Team:HZAU-China/Modeling/Cellular automata

• Overview
• Gray logistic
• Immune responce
• Cellular automata

Aim:

To know how the number of immunized dogs changes over time.

Steps:

1.Define the cellular automata;

2.Determine the related parameters of cellular automata;

3.Determine the rules of cellular automata;

4.Analyze the results of the model.

Background:

Cellular automata are discrete dynamical systems that can simulate complex behaviors by animating cells on a lattice based on simple, local rules. There are numerous applications of cellular automata, such as simulating traffic flows, network transmission and digital music. In our project, cellular automata are used for modeling the spread of immunity in stray dogs.

Definition of the cellular automaton

Our cellular automata contains cellular, the state of the cellular, neighborhood and the rules of the cells’ states updated over time.

A=(T,S,PRO,N)where T stands for a cell to maintain its current state, S stands for the state of the cell, PROstands for cells’ ability for spreading immunity, and N is the number of the cells.

Cell: An individual stray dog.

Cellular Space: A collection of cells distributed in a 2-dimentional space. The cellular space is divided into square lattice. Suppose the size of the cellular space is N = m*m where m is the number of rows (columns) and 100 in our model.

Cellular State: Assume that the state variable of the cell is Sij(t)where i and j indicate row i and column j in the cellular space and t is time.Shas 3 values of {0, 1, 2} where 0 represents a dog without immunity and 1 represents that a dog is in the process of obtaining immunity and 2 represents a state that a dog has been immunized to the rabies virus.

Neighbor: In our model, the dimension of the cellular space is two. Around each cell, there are eight cells as neighbors. So the current states of the present cell and its 8 neighbors determine its state of the next moment.

图片介绍

Initial Configuration::A certain number of dogs that have been immunized by our engineered bacteria are put randomly in an area at first moment.

The ability for spreading immunity:: A dog has the ability to spread immunity if it is in the process of obtaining immunity or has been immunized. The variable for the expression of this ability is PRO whose values are between 0 and 1.

Value:: If PRO increases to value1, the state-1 will be turned to state-2. If PRO decreases to value2, the state-2 will be turned to state-1.

The duration for a cell to maintain its state: The time for a cell to maintain its state in state-1 and state-2 are expressed as T1 and T2, respectively. Staying in different state, a cell has different ability to spread the engineered bacteria.

Evolution rules

Free Walking: A certain percentage of cells will walk freely before every update. The cells which will travel are randomly selected. The travel distance is a random integer ranging from 1 to 20.

Updating RulesThe state transition of a cell depends on the current states of itself and its neighbors. The impact of a neighbor on the transition rate (PRO) of the present cell is inversely proportional to their Euclidean distance. The Euclidean distance of adjacent cells is 1 while that of a diagonal is $radical\; 2$. k is the spread coefficient of adjacent cells.The calculation formula of PRO is as follows:

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