Team:HZAU-China/Modeling/Gray logistic

• Overview
• Gray logistic
• Immune responce
• Cellular automata

Aim:

To know the growth curve in the dog’s blood

Steps:

1.Do experiment to measure the number of bacteria;

2.Establish the gray logistic model to simulate the growth of bacteria;

3.Determine the parameter through the experiment;

4.Test the predicted results.

Results:

The gray logistic model gets the good forecasting result.And the model precision is excellent.

Background:

The color of blood is so deep that it is not fit to measure the OD value to determine the growth of bacteria in the blood.Then we choose dilution-plate method to detect the number of total bacteria. So we coated a large number of plates.The logistic model of population can well predict the increase of population.

Establish the logistic model:

In the environment of the blood,the number of bacteria have a maximum value K.And when the number of bacteria approach K,the growth rate is next to nil.Then the population growth equation is as follows:

The solution of the equation is :

N0 is the number of bacterial population.r is population growth rate.To make it convenient to calculate,we simplify the equation;A,B and r are unknown parameters. is the logarithm of the CFU of Bacillus subtilis.

Using the gray system theory to determine the parameters:

To determine the parameters of the equation,we use the gray system theory.The equation can be rewritten:;;;

Using the matrix equation in linear algebra we could determine the parameters α and β .,,

From the result ,we know the value of posterior-variance is 0.1931.The posterior-variance is lower than 0.35 so that the model precision is excellent.

In a conclution,Our engineering bacterium growth curve is in the dog's blood is;is the logarithm of the CFU of Bacillus subtilis.