YAN Zheng-ren ; WEI Yu-rui ; LIU Qi-shan

(South China Normal University China Guangzhou)

Optimization and Application of SAICQD Viral Model

YAN Zheng-ren ; WEI Yu-rui ; LIU Qi-shan

(South China Normal University China Guangzhou)

We build a model called SAICQD which is based on the classic model—SAIC to deal with the problems, including how to describe the force of virus, and how to predict the effect of vaccine or drug and how to make full limited resource to overcome virus. Our model divides people into six groups, then through the relationships and characters of these groups and some limited conditions. We build some differential equations to predict the trend of these people, to figure out, under virus and medical treatment, what the fate of disaster area will be. This article take Ebola as study object.

SAIC; Viral model; Ebola

1 Introduction

As for the SAICQD models. Letters of the name stand for susceptible, antidotal, infected, contaminated, quarantine and dead. In order to line among these six different conditions, we introduce eight different rates according to the reality: the contaminating rate of common people, the contaminating rate of doctors and nurses, the infection rate, the immunization rate, the kill-virus rate, the quarantine rate, the mortality rate and a parameter which is influenced by medical level. Though these group and these parameters which some are optimized, we can figure out the amount change of different groups of people, to see the influence of medicine treatment and Ebola. Especially the immunization rate and the kill-virus rate, which are considered the restriction of traffic, location, and medicine manufacture, can intuitively reflect the medicine treatment.

2 Model : The Modified SAICQD Model

2.1 The meaning of the variable

SignDefinition S(t)The number of the people who are in susceptible condition in th day. C(t)The number of the contaminated people who are still in incubation period in th day. A(t)The number of the people who are in antidotal condition in th day. I(t)The number of the people who are in infected condition in th day. Q(t)The number of the people who have been quarantined in infected condition in th day. D(t)The number of dead people in th day.

α1The contaminating rate(the number of the people who are infected by a patient per unit day). α2The contaminating rate(the number of the medical staff who are infected by a patient per unit day). The infection rate(the proportion of the number of people whose physical condition changes from β contaminated to infected in C(t) per unit day). The immunization rate(the proportion of the number of people whose physical condition changes ω from susceptible to antidotal in S(t) per unit day). The kill-virus rate(the proportion of the number of people whose physical condition changes δ from infected to antidotal in I(t) per unit day). qThe quarantine rate(the proportion of the number of infected people who are quarantined in I(t) per unit day). dThe mortality rate(the proportion of the number of infected people who would be dead in I(t) per unit day). mA parameter which is influenced by medical level(i.e., the proportion of the number of contaminated people who would be quarantined in C(t) per unit day).

2.2 The SAICQD Model

2.2.1 Introduction

In SAICQD model, parameter δ and are const, but they can’t describe the influence of drug and vaccine supply. So, we make a improvement of the model. We replace δ and ω with u(t) and v(t).In order to make the model more conform to reality, we replace β and d with c(t) and dr(t).So the new model an be shown as Figure 2.

Figure 2.The Modified SAICQD Model

2.2.2 Establishment of the Model

According to the Figure 2 and the analysis of the SAICQD model, the modified SAICQD model can be described as the following equation:

At first, we assume that there is a hospital in the area where the Ebola outbreaks. And we assume that H(t) is the drug storage capacity and His the vaccine storage capacitytin th day. Because the limitation of the location of delivery and the feasible delivery system, we assume that the hospital can gain a certain amount of drug and vaccine everyday, and is the number of drug which the hospital can gain everyday and is the number of vaccine which the hospital can gain everyday. From Figure 3 and the above assumptions, we can learn that the hospital would gain the drug and vaccine whose number are and everyday. At the same time, there may be residual drug and vaccine after a day. So the drug storage capacity in th day is the summation of the number of the residual drug in th day. And the number of patient would decide the drug demand in th day. On the one hand, we hope all patients(Q+C+I) can be treated,but infected and contaminated people are not be noticed. On the other hand, part of quarantine people are advanced, can not cure. In short we use to represent kill-virus rate.So, there have two situations. One is that the number ofpatients is larger than H(t),so u(t) is the proportion of the drug storage capacity in th day in the summation of the number of quarantine people, contaminated people and infected people, i.e.,.Another is thatH(t)is enough to cure all quarantine people, sou(t)is the proportion of the number of quarantine people in th day in the summation of the number of quarantine people, contaminated people and infected people,i.e..After determining the number of the quarantine people and ,we can calculate the number of the antidotal people. And then we can also get the following equation:

Figure 3.

