Spatial heterogeneity and infection patterns on epidemic transmission disclosed by a combined contact-dependent dynamics and compartmental model

Abstract

Epidemics, such as COVID-19, have caused significant harm to human society worldwide. A better understanding of epidemic transmission dynamics can contribute to more efficient prevention and control measures. Compartmental models, which assume homogeneous mixing of the population, have been widely used in the study of epidemic transmission dynamics, while agent-based models rely on a network definition for individuals. In this study, we developed a real-scale contact-dependent dynamic (CDD) model and combined it with the traditional susceptible-exposed-infectious-recovered (SEIR) compartment model. By considering individual random movement and disease spread, our simulations using the CDD-SEIR model reveal that the distribution of agent types in the community exhibits spatial heterogeneity. The estimated basic reproduction number R0 depends on group mobility, increasing logarithmically in strongly heterogeneous cases and saturating in weakly heterogeneous conditions. Notably, R0 is approximately independent of virus virulence when group mobility is low. We also show that transmission through small amounts of long-term contact is possible due to short-term contact patterns. The dependence of R0 on environment and individual movement patterns implies that reduced contact time and vaccination policies can significantly reduce the virus transmission capacity in situations where the virus is highly transmissible (i.e., R0 is relatively large). This work provides new insights into how individual movement patterns affect virus spreading and how to protect people more efficiently.

Fig 1. The classic SEIR model and the modified SEIR model.

(a) The classic SEIR model. Green, black, red, and blue squares represent susceptible, exposed, infectious, and recovered compartments, respectively. Susceptible individuals in the system successively go through these four stages. Transitions between adjacent stages occur according to certain probabilities. (b) The modified SEIR model. When the distance between the susceptible P_S and P_I is within the airborne transmission distance d, P_S can be infected by P_I and transformed into P_E at the probability p_SE. The latent period is within 4–10 days. During this period, P_E converts with probability p_EI to infectious P_I that can transmit the virus. Any P_E that remained unconverted to a P_I after 10 days is classified as a recover P_R. Meanwhile, P_I could also be cured with probability p_IR and transformed into P_R.

Fig 2. Schematic diagram of the CDD model.

(a) Stochastic motion process characterized by the CDD model. The upper panel represents that the direction of motion of an agent varies within [-45°, 45°] of the original direction when there are other agents within r_c. The lower panel gives an example of the change in motion state of two individuals before and after contact: i is the central agent, and the blue circle represents its contact range of radius r_c. Individuals i and j fall in contact when their distance is less than r_c, and the agents are in a uniform linear motion before contact. The red, green, and black solid vectors represent the velocities of i and j and the relative velocity j relative to i, respectively. The gray dashed line represents the trajectory of j relative to i. (b) The reassigned velocity v_rnd that satisfies a normal distribution with a mean of v and a standard deviation of $\frac{1}{3}\mathit{v}$. θ_rnd is the change in the reassigned direction (in the range of -45° to 45°), which satisfies a normal distribution with a mean of 0° and a standard deviation of 15°.

Fig 3. System setup of the CDD-SEIR model.

(a) Schematic diagram of a representative system obtained from the simulations. The system contains 200 particles with a box length of $100\sqrt{2}$ m and density of (100 m²)^-1. P_S, P_E, P_I and P_R are represented by green, black, red and blue particles, respectively. (b) The profiles of the temporal population of P_S (in green), P_E (in black), P_I (in red) and P_R (in blue) during epidemic transmission. (c) Representative snapshots of the infection within a community at days 0, 10, 20, 30, and 40 are shown.

Fig 4. Contact pattern of the CDD model.

(a) The mean (blue) and median (red) of the contact time ω influenced by the velocity v of agents in the CDD model. (b) The percentages of ω in different time ranges for the velocity v of agents in the CDD model. The blue line represents ω < 0.5*r_c /v, the red line represents 0.5*r_c /v ≤ ω < 4*r_c /v, and the green line represents ω ≥ 4*r_c/v. (c) The mean free time τ in the CDD model (solid line) and the CDD-NC model (dashed line) for different velocities.

Fig 5. Infection spreading in different p_SE and the basic reproduction number R₀ influenced by p_SE and the velocity v of agents in the system.

The dynamic progress of the infection spreading when p_SE = 1.0*10⁻⁴ (a) and p_SE = 2.0*10⁻⁴ (b). Snapshots of Days 0, 20, and 40 are shown in order. (c) The influence of p_SE and v of agents on R₀ in the system. The points represent the calculated data, and the solid line represents the fitting curve to eliminate noise.

Fig 6

The basic reproduction number R₀ (a), the heterogeneity variable ζ (b) and the critical velocity v_c of R₀ and ζ (c) influenced by five parameters (n_I,0, d, p_SE, p_EI,Day7, and p_IR) and the velocity v of agents in the system. In each plot, the results corresponding to the five values used for the parameter that changed are represented by five different colored curves. The points represent the calculated data, and the solid line represents the fitting curve to eliminate noise. The horizontal coordinates in (a) and (b) are on a logarithmic scale.

Fig 7

The slope k (a) and the saturated reproduction number R_S (b) influenced by five parameters (n_I,0, d, p_SE, p_EI,Day7, and p_IR) in the system.

Fig 8. Contact time required for infection t_SE at low velocity (a, v = 0.001 m/s) and medium velocity (b, v = 0.02 m/s), number of contacts between P_S and P_I agents before infection T_SI, the mean (blue) and median (orange) of the average time per contact between P_S and P_I agents before infection ω_SI with the probability of infection p_SE compared with the contact time ω calculated by the CDD model.

At a low velocity (v = 0.001 m/s), infection tends to occur through a small number of long-term contacts.