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Environmental System Modeling
A cellular model for describing relationships between adjacent environments is shown in Fig. 1 In this
model, each environment is decomposed into interconnected localized regions. Each region is related to other regions through two mechanisms:
- Disease State
: The state of a localized region is captured by the vector [ S(t) , I(t) , R(t) ], which
represents the total susceptible population, infected population, and the recovered (or deceased) population that are no longer susceptible, respectively. A more elaborate state vector includes
age, gender, and racial distribution of the infected patients. The state information of one localized region affects adjacent regions through carriers, such as air, insects, or animals.
- Feature Vector
: The feature vector includes environmental factors at or before time n for location (i,j). Note that the feature vector is influenced by neighboring feature vectors at or
before time n, and thus, allows other models to be incorporated into the system.
Although the proposed model uses rectangular grids, other types of grids (such as hexagonal and non-regular grids) are also possible. Furthermore, each region is assumed to interact only with the eight
adjacent regions. Consequently, this model is not applicable for diseases with airborne paths (where the vectors are birds, that can fly for long distances, or humans, that travel by car or aircraft).
In the following, we outline the general formulation of the environmental model: We first denote the  state vector at location (i,j) and time n as  . The  environmental vector,  , is constructed by concatenating the eight state vectors adjacent to location (i,j): 
the relationship among the adjacent state vectors and the other environmental factors, characterized by  is modeled as follows: 
In this equation,  is a  vector. Note that this model reduces to a linear time-invariant model if the equation above is simplified to 
where  is an  matrix,  is an  matrix, while  is an  matrix. A linear time-varying model, in contrast, is expressed as 
Figure 1: Environmental model consisting of interconnected sub-models.
An sample cell is shown in Fig. 2. In this example, the model is expressed hierarchically. External inputs
from adjacent state vectors as well as local environmental feature vectors are sent through the first stage modules to generate intermediate results. The last stage combine all of the intermediate results and
generates the state vector for time n+1. 
Figure 2: Structure of the environmental model at the cell level.
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