Team:Shenzhen SFLS/MODELING
Model
1.SIR Model
Purpose: To dynamically represent the evolutionary process and deduction of hepatitis C (HCV) by means of a mathematical model.
With dynamic models，We can…
Describe the evolution of HCV over time (space).
Analyze the changing pattern of HCV.
Predict the spread of HCV.
Study the means of controlling HCV.
Assumption
This model only considers the initial stages of HCV outbreaks. This provides a basic understanding of how this virus spreads. Further discussion will be made in the extended model.
• The Cured will not be infectious. In fact, in our human practice survey, we’ve learned that:
1.Hepatitis C is extremely hard to be cured.
2.if the once infected people were cured, the residual antibodies can prevent them from secondary infection.
•We assume that the total population in this model remains constant(With no births or deaths).
•In spite of simplifying the model, this assumption is also quite reasonable in a certain period of time.
•We assume that the population spread randomly in a region, so as to build up a reasonably and constant relationship between susceptible and uninfected populations
• We assume that the viruse can only transmit between humans. In other words, HCV transmission between humans and animals or betweenanimals is not within our consideration
• We assume that sex and age are not associated with HCV transmission.Though the possibility of infecting HCV do varies from time to time, the difference is actually quite small.
•Because the hepatitis C vaccine has not yet been developed, all the “vaccine”in this page refers to drugs.
Symbol Definition N the total population involved in this model(a constant) S the ratio of the Susceptible to N E the ratio of the Latent to N I the ratio of the Infected to N D the ratio of the Deceased to N a the rate at which susceptible become infected b the rate at which exposed become infected q the rate at which infectious become deceased m the ratio of the inflow(outflow) population to N. p he ratio of the susceptibility to the inflow population Total case D+I
Figure 1: The summary of conversion relationship among S, E, I, D, m
dS/dt is the ratio of the susceptibility to the inflow, while the last term, S/(S+E) m is accord with the ratio to the outflow. This is written as:
The equation that describes the ratio of the exposed increases by adding what was just removed from the susceptible (aIS) and deceases through subtracting those turns to be infectious(bE). The third term and the last term represent the ratio of the exposed to the inflow and to the outflow respectively. This is written as:
The equation that describes the ratio of the infected group increases by adding what was just removed from the exposed, bE. And it reduces in one way, people are killed by the virus,qI. This is written as:
The deceased group’s ratio is increased by those who die due to the virus. This is written as:
1.1 Simulation and Analysis
According to the data collected by the World Health Organization, we can calculate the constants that are listed in table.2:
a b m q p 1 0.43% 0.001% 2.2% 3%
Table 2: Factors
And the initial state of SEIRD is in table3,
S E I D 99.999% 0% 0.00010% 0%
Table 3: Initial State of SEID
According to the figure.2, the susceptible plummet immediately in view of how infectious the virus is. And at the same time, both the exposed and the infected group start to grow. The ratio of the deceased group begin to rise, following the infected people ˛a´ rs climb, and continues to rise until it hits a plateau. At the end, the susceptible, exposed, and infectious falls to zero, as they turn to be deceased.
Note that our model only focuses on the situation before the government or public intervened. Further research after some measures, like applying vaccine or drug, is implemented, will be discussed in the extended model.
1.2 Advantages and Disadvantages
1.2.1 Advantages
1. Our model is accord to the data in a degree, which means that it is a quite realistic and useful model. In other words, it can well reflect the trend of the transmission of HCV, especially when the outbreak is with no control from the government and the public.
2. Based on the previous studies，we extend the traditional SIR to the novel SEIRD model. And this extension simplifies our model substantially and make our analysis available, as it treat the problem from a macroscopic scale. So these points are quite valuable in our model.
3. The arguments have obvious physical meaning. Therefore it can give us a clear understanding about the spread of the virus, bringing us more convenience for further analysis.
1.2.2 Disadvantages
1. Only macroscopic. This model only focuses on the problem from a macroscopic view, leaving alone inhomogeneities from a local angle. In other words, such model only reflects the average trend.
2. The model assumes that neither the government nor the public pays enough attention to such an issue in the total process. This may be true at the beginning of the outbreak of HCV virus. Once the drug is in use, a large amount of additional arguments must be considered, though the model can hardly reflect the actual process of virus spreading.
