Program 2.6 The SEIR model, adding a latent period
We now introduce a refinement to the SIR model (Program 2.2) which takes into account a latent period. The process of transmission often occurs due to an initial inoculation with a very small number of pathogen units (e.g., a few bacterial cells or virions). A period of time then ensues during which the pathogen reproduces rapidly within the host, relatively unchallenged by the immune system. During this stage, pathogen abundance is too low for active transmission to other susceptible hosts, and yet the pathogen is present. Hence, the host cannot be categorized as susceptible, infectious, or recovered; we need to introduce a new category for these individuals who are infected but not yet infectious. These individuals are referred to as Exposed and are represented by the variable E in SEIR models.
Parameters
| μ | is the per capita death rate, and the population level birth rate. |
| β | is the transmission rate and incorporates the encounter rate between susceptible and infectious individuals together with the probability of transmission. |
| γ | is called the removal or recovery rate, though often we are more interested in its reciprocal (1/γ) which determines the average infectious period. |
| σ | is the rate at which individuals move from the exposed to the infectious classes. Its reciprocal (1/σ) is the average latent (exposed) period. |
| S(0) | is the initial proportion of the population that are susceptible. |
| E(0) | is the initial proportion of the population that are exposed (infected but not infectious) |
| I(0) | is the initial proportion of the population that are infectious |
All rates are specified in days.
Requirements.
All parameters must be positive, and S(0)+E(0)+I(0) ≤ 1.
Files
C++ Program, Python Program, Fortran Program, Parameters, MATLAB Code.