Introduction
The paper by Nipa and Allen (2020) focused on disease emergence in multi-patch stochastic epidemic models with demographic and seasonable variability. The investigation uses stochastic models in formulating variability that is both seasonal and demographic. Estimating a disease outbreak is through multi-type branching and application of backward Kolmogorov differential equation.
Description of the Problem
The study sought to solve the problem of infectious disease outbreaks. Some of the outbreaks are a result of seasonable changes, which have an impact on the transmission of pathogens. The paper focused on a multi-patch setting, especially when the movement between and transmission within patches are seasonal (Nipa & Allen, 2020). Other modeling studies have not focused on discrete patches that lack seasonal variations. The variables are time, number of initial infected individuals, and location.
0 < Pext (i,T) < 1
Poutbreak (i, T) = 1 – Pext ( i, T)
Methods
ODE Multi-Patch Model
Among the methods is ODE Multi-Patch Model, which considers movement among individuals between patches. The model has computations involving susceptible and infected individuals, births, natural deaths and others related to diseases (Nipa & Allen, 2020). There is also the element of the patch, population size, and transmission and dispersal rates.
Time-Nonhomogeneous Stochastic Process
It is a process that bases its foundation on the ODE model. It works by ensuring random variables are discrete, whereas time is continuous. This can be further divided into Branching Process Approximation and Numerical Methods. Branching Process Approximation is applied to the states that are infected while ensuring the time-nonhomogeneous process is in use (Nipa & Allen, 2020). The changes are then observed, recorded, and analyzed. In regards to Numerical Methods, estimation of probability takes place through the use of a differential equation. Other methods include Two and Three Patches.
New Results
Nipa and Allen (2020) found that seasonability in dispersal and transmission impacts the time and place with a significant risk for an outbreak. For instance, if a high-risk area has an infection during a time of large transmission rate, there is a high probability of an outbreak and vice versa.
Possible Extensions
There should be further studies in additional stages or levels of infection, incidence rate involving mass action, and arrangements of patches and population densities that are dependent on patch, among other areas (Nipa & Allen, 2020). The studies will help in controlling viral infections such as COVID-19, MERS, SARS and others.
Reference
Nipa, K. F., & Allen, L. J. (2020). Disease emergence in multi-patch stochastic epidemic models with demographic and seasonal variability. Bulletin of Mathematical Biology, 82(12), 1-30.