2 Revised SEIlIsRD Model
2.1 Model Overview
In this part, we take the control measures into account,( like quarantine, from the government)and the use of vaccine by extending the SEIRD model to the SEIlIsRD model. The impact ofthe control measures and of the vaccine for the exposed(E) and the susceptible(Il) are discussed separately. Finally, we solve the problem about how to distribute the vaccine between E and Il to consume minimum amount of vaccine with two evaluation criteria.Note that once the situation is under control, the patients will be distinguished based on the severity in order to treat them with different medical measures. Therefore, it is necessary to divide the infectious group into slightly infected and the seriously infectious.
2.2 Assumptions
• No inflow or outflow.In order to suppress the spread of disease, the government may prohibit any inflow or outflow.
•The control and regulations of the government can reduce the rate at which susceptible become infected.
For example, the infected are quarantined, which reduces the number of people they contact with. Thus, the rate mentioned above is brought down.
•The transportation time of vaccine can be neglected because we only model in a small region, such a city, in this part.
•Slightly infected patients and seriously infectious patients can be distinguished.
Since patients have different symptoms of illness and the detection means of HCV virus is quite advanced, we assume it is possible to distinguish between two groups.
• The vaccine can only cure those who are exposed and slightly infected.
Once became seriously infected, it is impossible to cure according to the characters of HCV virus.
• The vaccine is effective immediately and has the same effect on both slightly infected patients and seriously infected patients so as to simplify our model.
• The vaccine can increase the rate at which exposed patients or slightly infectied patients become recovered(t1, t2), while has no effect on the rate at which the exposed become the lightly infectious(b1) or the slightly infected become the seriously infected(b2)
• The distribution of the vaccine between the exposed and the lightly infectious will not cause any social disturbance.
It is obvious that such consideration is not the point this model should concentrate on.
2.3 Terms and Definitions
2.4 Creating the Differential Equation
overall conversion relationship among S, E, Il, Is, R, D, summarized in table.4
Table 4: Terms and Definitions
Symbol Definition
Il the ratio of the slightly Infected to N
Is the ratio of the seriously Infected to N
a the rate at which susceptible become infected
b1 the rate at which exposed become slightly infected
b2 the rate at which slightly infected become seriously infectious
t1 the rate at which exposed become recovered
t2 the rate at which slightly infected become recovered
q the rate at which seriously infected become deceased
Egoal the optimal goal of E under certain constraint
The summary of the conversion relationship among S, E, Il, Is, R, D:
2.5Advantages and Disadvantages
2.5.1Advantages
1. Many factors, like the quarantine and usage of the vaccine, in the actual situation are considered in this model. The influence of a, t1 and t2 to the spread of virus is analyzed in the revised SEIRD model. We proposed optimal allocation of vaccine under different constrains.
2. We divide the group of I into Il, Is and the group of R into the recovered and the deceased. These revisions make the model more realistic , but do not add much difficulty in solving the differential equations.
2.5.2 Disadvantages
1. In fact, the effect of the vaccine is different for the exposed and the slightly infected while it is assumed to be the same in the model.
2. The locations of delivery and possible delivery system are not taken into account since we only analyze a small region, like a city. And in this model we are not focusing on solving the transportation system problems.
3.Dynamic Programming Model
3.1 Model Overview
In this part, we try to apply the model mentioned above to other 3 hardest-hit countries. To facilitate our discussion more effectively, we divide 3 countries into 6 parts. First, we prove the effectiveness of the model when it is applied in other countries. Then the manufacturing speed of vaccine in these 6 parts is determined with constant distribution ratio between the exposed and the lightly exposed. Finally, by dynamic programming the distribution ratio of vaccine in each cycle, we make the model more realistic and full use of the ability of producing at the same time.
3.2 Assumptions
• The HCV virus only outbreaks in the densely populated places. The Sparsely populated places are safe. As we know, in sparsely populated place, the number of people one will meet is quite small. Therefore, it is hard to break out due to lack of close contact between people.
• Other assumptions are the same as 2.2.
3.3 Simulation and Analysis
3.3.1 Proof of Model Effectiveness in Other Countries
We divided the three hardest-hit countries, into 6 parts, including A, B, C, D, E and F. This division is based on the severity and the frequency of the outbreak. Such devision is necessary because those areas without outbreak of HCV virus are not in our consideration.
Table.10 are the initial state in the extended model in the first period
According to the data about the manufacturing ability of vaccines, we find that the maximum number of vaccines produced one day(Tmax) is 8000, which is reasonable. Based on Tmax, we redistribute the ratio of t1 and t2 through dynamic programming.
