Modelling under-reporting in epidemics

Under-reporting of infected cases is crucial for many diseases because of the bias it can introduce when making inference for the model parameters. The objective of this paper is to study the effect of under-reporting in epidemics by considering the stochastic Markovian SIR epidemic in which various reporting processes are incorporated. In particular, we first investigate the effect on the estimation process of ignoring under-reporting when it is present in an epidemic outbreak. We show that such an approach leads to under-estimation of the infection rate and the reproduction number. Secondly, by allowing for the fact that under-reporting is occurring, we develop suitable models for estimation of the epidemic parameters and explore how well the reporting rate and other model parameters can be estimated. We consider the case of a constant reporting probability and also more realistic assumptions which involve the reporting probability depending on time or the source of infection for each infected individual. Due to the incomplete nature of the data and reporting process, the Bayesian approach provides a natural modelling framework and we perform inference using data augmentation and reversible jump Markov chain Monte Carlo techniques.

Authors and Affiliations

1. Biomathematics and Statistics Scotland, Kings Buildings, Edinburgh, EH9 3JZ, UK

   Kokouvi M. Gamado, George Streftaris & Stan Zachary

2. School of Mathematical and Computer Sciences, Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS, UK

   George Streftaris & Stan Zachary

Corresponding author

Correspondence to Kokouvi M. Gamado.

Appendix

1.1 A.1: Eversible jump MCMC algorithm

An individual (say \(k\)) will always have one of the following states in this algorithm:

• \(0\) - Susceptible; i.e \(k \in \mathcal S \)

• \(1\) - Removed before time \(T\) and reported; i.e \(k \in \mathcal R _r\)

• \(2\) - Infected but not removed before time \(T\) (censored); i.e \(k \in \mathcal I \cap \bar{\mathcal{R }}\)

• \(3\) - Removed before time \(T\), but not reported; i.e \(k \in \mathcal R _u\).

The possible algorithm transitions are presented schematically as follow:

We now describe the algorithm in details:

• Choose an individual at random (let us say \(k\)).

• If the state of \(k\) is \(1\) (meaning that the individual was removed before \(T\) and reported), we update its infection time uniformly in \((T_0, r_k)\). The proposed infection time is accepted with probability:

  $$\begin{aligned} A_{1\rightarrow 1} = \min \left\{ 1, \frac{L(\beta , \gamma , p; \pmb {s}^{(new)}, \pmb {r})}{L(\beta , \gamma , p; \pmb {s}^{(old)}, \pmb {r})}\right\} . \end{aligned}$$

  More efficiently we make use of the model assumption by proposing the new infection time so that \((s_k - r_k) \sim \text {Exp}(\gamma )\) where \(s_k\) is the proposed infection time. In this case the acceptance probability is:

  $$\begin{aligned} A_{1\rightarrow 1}' = \min \left\{ 1, \frac{L(\beta , \gamma ; \pmb {s}^{(new)}, \pmb {r})}{L(\beta , \gamma ; \pmb {s}^{(old)}, \pmb {r})}*\frac{\exp \{-\gamma (r_k - s_k')\}}{\exp \{-\gamma (r_k - s_k)\}}\right\} \end{aligned}$$

  where \(s_k'\) is the current infection time of the individual \(k\). There is no change in state.

• If the state of \(k\) is \(0\) (susceptible individual) we propose, each with probability \(1/2\), to add a new infection time, or add a pair of infection and removal times:

  • Generate an infection time \(s_k\) uniformly in \((T_0, T)\) and add it with probability

    $$\begin{aligned} A_{0\rightarrow 2} = \min \left\{ 1, \frac{L(\beta , \gamma , p; \pmb {s}^{(new)}, \pmb {r})}{L(\beta , \gamma , p; \pmb {s}^{(old)}, \pmb {r})}\frac{2(T - T_0)}{3}\right\} . \end{aligned}$$

    The \(1/3\) term is the probability of proposing the reverse move. If accepted, the state of the individual becomes \(2\) (which characterises individuals that are infected but not removed before \(T\)), i. e

    $$\begin{aligned} |S| = |S| - 1 \ \ \ \ \ \text { and } \ \ \ \ \ |\mathcal I \cap \bar{\mathcal{R }}| = |\mathcal I \cap \bar{\mathcal{R }}| + 1. \end{aligned}$$

  • Propose a removal time \(r_k\) uniformly in \((T_0, T)\) and an infection time \(s_k\) in \((T_0, r_k)\) and add the pair with probability

    $$\begin{aligned} A_{0\rightarrow 3} = \min \left\{ 1, \frac{2(T-T_0)(r_k - T_0)}{3}\frac{L(\beta , \gamma , p; \pmb {s}^{(new)}, \pmb {r}^{(new)})}{L(\beta , \gamma , p; \pmb {s}^{(old)}, \pmb {r}^{(old)})}\right\} . \end{aligned}$$

    If this move is accepted, the state of the individual \(k\) becomes \(3\) (which represents individuals that are removed before time \(T\) but not reported), i.e

    $$\begin{aligned} |S| = |S| - 1 \ \ \ \ \ \text { and } \ \ \ \ \ |\mathcal R _u| = |\mathcal R _u| + 1. \end{aligned}$$

• If state of \(k\) is \(2\) (infected but not removed) we update the infection time, or add a removal time, or delete the infection time, each with probability \(1/3\):

  • Update the added infection time by proposing a new infection time uniformly in \((T_0, T)\). The acceptance probability is \(A_{2\rightarrow 2} = A_{1\rightarrow 1}\). There is no change in state.

