Effects of distribution of infection rate on epidemic models

A goal of many epidemic models is to compute the outcome of the epidemics from the observed infected early dynamics. However, often, the total number of infected individuals at the end of the epidemics is much lower than predicted from the early dynamics. This discrepancy is argued to result from human intervention or nonlinear dynamics not incorporated in standard models. We show that when variability in infection rates is included in standard susciptible-infected-susceptible ($\mathrm{SIS}$) and susceptible-infected-recovered ($\mathrm{SIR}$) models the total number of infected individuals in the late dynamics can be orders lower than predicted from the early dynamics. This discrepancy holds for $\mathrm{SIS}$ and $\mathrm{SIR}$ models, where the assumption that all individuals have the same sensitivity is eliminated. In contrast with network models, fixed partnerships are not assumed. We derive a moment closure scheme capturing the distribution of sensitivities. We find that the shape of the sensitivity distribution does not affect $R_0$ or the number of infected individuals in the early phases of the epidemics. However, a wide distribution of sensitivities reduces the total number of removed individuals in the $\mathrm{SIR}$ model and the steady-state infected fraction in the $\mathrm{SIS}$ model. The difference between the early and late dynamics implies that in order to extrapolate the expected effect of the epidemics from the initial phase of the epidemics, the rate of change in the average infectivity should be computed. These results are supported by a comparison of the theoretical model to the Ebola epidemics and by numerical simulation.

Figure 1

Infected individuals as a function of time. Initial time dynamics based on Eq. (19) for the $\mathrm{SIS}$ model and Eq. (20) for the $\mathrm{SIR}$ model. The results of a Monte Carlo simulation, the full moment closure scheme, and the approximation of the initial dynamics were compared, and a good fit was obtained. For all four distribution studied, $N=100000$, $\gamma =1$, and $E_N\left(\beta \right)=2\times {10}^{-5}$. The rising line is the initial time approximation, and the two other, highly similar, lines are the full moment closure solution and the simulation results.

Figure 2

$r\left(t\right)$ [Eq. (23) for the $\mathrm{SIS}$ model and Eq. (26) for the $\mathrm{SIR}$ model]. The $x$ axis is time since the initial infections. These values agree for all distributions with the values obtained from Monte Carlo simulations. Note that in both the Gaussian and the uniform distributions, the values are close to those obtained with constant infectivity. However, in the SF distribution, $r\left(t\right)$ deviates from the values obtained in the constant infectivity case. The source of the difference is in the drastic change in $E_I\left(\beta \right)$ in the SF model over time. This affects the denominator of Eq. (24) for the $\mathrm{SIS}$ and $\mathrm{SIR}$ models. For all four cases, $N=100000$, $\gamma =1$, and $E_N\left(\beta \right)=2\times {10}^{-5}$.

Figure 3

Plot of $E_I\left(\beta \right)\left(t\right)$ and $E_S\left(\beta \right)\left(t\right)$ as a function of time for the $\mathrm{SIS}$ model and for the $\mathrm{SIR}$ model for all distributions tested. While in the uniform and Gaussian distribution the value of $E_I\left(\beta \right)$ is close to $E_S\left(\beta \right)$, in the SF model this is not the case, and early in the dynamics $E_I\left(\beta \right)\gg E_S\left(\beta \right)$. The source of the difference between the SF and all other distributions is the presence of very high $\beta$ values in the population (even if they are rare). People with such high values of $\beta$ are almost automatically infected, and they increase $E_I\left(\beta \right)$ sharply. The bottom plot represents the relative infection rate $\mathrm{\Delta }I/I$ as a function of time. The total decrease in the expected value of $\beta$ in the $S$ compartment and the increase in the expected value of $\beta$ in the $I$ compartment induce a rapid decrease in this rate in the SF distribution.

Figure 4

Comparison between the analytic results and simulation results for the $\mathrm{SIS}$ steady-state value in the four distributions studied (upper plot), and the total number of removed individuals in $\mathrm{SIR}$ (lower plot). The SF model distribution has the lowest steady-state concentration of infected individuals in $\mathrm{SIS}$ and the lowest number of removed individuals in $\mathrm{SIR}$. This results from the removal of individuals with high infection probability from the susceptible pool. This effect is directly quantifiable through the higher moments of $\beta$ in the population and the resulting change in the first moment of $\beta$ in the susceptible and infected pool.

Figure 5

Observed number of Ebola infections in Sierra Leone (upper plot) and Guinea (bottom plot) and a comparison of the dynamics with the best fit obtained using the different distributions.

Figure 6

Decrease of the relative infection rate $\mathrm{\Delta }I/I$ as a function of time (top graph). This rapid decrease even in the early stage of the disease suggests that highly infectious individuals are rapidly removed from the population. In the bottom graph, an $F$ test was calculated between the best fit in the constant case and all the other distributions to incorporate the different number of free parameters. Three asterisks between the classic model and any of the distributions represent $p<0.001$. In Guinea, all three distributions are much better than the constant case, but there is no significant difference between the quality of the nonconstant distributions. In Liberia, the SF and Gaussian distributions have a significantly better fit than the uniform distribution, and the same result is obtained for Sierra Leone.