10 +55,8 @@ vector diagSPD_EQ(real alpha, real rho, real L, int M) {
   * @ingroup estimates_smoothing
   */
 vector diagSPD_Matern12(real alpha, real rho, real L, int M) {
-  vector[M] indices = linspaced_vector(M, 1, M);
-  real factor = 2;
-  vector[M] denom = rho * ((1 / rho)^2 + pow(pi() / 2 / L * indices, 2));
-  return alpha * sqrt(factor * inv(denom));
+  vector[M] denom = 1 / rho + rho * matern_indices(M, L);
+  return alpha * sqrt(2 ./ denom);
 }
 
 /**
@@ -53,10 +71,9 @@ vector diagSPD_Matern12(real alpha, real rho, real L, int M) {
   * @ingroup estimates_smoothing
   */
 vector diagSPD_Matern32(real alpha, real rho, real L, int M) {
-  vector[M] indices = linspaced_vector(M, 1, M);
-  real factor = 2 * alpha * pow(sqrt(3) / rho, 1.5);
-  vector[M] denom = (sqrt(3) / rho)^2 + pow((pi() / 2 / L) * indices, 2);
-  return factor * inv(denom);
+  real factor = 2 * alpha * (sqrt(3) / rho)^1.5;
+  vector[M] denom = 3 / square(rho) + matern_indices(M, L);
+  return factor ./ denom;
 }
 
 /**
@@ -71,11 +88,9 @@ vector diagSPD_Matern32(real alpha, real rho, real L, int M) {
   * @ingroup estimates_smoothing
   */
 vector diagSPD_Matern52(real alpha, real rho, real L, int M) {
-  vector[M] indices = linspaced_vector(M, 1, M);
   real factor = 16 * pow(sqrt(5) / rho, 5);
-  vector[M] denom =
-    3 * pow((sqrt(5) / rho)^2 + pow((pi() / 2 / L) * indices, 2), 3);
-  return alpha * sqrt(factor * inv(denom));
+  vector[M] denom = 3 * pow(5 / square(rho) + matern_indices(M, L), 3);
+  return alpha * sqrt(factor ./ denom);
 }
 
 /**
@@ -110,11 +125,8 @@ vector diagSPD_Periodic(real alpha, real rho, int M) {
   * @ingroup estimates_smoothing
   */
 matrix PHI(int N, int M, real L, vector x) {
-  matrix[N, M] phi = sin(
-    diag_post_multiply(
-      rep_matrix(pi() / (2 * L) * (x + L), M), linspaced_vector(M, 1, M)
-    )
-  ) / sqrt(L);
+  row_vector[M] k = linspaced_row_vector(M, 1, M);
+  matrix[N, M] phi = sin((pi() / (2 * L) * (x + L)) * k) / sqrt(L);
   return phi;
 }
 
@@ -130,10 +142,9 @@ matrix PHI(int N, int M, real L, vector x) {
   * @ingroup estimates_smoothing
   */
 matrix PHI_periodic(int N, int M, real w0, vector x) {
-  matrix[N, M] mw0x = diag_post_multiply(
-    rep_matrix(w0 * x, M), linspaced_vector(M, 1, M)
-  );
-  return append_col(cos(mw0x), sin(mw0x));
+  row_vector[M] k = linspaced_row_vector(M, 1, M);
+  matrix[N, M] w0xk = (w0 * x) * k;
+  return append_col(cos(w0xk), sin(w0xk));
 }
 
 /**
@@ -210,7 +221,7 @@ vector update_gp(matrix PHI, int M, real L, real alpha,
     } else if (nu == 2.5) {
       diagSPD = diagSPD_Matern52(alpha, rho, L, M);
     } else {
-      reject("nu must be one of 1/2, 3/2 or 5/2; found nu=", nu);
+      reject("nu must be one of 0.5, 1.5, or 2.5; found nu=", nu);
     }
   }
   return PHI * (diagSPD .* eta);
diff --git a/inst/stan/functions/observation_model.stan b/inst/stan/functions/observation_model.stan
index d903a04a6..e149be104 100644
--- a/inst/stan/functions/observation_model.stan
+++ b/inst/stan/functions/observation_model.stan
@@ -106,6 +106,25 @@ void truncation_lp(array[] real truncation_mean, array[] real truncation_sd,
   }
 }
 