Such as the above analysis, we can also get two equations about and as following:

As for ,we can get the following equation by data fitting.

Where:

y1,y2,y3are the parameters which we can gain by data fitting.

Because it may be several days in incubation period, may be different in different day. So, we can get the following equation:

And because that self-protection awareness，educational level and medical equipment can make a difference to the quarantine rate m, we connect these three factors to GDP. If we would like to find out who is in the incubation period, the technical requirement should be advanced. So, we assume the United States quarantine rate(mUSA) is 0.5,by comparing with GDP( GandGUSA),we can calculate the .

2.3 Model Testing

2.3.1 Amount of People

There is no harm in taking Liberia as Model 2 Testing. We know Liberia is in serious condition of Ebola, with death of thousands of people. By some data from WHO statistics[6][9], we substitute these data, as follows, into model,.

Table 1

We regard the amount of total population as amount of all susceptible people(S). And it is also reasonable to assume that there is no antidotal people (A) and the amount of infected(I), contaminated(C) and quarantine(Q) people are equal.

From our SAICQD model result, we can predict the population change, like follows:

Figure 4. Amount Change of Different Kinds of Groups

Figure 5. Amount Change of Different Kinds of Groups

We can see that, under vaccines and drugs control, the outbreak will be restrained within 60 days. It can be positive to predict the medicine is really useful: from Figure 4, most susceptible people(S) will be transformed into antidotal people (A). It means most people can avoid the Threat of Ebola, but still some people will be infected, contaminated, quarantine, or even dead. From figure 5, clearly, from the begin of control action, the outbreak is still severe, even worse, because the amount of contaminated and quarantine people can not stop climbing, due to large infected and contaminated population base. But, soon, about on 3rd days, the amount of all dangerous people(I,C), decrease fast, while quarantine group from begin. On 60th day, dangerous people are nearly all be control. However we can not deny that Ebola outbreak is tragic.

3.Conclusion

Based on the reasonable conditions, the modified SAICQD model can describe, predict and figure out the effect of drugs and Ebola’ s force. If data permitted, we believe our model can help make plan to eradicate Ebola and make full use of drugs and vaccines. And the resource optimization model can calculate the minimum value of the drug and vaccine costs,so that we can save more money to manufacture more drugs and vaccine,which can do good to eradicate Ebola. What's more, we advice that it is important to improve the self-protection awareness and educational level, which is good for us to protect ourselves. And other countries could sent some medical equipment for them.

[1] Jaime Astacio,DelMar Briere,Milton Guillen,Josue Martinez,Francisco Rodriguez,Noe Valenzuela-Campos. MATHEMATICAL MODELS TO STUDY THE OUTBREAKS OF EBOLA.1996.

[2] Chowell, Nishiura. Transmission dynamics and control of Ebola virus disease (EVD): a review.2014.

[3] Ebola by the Numbers: The Size, Spread and Cost of the Outbreak. http://www.scientificamerican.com/article/ ebola-by-the-numbers-the-size-spread-and-cost-of-theoutbreak/?mobileFormat=true

[4] wikipedia. http://zh.wikipedia.org/zh/West African Ebola virus epidemic.

[5] http://zh.wikipedia.org/wiki/America.

[6] http://zh.wikipedia.org/wiki/Liberia

[7] QiYuan Jiang.JinXing Xie.Jun Ye.2011.Shuxue Moxing. Gaodeng Jiaoyue Press.(in Chinese)

[8] XueJun Wu.Kai Zhou.JunQuan Song.2009.Shuxuejianmo jingsaifudaojiaocheng. Zhenjiang University Press.(in Chinese)

[9] Web of WHO. http://www.who.int/mediacentre/factsheets/ fs103/zh/

[10] Chunming Zhang.Optimal Control Models of Computer Viruses and HIV Viruses[D].Chongqing University,2011(in Chinese)

[11] Xu-long Zhang,Xiaofan Yang.Optimal control model for computer viruses[J].Application Research of Computers,2011(in Chinese)

O572.25

A

1009-5624-（2016）02-0101-04