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Spatially-implicit modelling of disease-behaviour interactions in the context of non-pharmaceutical interventions

  • * Corresponding author: N. Ringa

    * Corresponding author: N. Ringa 
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  • Pair approximation models have been used to study the spread of infectious diseases in spatially distributed host populations, and to explore disease control strategies such as vaccination and case isolation. Here we introduce a pair approximation model of individual uptake of non-pharmaceutical interventions (NPIs) for an acute self-limiting infection, where susceptible individuals can learn the NPIs either from other susceptible individuals who are already practicing NPIs ("social learning"), or their uptake of NPIs can be stimulated by being neighbours of an infectious person ("exposure learning"). NPIs include individual measures such as hand-washing and respiratory etiquette. Individuals can also drop the habit of using NPIs at a certain rate. We derive a spatially defined expression of the basic reproduction number $R_0$ and we also numerically simulate the model equations. We find that exposure learning is generally more efficient than social learning, since exposure learning generates NPI uptake in the individuals at immediate risk of infection. However, if social learning is pre-emptive, beginning a sufficient amount of time before the epidemic, then it can be more effective than exposure learning. Interestingly, varying the initial number of individuals practicing NPIs does not significantly impact the epidemic final size. Also, if initial source infections are surrounded by protective individuals, there are parameter regimes where increasing the initial number of source infections actually decreases the infection peak (instead of increasing it) and makes it occur sooner. The peak prevalence increases with the rate at which individuals drop the habit of using NPIs, but the response of peak prevalence to changes in the forgetting rate are qualitatively different for the two forms of learning. The pair approximation methodology developed here illustrates how analytical approaches for studying interactions between social processes and disease dynamics in a spatially structured population should be further pursued.

    Mathematics Subject Classification: Mathematical biology (92Bxx).

    Citation:

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  • Figure 1.  Typical network distributions of susceptible contacts, $S$, neighbors who practice social distancing techniques, $S_p$ (as well as the respective calculations of the basic reproduction number) around the initial infection source, where all other members of the host population are fully susceptible (i.e. state $S$). The population size is $N=40000$, each individual has $n=4$ neighbors and model parameters are $\tau=0.75$ $day^{-1}$, $\tau_p=0.1$ $day^{-1}$, $\sigma=0.25$ $day^{-1}$, $\xi=\rho=0.5$ $day^{-1}$ and $\kappa=0.01$ $day^{-1}$

    Figure 2.  The basic reproduction number as a function of social learning from protective contacts at a rate $\xi$, and from infectious contacts at a rate $\rho$, where the transmission rate to protective individuals is $\tau_p=0.1$ $day^{-1}$ (a) and $\tau_p=0.5$ $day^{-1}$ (b). In all these plots $N = 40000,n=4$, $\tau=0.75$ $day^{-1}$, $\sigma=0.25$ $day^{-1}$, $C_{S_pS_p}=0,C_{S_pS}=3/4$, $\kappa= 0$ $day^{-1}$ and $s_p=1/N$

    Figure 3.  Infection peak versus initial distribution of single infected individuals with 4 state $S_p$ neighbors (a, d, g, j), time series for susceptible individuals who protect (b, e, h, k) and time series for infectious individuals (c, f, i, l), varying the number of 1 infected node plus 4 $S_p$ neighbors at the beginning of the outbreak (the rest of the population is fully susceptible). In (a to f) $\xi=0.25$ $day^{-1}$, $\rho=0$ $day^{-1}$; in (g to l) $\xi=0$ $day^{-1}$, $\rho=0.25$ $day^{-1}$; in (a, b, c and g, h, i) $\tau_p=0.6$ $day^{-1}$; in (d, e, f and j, k, l) $\tau_p=0.1$ $day^{-1}$. Model parameters common to all graphs are $\tau=0.8$ $day^{-1}$, $\sigma=0.25$ $day^{-1}$ and $\kappa=0$ $day^{-1}$