  • Propose to add a removal time chosen uniformly in \((s_k, T)\) with probability

    $$\begin{aligned} A_{2\rightarrow 3} = \min \left\{ 1, \frac{L(\beta , \gamma , p; \pmb {s}^{(new)}, \pmb {r})}{L(\beta , \gamma , p; \pmb {s}^{(old)}, \pmb {r})}(T - s_k)\right\} . \end{aligned}$$

    The state of \(k\) becomes \(3\) if the move is accepted, i.e

    $$\begin{aligned} |\mathcal I \cap \bar{\mathcal{R }}| = |\mathcal I \cap \bar{\mathcal{R }}| - 1 \ \ \ \ \ \text { and } \ \ \ \ \ |\mathcal R _u| = |\mathcal R _u| + 1. \end{aligned}$$

  • Delete the added infection time with probability

    $$\begin{aligned} A_{2\rightarrow 0} = \min \left\{ 1, \frac{L(\beta , \gamma , p; \pmb {s}^{(new)}, \pmb {r})}{L(\beta , \gamma , p; \pmb {s}^{(old)}, \pmb {r})}\frac{3}{2(T - T_0)}\right\} . \end{aligned}$$

    This individual becomes susceptible (state \(0\)) if the move is accepted, i.e

    $$\begin{aligned} |\mathcal I \cap \bar{\mathcal{R }}| = |\mathcal I \cap \bar{\mathcal{R }}| - 1 \ \ \ \ \ \text { and } \ \ \ \ \ |\mathcal S | = |\mathcal S | + 1. \end{aligned}$$

• If state of \(k\) is \(3\) we either propose, with probability \(1/3\), to delete the added removal time, or update the pair of infection and removal times, or delete the pair of infection and removal times:

  • Delete the removal time previously added with probability

    $$\begin{aligned} A_{3\rightarrow 2} = \min \left\{ 1, \frac{L(\beta , \gamma , p; \pmb {s}^{(new)}, \pmb {r})}{L(\beta , \gamma , p; \pmb {s}^{(old)}, \pmb {r})} \frac{1}{T - s_k}\right\} . \end{aligned}$$

    The state becomes \(2\) when this removal is accepted, i.e

    $$\begin{aligned} |\mathcal R _u| = |\mathcal R _u| - 1 \ \ \ \ \ \text { and } \ \ \ \ \ |\mathcal I \cap \bar{\mathcal{R }}| = |\mathcal I \cap \bar{\mathcal{R }}| + 1. \end{aligned}$$

  • Update the pair of infection and removal times of \(k\) (with \(T_0 \le s_k < r_k \le T\)) with probability

    $$\begin{aligned} A_{3\rightarrow 3} = \min \left\{ 1, \frac{L(\beta , \gamma , p; \pmb {s}^{(new)}, \pmb {r})}{L(\beta , \gamma , p; \pmb {s}^{(old)}, \pmb {r})} \frac{r_k - T_0}{r_k' - T_0}\right\} \end{aligned}$$

    where \(r_k'\) is the removal time of individual \(k\) before the new proposed one \(r_k\). There is no change in state.

  • Delete the pair of infection and removal times with probability

    $$\begin{aligned} A_{3\rightarrow 0} = \min \left\{ 1, \frac{L(\beta , \gamma , p; \pmb {s}^{(new)}, \pmb {r})}{L(\beta , \gamma , p; \pmb {s}^{(old)}, \pmb {r})} \frac{3}{2(T - T_0)(r_k - T_0)}\right\} . \end{aligned}$$

    The state of the individual \(k\) becomes \(0\) if the deletion is accepted, i.e

    $$\begin{aligned} |\mathcal R _u| = |\mathcal R _u| - 1 \ \ \ \ \ \text { and } \ \ \ \ \ |\mathcal S | = |\mathcal S | + 1. \end{aligned}$$

In the case of completed epidemic, the set of possible states becomes \(\left\{ 0, 1, 3 \right\} \) where state \(0\) stands for susceptible individuals, \(1\) for removed and reported individuals and \(3\) for removed but non-reported individuals. This reduces the \(8\) steps of the algorithm above to \(4\) with simple changes.

1.2 A.2: M-H within Gibbs algorithm for the time-dependent reporting probability

• Update \(\beta \) and \(\gamma \) following Gibbs steps using Eqs. (6) and (7);

• Update Event times following RJMCMC algorithm described in Appendix A.1;

• For each accepted event times, count the number of removed individuals before and after the change point \(a_1\);

• Identify the number of reported and unreported cases before and after \(a_1\);

• Update the reporting probabilities following Eqs. (10);

• Repeat the above steps until convergence.

1.3 A.3: Auto-correlation functions for \(\beta \), \(\gamma \), \(p_1\) and \(p_2\) in the case of dynamic reporting

ACFs for \(\beta \), \(\gamma \), \(p_1\) and \(p_2\) after burn-in period of 1,000 iterations and a thinning of \(20\) samples, in the case of completed epidemic with reporting depending on the source of infection and using \((\mathcal U (0, 1), \mathcal B (8, 2))\) (a), (c), (e) and (g) and \((\mathcal B (10, 10), \mathcal B (24, 6))\) (b), (d), (f) and (h) for \((p_1, p_2)\)

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1.4 A.4: Correlation between \(\beta \), \(\gamma \), \(p_1\) and \(p_2\) in the case of dynamic reporting

Correlation between the model parameters \(\beta \), \(\gamma \), \(p_1\) and \(p_2\) in the case of dynamic reporting using \((\mathcal U (0, 1), \mathcal B (8, 2))\) prior for \((p_1, p_2)\)

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