+/**
+ * Negative binomial overdispersion for the reporting model
+ *
+ * Converts the reporting overdispersion parameter into the `phi` of the
+ * negative binomial. When no overdispersion is modelled a large value is
+ * returned so the negative binomial behaves like a Poisson.
+ *
+ * @param reporting_overdispersion Real value for reporting overdispersion.
+ * @param model_type Integer indicating the model type (0 for Poisson, >0 for
+ * Negative Binomial).
+ *
+ * @return The negative binomial overdispersion `phi`.
+ *
+ * @ingroup observation_model
+ */
+real reporting_phi(real reporting_overdispersion, int model_type) {
+  return model_type ? inv_square(reporting_overdispersion) : 1e5;
+}
+
 /**
  * Update log density for reported cases
  *
@@ -127,7 +146,7 @@ void report_lp(array[] int cases, array[] int case_times, vector reports,
   int n = num_elements(case_times); // number of observations
   vector[n] obs_reports = reports[case_times]; // reports at observation time
   if (model_type) {
-    real phi = inv_square(reporting_overdispersion);
+    real phi = reporting_phi(reporting_overdispersion, model_type);
     if (weight == 1) {
       cases ~ neg_binomial_2(obs_reports, phi);
     } else {
@@ -197,7 +216,7 @@ vector report_log_lik(array[] int cases, vector reports,
       log_lik[i] = poisson_lpmf(cases[i] | reports[i]) * weight;
     }
   } else {
-    real phi = inv_square(reporting_overdispersion);
+    real phi = reporting_phi(reporting_overdispersion, model_type);
     for (i in 1:t) {
       log_lik[i] = neg_binomial_2_lpmf(
         cases[i] | reports[i], phi
@@ -257,10 +276,7 @@ int neg_binomial_2_safe_rng(real mu, real phi) {
 array[] int report_rng(vector reports, real reporting_overdispersion, int model_type) {
   int t = num_elements(reports);
   array[t] int sampled_reports;
-  real phi = 1e5;
-  if (model_type) {
-    phi = inv_square(reporting_overdispersion);
-  }
+  real phi = reporting_phi(reporting_overdispersion, model_type);
 
   for (s in 1:t) {
     sampled_reports[s] = neg_binomial_2_safe_rng(reports[s], phi);
diff --git a/tests/testthat/test-stan-guassian-process.R b/tests/testthat/test-stan-guassian-process.R
index 085285e67..08ed026b2 100644
--- a/tests/testthat/test-stan-guassian-process.R
+++ b/tests/testthat/test-stan-guassian-process.R
@@ -24,6 +24,18 @@ test_that("diagSPD_EQ returns correct dimensions and values", {
   expect_equal(result, expected_result, tolerance = 1e-8)
 })
 
+test_that("matern_indices returns correct dimensions and values", {
+  L <- 1.0
+  M <- 5
+  result <- matern_indices(M, L)
+  expect_equal(length(result), M)
+  expect_true(all(result > 0))
+  # Check specific values for known inputs
+  indices <- linspaced_vector(M, 1, M)
+  expected_result <- (pi / (2 * L) * indices)^2
+  expect_equal(result, expected_result, tolerance = 1e-8)
+})
+
 test_that("diagSPD_Matern functions return correct dimensions and values", {
   alpha <- 1.0
   rho <- 2.0
diff --git a/tests/testthat/test-stan-observation_model.R b/tests/testthat/test-stan-observation_model.R
index a871d1f58..d0ada777e 100644
--- a/tests/testthat/test-stan-observation_model.R
+++ b/tests/testthat/test-stan-observation_model.R
@@ -1,6 +1,17 @@
 skip_on_cran()
 skip_on_os("windows")
 
+test_that("reporting_phi returns overdispersion or Poisson fallback", {
+  reporting_overdispersion <- 0.5
+  # Negative binomial: phi is the inverse square of the overdispersion
+  expect_equal(
+    reporting_phi(reporting_overdispersion, 1),
+    1 / reporting_overdispersion^2
+  )
+  # Poisson: large phi so the negative binomial behaves like a Poisson
+  expect_equal(reporting_phi(reporting_overdispersion, 0), 1e5)
+})
+
 test_that("day_of_week_effect applies day of week effect correctly", {
   reports <- c(100, 200, 300, 400, 500, 600, 700, 800, 900, 1000)
   day_of_week <- c(1, 2, 3, 1, 2, 3, 1, 2, 3, 1)