    Figure 4.  Infection peak versus rate of disease transmission to protective individuals, $\tau_p$, and the initial distribution of single infected individuals with 4 state $S_p$ neighbors (and all other members of the host population are fully susceptible, $S$), where $\xi=0.25$ $day^{-1}$, $\rho=0$ $day^{-1}$ (a) and $\xi=0$ $day^{-1}$, $\rho=0.25$ $day^{-1}$ (b). Other model parameters are $\tau=0.8$ $day^{-1}$, $\sigma=0.25$ $day^{-1}$ and $\kappa=0$ $day^{-1}$

    Figure 5.  Cumulative infections as a function of social learning from both infectious and state $S_p$ neighbors at rates $\rho$ and $\xi$, respectively, where the initial conditions are 1 infected node and 1 state $S_p$ neighbor while the rest of the population is fully susceptible (i.e. state $S$), and $\tau_p=0.1$ $day^{-1}$ (a), $\tau_p=0.2$ $day^{-1}$ (b), $\tau_p=0.3$ $day^{-1}$ (c). Other model parameters are $\tau=0.8$ $day^{-1}$, $\sigma=0.25$ $day^{-1}$ and $\kappa=0$ $day^{-1}$.

    Figure 6.  Cumulative infections as a function of social learning from both infectious and state $S_p$ neighbors at rates $\rho$ and $\xi$, respectively, where the initial conditions are 1 infected node and 1 state $S_p$ neighbor while the rest of the population is fully susceptible (i.e. state $S$). Model parameters are $\tau=0.8$ $day^{-1}$, $\sigma=0.25$ $day^{-1}$ and $\kappa=0$ $day^{-1}$

    Figure 7.  Infection peak versus the rate at which protective susceptible individuals forget, $\kappa$, varying regimes for social contagion parameters $\xi$ and $\rho$. Initial conditions are 1 infected node and 2 state $S_p$ neighbors while the rest of the population is fully susceptible (i.e state $S$). Other model parameters are $\tau=0.8$ $day^{-1}$, $\tau_p=0.3$ $day^{-1}$ and $\sigma=0.25$ $day^{-1}$

    Figure 8.  Cumulative infections as a function of the initial number of state $S_p$ individuals and the time at which the infection is introduced, varying $\xi$ and $\rho$, for the scenario of exposure learning only (dark grey surface) and social learning only (light grey surface). Other model parameters are $\tau=0.8$ $day^{-1}$, $\tau_p=0.001$ $day^{-1}$, $\sigma=0.25$ $day^{-1}$ and $\kappa=0$ $day^{-1}$

    Table 1.  Summary of expressions of the basic reproduction number $R_0$ developed in this paper

    (a) General expression of $R_0$ Equation (10)
    (b) Expression of $R_0$ used in simulation results in this manuscript: obtained by assuming that initially the proportion of susceptible individuals who practice NPIs is very small $s_p\approx O(1/N)$ Equation (17)
    (c) Simplification of $R_0$ in Part (b) above by further assumptions: adoption of NPIs is through social learning only (i.e. $\xi>0$, $\rho=0$); once adopted NPIs are practised consistently (i.e. $\kappa = 0$); at initial stage there is 1 state $I$ with 1 state $S_p$ contact who has 1 state $S_p$, and the rest of the population is of state $S$ Equation (18)
    (d) Simplification of $R_0$ in Part (c) above by a further assumption: high efficacy NPIs (i.e. $\tau_p\approx 0$) Equation(19)
    (e) Simplification of $R_0$ in Part (d) above by cancelling out insignificant terms dependent on the parameter regine: $N=40000$; initially $s_p=2/N$; $n=4$; $\tau = 1$; $\tau_p=0.0025$; $\sigma=0.25$; $\xi= 0.25$ Equation (20)
    (f) Simplification of $R_0$ in Part (b) above by further assumptions: adoption of NPIs is through exposure learning only (i.e. $\xi=0$, $\rho>0$); other conditions are as in Part (c) above Equation(21)
    (g) Simplification of $R_0$ in Part (f) above by a further assumption: high efficacy NPIs (i.e. $\tau_p\approx 0$) acquired through exposure learning only Equation (22)
    (h) Simplification of $R_0$ in Part (g) above by cancelling out insignificant terms dependent on the parameter regine: $N=40000$; initially $s_p=2/N$; $n=4$; $\tau = 1$; $\tau_p=0.0025$; $\sigma=0.25$; $\xi= 0.25$ Equation (23)
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