Impact of the infection period distribution on the epidemic spread in a metapopulation model

Abstract

Epidemic models usually rely on the assumption of exponentially distributed sojourn times in infectious states. This is sometimes an acceptable approximation, but it is generally not realistic and it may influence the epidemic dynamics as it has already been shown in one population. Here, we explore the consequences of choosing constant or gamma-distributed infectious periods in a metapopulation context. For two coupled populations, we show that the probability of generating no secondary infections is the largest for most parameter values if the infectious period follows an exponential distribution, and we identify special cases where, inversely, the infection is more prone to extinction in early phases for constant infection durations. The impact of the infection duration distribution on the epidemic dynamics of many connected populations is studied by simulation and sensitivity analysis, taking into account the potential interactions with other factors. The analysis based on the average nonextinct epidemic trajectories shows that their sensitivity to the assumption on the infectious period distribution mostly depends on R0, the mean infection duration and the network structure. This study shows that the effect of assuming exponential distribution for infection periods instead of more realistic distributions varies with respect to the output of interest and to other factors. Ultimately it highlights the risk of misleading recommendations based on modelling results when models including exponential infection durations are used for practical purposes.

Figure 1. Comparison of probabilities of no secondary cases in a two-population model.

On y-axis refers to (closed circles) and (open circles). On x-axis Pexp refers to . For each of the three probabilities, 100 points with different parameter combinations were generated ( and were drawn from exponential distributions and was taken equal to 1). The points represent means over 100000 Monte-Carlo simulations of a time continuous event-driven approach: one infectious individual is introduced in one population and the probability of no secondary cases is calculated based on the time spent in each population. Estimated average standard deviation for computed values was below . All parameters and variables are explained in Table 1.

Figure 2. Comparison of probabilities of no secondary cases in a two-population model where individuals move only once.

Variation of is represented as a function of and which vary on plausible ranges of values, under the constraint . All parameters and variables are explained in Table 1.

Figure 3. Dynamics of global number of cases and cumulative incidence described by a stochastic metapopulation model based on a completely connected network.

Early extinct trajectories were not considered. The mean (blue curve) was calculated over major epidemics only (corresponding to a final attack rate greater than 5). Simulations are performed using a time-continuous event-driven approach with Exp() (top panel), (middle panel) and (bottom panel). Parameters values are given in the subsection Examples of Results.

Figure 4. Dynamics of infected populations and cumulative incidence (in number of populations) described by a stochastic metapopulation model based on a completely connected network.

Early extinct trajectories were not considered. The mean (blue curve) was calculated over major epidemics only (corresponding to a final attack rate greater than 5). Simulations are performed using a time-continuous event-driven approach with Exp() (top panel), (middle panel) and (bottom panel). Parameters values are given in the subsection Examples of Results.

Figure 5. Dynamics of global number of cases and cumulative incidence described by a stochastic metapopulation model based on a scale-free network with mean degree of connectivity equal to 10.

Early extinct trajectories were not considered. The mean (blue curve) was calculated over major epidemics only (corresponding to a final attack rate greater than 5). Simulations are performed using a time-continuous event-driven approach with Exp() (top panel), (middle panel) and (bottom panel). Parameters values are given in the subsection Examples of Results.

Figure 6. Dynamics of infected populations and cumulative incidence (in number of populations) described by a stochastic metapopulation model based on a scale-free network with mean degree of connectivity equal to 10.

Early extinct trajectories were not considered. The mean (blue curve) was calculated over major epidemics only (corresponding to a final attack rate greater than 5). Simulations are performed using a time-continuous event-driven approach with Exp() (top panel), (middle panel) and (bottom panel). Parameters values are given in the subsection Examples of Results.

Figure 7. Distribution of the final epidemic size.

Calculation was performed on 300 simulations of a stochastic metapopulation model based on a completely connected graph (left panel) and on a scale-free network with mean degree of connectivity equal to 10 (right panel), with Exp() (top graphs), (middle graphs) and (bottom graphs). Parameters values are given at page 10.

Figure 8. Results of ANOVA on 960 simulated scenarios of epidemic spread with parameter values given in Table 2.

Dependent variables (on x-axis) are logarithm of means (over the non early extinct dynamics) of global variables (directly referring to individuals regardless of their population of origin): size and duration of the epidemic, size and date of the epidemic peak. For each of these outputs three variants are considered with respect to the distribution of infection duration: (difference between the value of the output simulated with the exponentially distributed infectious period and the value corresponding to the gamma distributed infectious period), (difference between the value of the output simulated with the gamma distributed infectious period and the value corresponding to a constant infectious period) and (difference between the value of the output simulated with the exponentially distributed infectious period and the value corresponding to a constant infectious period). Different pattern fills correspond to contributions of five input factors (mean infection duration, network, transmission rate, and migration intensity) to the variation in outputs amongst scenarios.

Figure 9. Results of ANOVA on 960 simulated scenarios of epidemic spread with parameter values given in Table 2.

Dependent variables (on x-axis) are logarithm of means (over the non early extinct dynamics) of global variables (referring to populations): size and duration of the epidemic, size and date of the epidemic peak. For each of these outputs three variants are considered with respect to the distribution of infection duration: (difference between the value of the output simulated with the exponentially distributed infectious period and the value corresponding to the gamma distributed infectious period), (difference between the value of the output simulated with the gamma distributed infectious period and the value corresponding to a constant infectious period) and (difference between the value of the output simulated with the exponentially distributed infectious period and the value corresponding to a constant infectious period). Different pattern fills correspond to contributions of five input factors (mean infection duration, network, transmission rate, and migration intensity) to the variation in outputs amongst scenarios.

Figure 10. Results of ANOVA on 960 simulated scenarios of epidemic spread with parameter values given in Table 2.

Dependent variables (on x-axis) are logarithm of variances (over the non early extinct dynamics) of global variables (directly referring to individuals regardless of their population of origin): size and duration of the epidemic, size and date of the epidemic peak and date of intra-population epidemic peak. For each of these outputs three variants are considered with respect to the distribution of infection duration: (difference between the value of the output simulated with the exponentially distributed infectious period and the value corresponding to the gamma distributed infectious period), (difference between the value of the output simulated with the gamma distributed infectious period and the value corresponding to a constant infectious period) and (difference between the value of the output simulated with the exponentially distributed infectious period and the value corresponding to a constant infectious period). Different pattern fills correspond to contributions of five input factors (mean infection duration, network, transmission rate, and migration intensity) to the variation in outputs amongst scenarios.