# Numerical Study on Zika Epidemic Early Warning Algorithms Driven by Dynamical Network Biomarker

Authors: Qingyi Zhan; Zhifang Zhang
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

This paper focuses on the numerical study and precise forecast of Zika epidemic. An early warning index algorithm of Zika epidemic is given, and the applications of this algorithm are investigated. We pay attention to the mathematical approaches to estimate the tendency of Zika epidemic in detail. A numerical experiment is provided to illustrate the effectiveness of the proposed method by the numerical calculation of early warning index of Brazil.


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## Body

## 1. Introduction

Nowadays, the forecast has played an important role in the theory and application of epidemic, especially in the research of the effective measures before the outbreak of the epidemic. Numerical computation is becoming the center position in the investigations of epidemic tendency which describes many natural phenomena, for example, in meteorology and biology [1–10]. However, we mainly concern the reliability and feasibility of numerical simulations by the network biomarker of Zika virus.This work is motivated for two facts. Firstly, some relative works have been finished. Lesterhuis W.J. and his team had made useful contributions to the discovery of the threshold value in the system by the compare of dynamical and static network biomarker in the complex disease [11]. The results of Dahlem M.A. show that the critical condition of the disease will be rapidly transformed into the illness state motivated by some factors, and the threshold value can be viewed as early warning signal of some complex disease [12]. Chen L.N. utilized dynamical network biomarker (DNB) to analyze the dynamical signal of the disease and provide three common characters of DNB [13]. These results are the foundations of forecast and numerical simulations by DNB. Secondly, a high degree of concern on Zika virus has been aroused in the world. As we know, Zika disease is a very serious epidemic which is a self-limited acute infectious disease and is mainly caused by Aedes mosquitoes. For example, at least 11000 confirmed cases of Zika have affected pregnant women, which leads to about 10000 cases of birth defects, such as microcephaly. The clinical features are mainly rash, fever, joint pain, or conjunctivitis, and it rarely causes death. Works on Zika continue to be an interesting topic. The results have illustrated that Zika virus has no protein structure (NS1) which is most related to the outbreak of Zika disease [14]. To the best of our knowledge, no investigations of the numerical computations of Zika virus protein in a finite time interval exist in the literature [6–10].In this work, we mainly focus on the early warning forecast of Zika by DNB. The protein of Zika virus is used to construct DNB, and the early warning index is also established by the property of DNB molecule. The early warning model is defined so that we can distinguish the critical value of Zika epidemic outbreak. The data is obtained by some country on Zika virus. In the numerical experiments, the method is applied to compute the year of Zika epidemic outbreak in one country, and the results show that it matches well with the time of Zika epidemic real outbreak in that country. And the results demonstrate that under certain appropriate assumptions the early warning index algorithm is well designed so that it can forecast the critical time exactly.The rest of this paper is organized as follows. Section2 deals with some preliminaries. In Section 3 the theoretical results of the early warning algorithm are summarized. Section 4 presents the details of the numerical implementations. Illustrative numerical experiments for the main results are included in Section 4. We demonstrate that the main theorem and numerical implementation methods can be applied to Brazil Zika forecast. Finally, Section 5 is addressed to summarize the conclusions of the paper.


## 2. Preliminaries

### 2.1. Results on the Protein of Zika Virus

Zika disease is highly contagious and is frequently found all over the world. Up to now, many advanced researches have been pursued in different aspects. Recently, the results in [14] show that the molecular structure of nonstructural protein, that is, NS2B, which is related to the outbreak of Zika epidemic, is the foundation of Zika virus research.


### 2.2. Resource of the Data

The data in this paper is selected from the Zika protein data in Brazil and United States, and they are shown on the internet website(https://www.ncbi.nlm.nih.gov/) As we know, Zika virus is single chain RNA virus, and more than 10000 basic groups encode into ten proteins. There are seven nonstructural proteins, such as NS2B, NS2A, NS4A, NS4B, NS1, NS3, and NS5, and three structural proteins, such as C proteins, PRM, and E protein.


## 2.1. Results on the Protein of Zika Virus

Zika disease is highly contagious and is frequently found all over the world. Up to now, many advanced researches have been pursued in different aspects. Recently, the results in [14] show that the molecular structure of nonstructural protein, that is, NS2B, which is related to the outbreak of Zika epidemic, is the foundation of Zika virus research.


