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Revision as of 14:18, 28 September 2018
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Model
We set up a mathematical model to predict the population dynamics of cockroaches before and after using our product. By doing so, we can estimate the lethal time of our cockroaches terminator, analyse the relationships among each relative factors so as to modify our product.
Assumptions
- 1. The number of cockroach has reached the highest value in stable stage
- 2. Ignore natural birth and death rates in our system
- 3. Infectious individuals can not recover
- 4. Other factors that may affect the experiment are ignored
Natural condition
In natural condition indoors, due to environmental resistance like food, water, space, the population of cockroaches is more likely to follow a S-shaped growth curve (sigmoid growth curve), which can be formalized mathematically by logistic function.
With infection
Our model was constructed based on SIR epidemic model (Susceptible, Infectious, Recovered) , following are some basic properties:
- 1. Naturally all cockroaches are susceptible individuals, they can infect by M.anisopliae becoming infectious individuals.
- 2. The number of individual being infected in a contact between a susceptible and an infectious subject is simulate by standard incidence .
- 3. The transition rate between Infectious and dead is γ, its reciprocal (1/γ) determines the average infectious period, which is estimate by experiment data.
Parameters
Model was simulate during 30 days, with total number of 60.
| Parameter | Value | Meaning |
|---|---|---|
| S(t) | the number of susceptible individuals over time | |
| I(t) | the number of infectious individuals over time | |
| D(t) | the number of dead individuals over time | |
| β | 0.75 | transmission rate, which is the probability of getting the infection in a contact between susceptible and an infectious |
| γ | 1/8 | mortality, which is the the transition rate between I and D, its reciprocal (1/γ) determines the average infectious period |
| S(0) | 55 | the initial number of susceptible individuals |
| I(0) | 5 | the initial number of infectious individuals |
| r | 0.3 | growth rate |
| N=S+I | population size | |
| K | 70 | carring capacity |
The system without so-called vital dynamics (birth and death) described above can be expressed by the following set of ordinary differential equations:
This system is non-linear, and the analytic solution does not exist, but we can compute the numerical solution by MATLAB. (see results)
Results
The following curves show dynamics number change of each kinds of individuals. We see that the infectious individuals grow fast before first 6 day, and then began to drop. The total number of cockroaches continuously going down. We specify the median lethal time (LT50), which in this condition is 11.1 days.
Sensitivity Analysis
We use sensitivity analysis to analyze the impacts of some important parameter values (α, β) on our model outcomes (LT50). The figures below show the tendency of dead number with respect to each parameter change.
1. change gamma
| change α | α | β | LT50 | ΔLT50 | Ratio |
|---|---|---|---|---|---|
| +20% | 0.750 | 0.150 | 10.100 | 0.090 | 0.450 |
| 0.075 | 0.125 | 11.100 | |||
| -20% | 0.750 | 0.100 | 12.700 | 0.138 | 0.690 |
2. change beta
| change β | α | β | LT50 | ΔLT50 | Ratio |
|---|---|---|---|---|---|
| +20% | 0.900 | 0.125 | 10.300 | 0.072 | 0.360 |
| 0.750 | 0.125 | 11.100 | |||
| -20% | 0.600 | 0.125 | 12.400 | 0.117 | 0.585 |
The last term Ratio is the normalized sensitivities-the ratio of the relative change of the output to the relative change of the parameter.