A Behaviour and Disease Model of Testing and Isolation

TL;DR

This work extends an existing "behaviour and disease"(BaD) model by incorporating the dynamics of symptomatic testing and isolation, including the influence of positive tests on perception of infection risk, and provides a dynamical systems analysis of the ordinary differential equations that define this model.

Abstract

There has been interest in the interactions between infectious disease dynamics and behaviour for most of the history of mathematical epidemiology. This has included consideration of which mathematical models best capture each phenomenon, as well as their interaction, but typically in a manner that is agnostic to the exact behaviour in question. Here, we investigate interacting behaviour and disease dynamics specifically related to decisions around testing and isolation. To carry out our investigation we extend an existing "behaviour and disease" (BaD) model by incorporating the dynamics of symptomatic testing and isolation, including the influence of positive tests on perception of infection risk. We provide a dynamical systems analysis of the ordinary differential equations that define this model, providing theoretical results on its behaviour early in a new outbreak (particularly its basic reproduction number) and endemicity of the system (its steady states and associated stability criteria). We then supplement these findings with a numerical analysis to inform how temporal and cumulative outbreak metrics depend on the model parameter values for epidemic and endemic regimes. We observe novel model outputs such as epidemics that have more observed cases detected through increased testing, but are less objectively severe in terms of total number of infections.

Figures (11)

• Figure 1: A compartmental diagram illustrating the BaD model for symptomatic testing. We distinguish between individuals who do not seek testing when symptomatic, referred to as non-behavers (labelled $N$) and those who intend to seek testing when symptomatic, referred to as behavers (labelled $B$). Changes in testing behaviour are governed by the rates of behavioural uptake ($\omega$) and abandonment ($\alpha$). Uptake due to perception of illness threat is with respect to those in the highlighted compartment $T$. The epidemiological states are susceptible ($S$), exposed ($E$), asymptomatic infectious (or pauci-symptomatics) labelled $A$, symptomatic infectious who are labelled $I$ if they do not seek or do not receive a positive test and $T$ if they seek and receive a positive test, and those who are recovered ($R$). The transitions between epidemiological states follow a standard SEIRS framework, with waning immunity where recovered individuals can become susceptible again. We define the model parameters in Table \ref{['tab:model_parameter']}.
• Figure 2: Impact of test effectiveness (${p_T}$), isolation effectiveness ($1-{q_T}$), initial condition of behaviour (${B(0)}$) and the behaviour-free reproduction number (${\mathcal{R}_0^{D}}$) on observed symptomatic (${T}$), undetected symptomatic (${I}$), and total symptomatic (${O=I+T}$) final sizes. The colour bars represent the final size as a percentage of the population and are consistent across each row. Note, we vary the behaviour-free reproduction number instead of $\mathcal{R}_0$ due to the dependence of $\mathcal{R}_0$ on the vertical axes. The dashed grey lines show $\mathcal{R}_0=1$ and the black cross indicates the baseline parameter values from Table \ref{['tab:model_parameter']}.
• Figure 3: The impact of key model parameters on the peak population proportion willing to test (${B}$) for different behaviour-free reproduction numbers (${\mathcal{R}_0^D}$).(a) The initial proportion willing to test and isolate ($B(0)$) varies through changing spontaneous uptake ($\omega_3$); (b) test effectiveness ($p_T$) varies; (c) isolation effectiveness ($1-q_T$) varies, and; (d) the social reproduction number ($R_0^B$) varies through changing the social influence of behaviour ($\omega_1$). Note we vary the behaviour-free reproduction number instead of $\mathcal{R}_0$ because of the dependence of $\mathcal{R}_0$ on the vertical axes. The dashed grey line shows $\mathcal{R}_0=1$ and the black cross shows the baseline parameter values from Table \ref{['tab:model_parameter']}.
• Figure 4: Phase diagrams for different behavioural parameters fixing ${\mathcal{R}_0^D=3.28}$ in the epidemic regime. Column one corresponds to changes in social influence ($\omega_1$), column two corresponds to changes in perception of illness threat ($\omega_2$), and column three corresponds to spontaneous uptake ($\omega_3$), respectively. The rows show the susceptible versus observed epidemic ($S$ vs. $T$), the susceptible versus true epidemic ($S$ vs. $O +A$), and the behaviour versus observed epidemic ($B$ vs. $T$) phase planes. The dashed line represents near removal of the behaviour construct, the dotted lines represent a reduction of the baseline value, the solid line represents the baseline values, and the dash-dot lines represent an increase in the baseline values. The solid lines all represent the baseline values from Table \ref{['tab:model_parameter']}, ensuring the solid lines are comparable across each column. Each simulation was run with the initial conditions $S_B(0) = 10^{-6}, I_N(0) = 10^{-6}, S_N = 1 - S_B - I_N$ and all other compartments empty: this ensured the early-stage dynamics of each model were approximately comparable.
• Figure 5: The effect of isolation and test effectiveness on endemic states. In each panel we display outputs for different combinations of isolation effectiveness ($1-q_T$, y-axis) and test effectiveness ($p_T$, x-axis). In all simulations we used a fixed behaviour-free reproduction number of $\mathcal{R}_0^D=3.28$. (a) Total endemic infection state in the population ($O+A$); (b) Symptomatic endemic infection state in the population ($O$); (c) Observed symptomatic endemic infection state in the population ($T$); (d) Undetected symptomatic infection endemic state in the population ($I$); (e) Proportion of the population willing to test and isolate if symptomatic in the steady state of the population ($B$); (f) Basic reproduction number ($\mathcal{R}_0$). For (a)-(e), colour bars show the percentage of the population in each state. For (f), the colour bar shows the change in reproduction number.

Theorems & Definitions (4)

• Proposition 3.1
• proof
• Proposition 3.2
• proof