## 2.2. Resource of the Data

The data in this paper is selected from the Zika protein data in Brazil and United States, and they are shown on the internet website(https://www.ncbi.nlm.nih.gov/) As we know, Zika virus is single chain RNA virus, and more than 10000 basic groups encode into ten proteins. There are seven nonstructural proteins, such as NS2B, NS2A, NS4A, NS4B, NS1, NS3, and NS5, and three structural proteins, such as C proteins, PRM, and E protein.


## 3. Design of the Early Warning Algorithm

### 3.1. A Selected Example

Here we choose NB2B protein as an example. Suppose that this protein is contained as t chains, which is denoted asp=x1,x2,…,xt, where i=1,2,…,t. xi∈A,V,L,I,P,F,W,M,D,E,G,S,T,C,Y,N,Q,R,K,H. Assume that one country reports m kind Zika virus proteins in the (y-1)-th year, whose amino acid series are Py-1,1,Py-1,2,…,Py-1,m.And it reports m kind Zika virus proteins in they-th year, whose amino acid series are Py,1,Py,2,…,Py,n. The number of amino acid Pij is denoted as Cij, where i=y-1,y, j=1,2,…,k, and k=maxm,n. We choose k-th amino acid in turns from the sequences Py-1,1,Py-1,2,…,Py-1,m and resemble them as a new series in the older turn Ly-1,k. Furthermore, we choose the amino acid to be denoted as xk which appears most frequently, where k=1,2,…,r,r=maxCy-1,1,Cy-1,2,…,Cy-1,m; otherwise, the first will be selected. Then xk will be interlinked in turns and to obtain a new amino acid series Ly-1=x1x2…xk.Therefore, it can be compared with the seriesPy,1,Py,2,…,Py,n successively, and the results can be shown as follows. If the condition r>Cy,j holds, we compare before Cy,j amino acid of the series Ly-1 with every series of y-th year. If the condition r≤Cy,j holds, we compare before r amino acid of Py,j with the series Ly-1. If the kind of these amino acids is the same, then it is assigned as 1; otherwise, it is assigned as 0. Therefore, we can obtain a new series of y-th year which has only the elements 0 and 1 and can be denoted as Qy,1,Qy,2,…,Qy,n.


### 3.2. Implementations of EWI Algorithm

Now we compute the following quantities.Step 1.
They-th year arithmetic mean value Ay is shown as follows. (1)Ay=1n∑j=1nfy,j
The standard deviationSTDy is defined as (2)STDy=∑j=1nfy,j-Ay2n,and the coefficient of variation CVy is defined as (3)CVy=STDyAy,where f(y,j) denotes the number of times of one appearing in Qij, j=1,2,…,n. The value of the parameters Ay, STDy, and CVy of other proteins is obtained by the same method.Meanwhile, we utilize the software SAS to test on all protein standard deviations in all considered years, and we choose the protein as the central, which shows obvious difference; that is, it is can be seen as the signal of dynamical network biomarker, and others can be defined as noncentral protein.As we know, every series contains N kind amino acids so that the times which every amino acid emerges in the y-th year can be calculated. And the result is as follows.Step 2.
(4) F x i = 1 n ∑ j = 1 n f x i y , j ,where fxi(y,j) denotes the times which xi emerges in the amino acid Py,j.Following from the above results, we can obtain a feature vector of amino acidV(y), and feature vectors of other amino acids are obtained by the similar method [15].Step 3.
The matrix of feature vector is defined as(5)V=V1y,V2y,…,VNy,where Vk(y) is the feature vector of thek-th protein, k=1,2,…,N. And the feature distance of protein in they-th year is defined as follows: (6)EDk1k2=Ak1-Ak22+⋯+Fk1xN-Fk2xN2,where k1,k2 denote two kinds of different proteins. Therefore, we can compute all feature distances betweenN kinds of protein.By the dynamical property of center protein and DBN, Early Warning Index (EWI) can be computed as follows.Step 4.
(7) E W I = C V 0 ¯ • E D 0 ¯ E D 0 ,where CV0¯ denotes the mean value of the coefficient of variation of Zika virus center protein, ED0¯ denotes the mean character distance of center proteins, and ED0 denotes the mean character distance of center proteins and noncenter proteins.We can apply EWI to determine the outbreak time of Zika epidemic as follows.Step 5.
If the following conditions are satisfied,(8)EWIy-1<1,EWIy+1>1EWIy-EWIy-1<EWIy+1-EWIy,then they-th year is called as the critical year of Zika epidemic outbreak, and they+1-th year is called the year of Zika epidemic outbreak, where EWIy-1, EWIy, and EWIy+1 are EWI of they-1-th,y-th, andy+1-th year, respectively [16, 17].


