Team:HZAU-China/Modeling/Gray logistic

• Overview
• Gray logistic
• Immune responce
• Cellular automata

Aim:

To know the growth curve of Bacillus subtilis in the dog’s blood.

Steps:

1. Experimentally measure the number of bacteria;

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

3. Determine the parameters through experiments;

4. Test the predicted results.

Results:

The gray logistic model gives good prediction 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. So we chose dilution-plate method to detect the number of total bacteria. We coated a large number of plates. If you want to know the details of the experiment, please click here{ text-decoration:underline;} . The logistic model of population can well predict the increase of population.

Establishing the logistic model:

In the blood environment, the number of bacteria has a maximum value K. When the bacteria number approaches K, the growth rate approaches zero. Then the population growth equation is as follows:

.

The solution of the equation is :,

where N₀ is the size of bacterial population and r is population growth rate. For convenience, we rewrite the above equation as where A=K, and r are unknown parameters. N is the logarithm of the colony-forming unit (CFU) of Bacillus subtilis.

Determining the parameters using the gray system theory:

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

,

,

;

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

,,

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

In conclution, the growth curve of our engineered bacterium in dog's blood is given by; where N(t) is the logarithm of the CFU of Bacillus subtilis.

Reference:

1.Shiqiang Zhang, China's Population Growth Model Based on Grey System Theory and Logisitic Model[C]. 2010:4. (In Chinese)

2.Xiaoyin Wang, Baoping Zhou 2010. Mathematical modeling and mathematical experiment. Beijing : Science press. (In Chinese)