Here, we provide a simple example to illustrate the process of dynamic programming. We divide 40 days into 2 parts(the first 20 days and the last twenty days).
At the beginning of the first twenty days, we distribute the ration of t2 and t1.
3.3.2 Static Programming the Vaccine Distribution ratio
The optimization problem is similar to part2. The only difference is that we need to take 6 parts into account and figure out the manufacture speed of vaccine each day. The ultimate goal is to use minimum number of vaccines under the condition that E is smaller than Egoal (10−5) in 40 days.
The problem can be expressed as:
;
The number of vaccine needed each day in one location is calculated through.
3.4 Advantages and Disadvantages
3.4.1 Advantages
1. We divided 3 countries into 6 small regions which share a similar degree of epidemic and similar outbreak time, in 3 hardest-hit countries to make the plan about how to distribute the vaccine more targetedly and reasonably. TThus, the priority of using vaccine is given to the hardest-hit areas.
2.Dynamic programming model is closer to the actual situation. By replanning the manufacture strategies in each cycle, we make full use of the ability of producing the vaccines.
3.4.2 Disadvantages
1. Differences between regions or countries has not been taken into account.
2. The time of transmission is also neglected in the model.
3.5Conclusion
3.5.1 The need of vaccines in the slightly infected groups should be given high priority as this strategy can minimize the need of vaccine. Note that the main goal here is to reduce the ratio of the exposed groups to a certain value in limited days. And the left vaccine can be distributed to the exposed group.
3.5.2 The number of vaccines distributed to certain area should be in accord with its severity of the desease. In other words, more serious the desease is, more vaccine such area should receive.
3.5.3 The distribution ratios between the exposed and the slightly infected groups, should be renewed in each cycle in order to adapt to the actual situation.
3.5.4The manufacturing speed of the vaccines can be optimized by dynamic programming.
4 Sensitivity Analysis
We test our model with different factors and the initial states in the differential equations. The analysis proves that our model is not unduly sensitive.
4.1 Different Factors
1. a: the rate at which susceptible become infected.The people the infected group meet may vary due to the degree of the control measures. Thus, we change a by up to10%( 0.7) to see what will happen. We observe a 3.6% increase in the total number of the vaccine needed(Tall). Hence, our model can be used under various degree of control measures from the government.
2. b1: the rate at which the exposed become the slightly infected.
b2: the rate at which the slightly infected the seriously infected.
If we increase both b1 and b2 by 10%, Tall rise 0.4% and 0.5% respectively, which is acceptable.
3. t1: the rate at which the exposed become recovered.
t2: the rate at which the slightly infected become the deceased.
t1 and t2 can be changed through the distribution of the vaccine. When t1 is increased from 0.45 to 0.49 by 10%, Tall increase 3.2%. When t2 is decreased from 1 to 0.9, Tall increases by 1.3%. Theses all indicates good robustness.
4.2 Different Initial States
4.2.1. E0: the initial percentage of the exposed group before vaccine is used.
We change E0(1.33%) by up to 10%, 2.7% increase of Tall is observed. Therefore, our model may suit in other areas with different E0.
4.2.2 Il0: the initial percentage of the slightly infected group before vaccine is used.
If 10% change is put in Il0, Tall changes by 4.1%, which is reasonable.
Is0: the initial percentage of the slightly infected group before vaccine is used.
If we increase Is0 by 10%, Tall will drop 2.3%, indicating good robustness of the model.
5 Future work
We only analyzed the vaccine production strategies of several hardest-hit countries in Fictional place. In fact, there is a potential outbreak of HCV virus in other counties with the development of globalization. Thus , how to take effective measures to prevent the HCV virus from outbreaking is worthy of further study. Even if we can only predict where it will outbreaks, it is meaningful. However, these considerations involve the interaction between different countries. Therefore, the differential equation should be corrected with a series of influential factors. Our group decides to make a follow-up exploration based on existing results and will try our best to obtain some meaningful results.
Moreover, network model can effectively reflect the interaction between individuals and the spreading of the virus in microscopic view. Valid immunization strategies based on individuals may also be worked out in this model. In the subsequent study, in order to obtain more accurate results, we will combine the model of differential equation and of the networking together.
Reference
传染病问题中的SIR模型
问题二SIR模型