## 3.1. A Selected Example

Here we choose NB2B protein as an example. Suppose that this protein is contained as t chains, which is denoted asp=x1,x2,…,xt, where i=1,2,…,t. xi∈A,V,L,I,P,F,W,M,D,E,G,S,T,C,Y,N,Q,R,K,H. Assume that one country reports m kind Zika virus proteins in the (y-1)-th year, whose amino acid series are Py-1,1,Py-1,2,…,Py-1,m.And it reports m kind Zika virus proteins in they-th year, whose amino acid series are Py,1,Py,2,…,Py,n. The number of amino acid Pij is denoted as Cij, where i=y-1,y, j=1,2,…,k, and k=maxm,n. We choose k-th amino acid in turns from the sequences Py-1,1,Py-1,2,…,Py-1,m and resemble them as a new series in the older turn Ly-1,k. Furthermore, we choose the amino acid to be denoted as xk which appears most frequently, where k=1,2,…,r,r=maxCy-1,1,Cy-1,2,…,Cy-1,m; otherwise, the first will be selected. Then xk will be interlinked in turns and to obtain a new amino acid series Ly-1=x1x2…xk.Therefore, it can be compared with the seriesPy,1,Py,2,…,Py,n successively, and the results can be shown as follows. If the condition r>Cy,j holds, we compare before Cy,j amino acid of the series Ly-1 with every series of y-th year. If the condition r≤Cy,j holds, we compare before r amino acid of Py,j with the series Ly-1. If the kind of these amino acids is the same, then it is assigned as 1; otherwise, it is assigned as 0. Therefore, we can obtain a new series of y-th year which has only the elements 0 and 1 and can be denoted as Qy,1,Qy,2,…,Qy,n.


## 3.2. Implementations of EWI Algorithm

Now we compute the following quantities.Step 1.
They-th year arithmetic mean value Ay is shown as follows. (1)Ay=1n∑j=1nfy,j
The standard deviationSTDy is defined as (2)STDy=∑j=1nfy,j-Ay2n,and the coefficient of variation CVy is defined as (3)CVy=STDyAy,where f(y,j) denotes the number of times of one appearing in Qij, j=1,2,…,n. The value of the parameters Ay, STDy, and CVy of other proteins is obtained by the same method.Meanwhile, we utilize the software SAS to test on all protein standard deviations in all considered years, and we choose the protein as the central, which shows obvious difference; that is, it is can be seen as the signal of dynamical network biomarker, and others can be defined as noncentral protein.As we know, every series contains N kind amino acids so that the times which every amino acid emerges in the y-th year can be calculated. And the result is as follows.Step 2.
(4) F x i = 1 n ∑ j = 1 n f x i y , j ,where fxi(y,j) denotes the times which xi emerges in the amino acid Py,j.Following from the above results, we can obtain a feature vector of amino acidV(y), and feature vectors of other amino acids are obtained by the similar method [15].Step 3.
The matrix of feature vector is defined as(5)V=V1y,V2y,…,VNy,where Vk(y) is the feature vector of thek-th protein, k=1,2,…,N. And the feature distance of protein in they-th year is defined as follows: (6)EDk1k2=Ak1-Ak22+⋯+Fk1xN-Fk2xN2,where k1,k2 denote two kinds of different proteins. Therefore, we can compute all feature distances betweenN kinds of protein.By the dynamical property of center protein and DBN, Early Warning Index (EWI) can be computed as follows.Step 4.
(7) E W I = C V 0 ¯ • E D 0 ¯ E D 0 ,where CV0¯ denotes the mean value of the coefficient of variation of Zika virus center protein, ED0¯ denotes the mean character distance of center proteins, and ED0 denotes the mean character distance of center proteins and noncenter proteins.We can apply EWI to determine the outbreak time of Zika epidemic as follows.Step 5.
If the following conditions are satisfied,(8)EWIy-1<1,EWIy+1>1EWIy-EWIy-1<EWIy+1-EWIy,then they-th year is called as the critical year of Zika epidemic outbreak, and they+1-th year is called the year of Zika epidemic outbreak, where EWIy-1, EWIy, and EWIy+1 are EWI of they-1-th,y-th, andy+1-th year, respectively [16, 17].


## 4. Applications of EWI Algorithm

This section is the applications of EWI algorithm. We apply the data from Brazil in the past 37 years. The change is analysed and the precise forecast is obtained.


### 4.1. EWI of Zika Epidemic in Brazil

Here we select the data of ten kinds of proteins of Zika virus from 1980 to 2016. By the former formula, we can calculate EWI of Brazil, and the results are shown as Table1 [17, 18].Table 1
Summaries of Zika virus EWI from 1980 to 2016 in Brazil.
Year EWI Year EWI Year EWI 1980 0.011234 1992 0.881891 2005 0.889432 1981 0.201478 1993 0.967212 2006 0.543217 1982 0.310231 1994 0.043254 2007 0.612523 1983 0.711321 1995 0.006547 2008 0.321876 1984 0.781221 1996 0.054334 2009 6.342679 1985 0.219911 1997 0.063429 2010 0.546751 1986 0.097215 1998 0.033459 2011 0.132156 1987 0.021018 1999 0.207752 2012 0.923995 1988 0.034662 2000 0.304561 2013 0.992189 1989 0.800772 2001 0.762337 2014 3.032341 1990 0.821221 2002 0.821356 2015 7.127854 1991 0.940983 2003 0.021355 2016 4.452137


### 4.2. Analysis on Early Warning Model of Brazil Zika Epidemic

As can be seen from Figure1, the value of EWI in the years 1983 and 1991 increases suddenly. However, EWI1982<1, and EWI1983<1. It follows from EWI algorithm that there were no Zika epidemics in Brazil at that time. This theoretical result matches the reality well.Figure 1
Variation tendency of EWI in Brazil in the time intervals[1981, 1983] and [1989, 1991].As can be seen from these numerical results shown as Figures2 and 3, in the years 2009 and 2014 EWI increased suddenly and it is obviously greater than the former. For example, EWI2013<1, while EWI2014>1. It follows from EWI algorithm that there were Zika epidemics in Brazil at that time. Therefore, the years 2009 and 2014 are called the years of Zika epidemic outbreak, and the years 2008 and 2013 are called the critical years of Zika epidemic outbreak. These theoretical results match the reality well. The numerically detected behavior of the system indeed reflects its real fact which can be described more accurately.Figure 2
Variation tendency of EWI in Brazil in the time intervals[2007, 2009] and [2011, 2013].Figure 3
Variation tendency of EWI in Brazil from 2014 to 2016.


### 4.3. Forecast on Zika Epidemic in Brazil

Based on these numerical results, we can predict Zika epidemic in the near future in Brazil. As can be seen from these numerical results shown as Figures2 and 3, in the year 2017, EWI maintains steadiness and does not increase suddenly and it is not obviously greater than the former. That is, EWI2017<EWI2016. If EWI2018>EWI2017 or EWI2019>EWI2017, it follows from EWI algorithm that there will be Zika epidemic in Brazil at that time. Otherwise, i.e., EWI2018<EWI2017 and EWI2019<EWI2017, and Zika epidemic will not break out in 2018 and 2019.


### 4.4. Comparison with Other Models and Methods

As we know, some existing work about this topic is also interesting, such as References [19, 20]. These studies used ecological niche models (ENMs) to map the possible distribution of Zika virus, utilizing a different combination of occurrence data, environmental predictors, and statistical approaches. And it is shown in [20] that the results obtained by generic and uniformed stochastic county-level simulations illustrate a basic consensus method, which can resolve conflicting models of potential outbreak geography and seasonality in the United States. However, our results illustrate that the forecast of the outbreak of Zika by dynamical network biomarker relies on the protein of Zika virus, and the early warning index is also established by the property of DNB molecule. This is to say, our method mainly depends on dynamical network biomarker, such that it can be less influenced by uncertain factors. The method in [20] focuses on the United States as a test system and depends on many random factors, including environmental variable and population-at-risk. Therefore, we have the same purpose, i.e., to forecast the outbreak of Zika epidemic, but by different methods.


## 4.1. EWI of Zika Epidemic in Brazil

Here we select the data of ten kinds of proteins of Zika virus from 1980 to 2016. By the former formula, we can calculate EWI of Brazil, and the results are shown as Table1 [17, 18].Table 1
Summaries of Zika virus EWI from 1980 to 2016 in Brazil.
Year EWI Year EWI Year EWI 1980 0.011234 1992 0.881891 2005 0.889432 1981 0.201478 1993 0.967212 2006 0.543217 1982 0.310231 1994 0.043254 2007 0.612523 1983 0.711321 1995 0.006547 2008 0.321876 1984 0.781221 1996 0.054334 2009 6.342679 1985 0.219911 1997 0.063429 2010 0.546751 1986 0.097215 1998 0.033459 2011 0.132156 1987 0.021018 1999 0.207752 2012 0.923995 1988 0.034662 2000 0.304561 2013 0.992189 1989 0.800772 2001 0.762337 2014 3.032341 1990 0.821221 2002 0.821356 2015 7.127854 1991 0.940983 2003 0.021355 2016 4.452137


## 4.2. Analysis on Early Warning Model of Brazil Zika Epidemic

As can be seen from Figure1, the value of EWI in the years 1983 and 1991 increases suddenly. However, EWI1982<1, and EWI1983<1. It follows from EWI algorithm that there were no Zika epidemics in Brazil at that time. This theoretical result matches the reality well.Figure 1
Variation tendency of EWI in Brazil in the time intervals[1981, 1983] and [1989, 1991].As can be seen from these numerical results shown as Figures2 and 3, in the years 2009 and 2014 EWI increased suddenly and it is obviously greater than the former. For example, EWI2013<1, while EWI2014>1. It follows from EWI algorithm that there were Zika epidemics in Brazil at that time. Therefore, the years 2009 and 2014 are called the years of Zika epidemic outbreak, and the years 2008 and 2013 are called the critical years of Zika epidemic outbreak. These theoretical results match the reality well. The numerically detected behavior of the system indeed reflects its real fact which can be described more accurately.Figure 2
Variation tendency of EWI in Brazil in the time intervals[2007, 2009] and [2011, 2013].Figure 3
Variation tendency of EWI in Brazil from 2014 to 2016.


## 4.3. Forecast on Zika Epidemic in Brazil

Based on these numerical results, we can predict Zika epidemic in the near future in Brazil. As can be seen from these numerical results shown as Figures2 and 3, in the year 2017, EWI maintains steadiness and does not increase suddenly and it is not obviously greater than the former. That is, EWI2017<EWI2016. If EWI2018>EWI2017 or EWI2019>EWI2017, it follows from EWI algorithm that there will be Zika epidemic in Brazil at that time. Otherwise, i.e., EWI2018<EWI2017 and EWI2019<EWI2017, and Zika epidemic will not break out in 2018 and 2019.


## 4.4. Comparison with Other Models and Methods

As we know, some existing work about this topic is also interesting, such as References [19, 20]. These studies used ecological niche models (ENMs) to map the possible distribution of Zika virus, utilizing a different combination of occurrence data, environmental predictors, and statistical approaches. And it is shown in [20] that the results obtained by generic and uniformed stochastic county-level simulations illustrate a basic consensus method, which can resolve conflicting models of potential outbreak geography and seasonality in the United States. However, our results illustrate that the forecast of the outbreak of Zika by dynamical network biomarker relies on the protein of Zika virus, and the early warning index is also established by the property of DNB molecule. This is to say, our method mainly depends on dynamical network biomarker, such that it can be less influenced by uncertain factors. The method in [20] focuses on the United States as a test system and depends on many random factors, including environmental variable and population-at-risk. Therefore, we have the same purpose, i.e., to forecast the outbreak of Zika epidemic, but by different methods.


## 5. Conclusion

The main result of this paper is the construction and applications of Zika epidemic EWI algorithm. It focuses on the mathematical approaches to estimate the tendency of Brazil Zika epidemic in the past 37 years. The results show that the methods are effective and the numerical experiments are performed and match the results of theoretical analysis. More simple and higher accuracy methods will be shown in our further work.



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