Application of the backstepping method to the prediction of increase or decrease of infected population
 Toshikazu Kuniya^{1}Email author and
 Hideki Sano^{1}
https://doi.org/10.1186/s1297601600416
© Kuniya and Sano. 2016
Received: 11 January 2016
Accepted: 29 April 2016
Published: 10 May 2016
Abstract
Background
In mathematical epidemiology, agestructured epidemic models have usually been formulated as the boundaryvalue problems of the partial differential equations. On the other hand, in engineering, the backstepping method has recently been developed and widely studied by many authors.
Methods
Using the backstepping method, we obtained a boundary feedback control which plays the role of the threshold criteria for the prediction of increase or decrease of newly infected population. Under an assumption that the period of infectiousness is same for all infected individuals (that is, the recovery rate is given by the Dirac delta function multiplied by a sufficiently large positive constant), the prediction method is simplified to the comparison of the numbers of reported cases at the current and previous time steps.
Results
Our prediction method was applied to the reported cases per sentinel of influenza in Japan from 2006 to 2015 and its accuracy was 0.81 (404 correct predictions to the total 500 predictions). It was higher than that of the ARIMA models with different orders of the autoregressive part, differencing and movingaverage process. In addition, a proposed method for the estimation of the number of reported cases, which is consistent with our prediction method, was better than that of the bestfitted ARIMA model ARIMA(1,1,0) in the sense of mean square error.
Conclusions
Our prediction method based on the backstepping method can be simplified to the comparison of the numbers of reported cases of the current and previous time steps. In spite of its simplicity, it can provide a good prediction for the spread of influenza in Japan.
Keywords
Backstepping method Age structure Prediction Influenza ARIMABackground
The spreading mechanism of infectious diseases in population has been studied in terms of mathematical modeling since the pioneering works by Kermack and McKendrick [1, 2] (see Hethcote [3] for a detailed review). In the eradication or reduction of diseases, the concept of control would play an important role (see, for instance, Anderson and May [4] and Smith et al. [5]). To design the appropriate control strategies, the prediction of epidemic size is essentially important. In recent years, Ferguson et al. [6], Hyder et al. [7] and Riley et al. [8] studied influenza and other diseases from this perspective. On the other hand, the qualitative analysis of mathematical models is also important for understanding the effect of control. Alexander et al. [9] and Mills et al. [10] published their study from this perspective.
For a long time, agestructured epidemic models have been studied by many authors (see, for instance, Iannelli [11], Inaba [12], Inaba and Nishiura [13] and Tudor [14]). Mathematically, these models can be regarded as the boundaryvalue problems of partial differential equations. On the other hand, in engineering, the backstepping method has recently been developed by Smyshlyaev and Krstic [15] to obtain the boundary feedback control for stabilizing the systems of partial differential equations and has widely been studied by many authors (see, for instance, Susto and Krstic [16] and Baccoli et al. [17]). The aim of this study is to make use of the backstepping method for epidemiological considerations. Specifically, we will develop a new method for the prediction of increase or decrease of infected population.
In the classical theory of the basic reproduction number \({\mathcal {R}}_{0}\) (see, for instance, Diekmann et al. [18] and van den Driessche and Watmough [19]), the number of newly infected individuals produced by a typical infected individual invading into a fully susceptible population is characterized as the threshold value. That is, if \({\mathcal {R}}_{0} > 1\), then the epidemic size will increase and if \({\mathcal {R}}_{0} < 1\), then it will decrease. In our method, a threshold criteria U(t) will be calculated for each time step t as a boundary feedback control. It will be shown that whether the newly infected population I(t,0) at each time step t will exceed U(t) or not determines the increase or decrease of infected population at the next time step t+1. That is, if I(t,0)>U(t) (I(t,0)<U(t)) at the current time step t, then the infected population will increase (decrease) at the next time step t+1. Thus, we can predict the increase or decrease of infected population by comparing I(t,0) and U(t) for each time step.
Methods
The model formulation

t≥0: the chronological time;

a∈ [ 0,a _{ † }]: the class age (that is, the time elapsed since the infection) of infected individuals;

a _{ † }>0: the maximum period of infectiousness;

I(t,a): the infected population of class age a at time t;

\(\gamma (a) \in L^{1}_{+}(0,a_{\dagger })\): the per capita recovery rate of infected individuals of class age a.
Note that the boundary condition for a=0 is not considered in (1) unlike usual class agestructured epidemic models. In fact, in this paper, we analytically derive the boundary feedback control U(t) by using the backstepping method and compare it with I(t,0) which will be obtained from the real data.
The backstepping method
has an equilibrium I ^{∗}(a), then I(t,a) converges to I ^{∗}(a) as t→+∞. Hence, in what follows, we will determine the integral kernel k(a,σ) such that (2) satisfies (3) and further (4) has the equilibrium I ^{∗}(a).
and if I(t,0)>U(t), then the infected population I(t,a) will increase and diverge, while if I(t,0)<U(t), then the infected population I(t,a) will decrease and converge to 0.
Remark on mathematical wellposedness
In fact, if the first two conditions are satisfied, we can show the existence of the differentiable solution I(t,a) by constructing a C _{0}semigroup generated by the differentiation operator −d/da−γ(a) with the domain \(\{I(\cdot) \in H^{1} (0,a_{\dagger }) ; I(0) = \int _{0}^{a_{\dagger }} \psi (a_{\dagger }\sigma) \mathrm {e}^{\int _{0}^{a_{\dagger }\sigma } \left (\gamma (a_{\dagger }\sigma \rho) \gamma (a_{\dagger }\rho) \right) \mathrm {d}\rho } I(\sigma) \mathrm {d}\sigma \}\). Since the aim of this paper is to develop a prediction method, we do not require the observed data to satisfy the condition (11). However, we remark that this mathematical rigorousness is essentially important from the analytical point of view.
The prediction method
Results and discussion
Prediction of increase or decrease of infected population for influenza in Japan
In this section, we apply our prediction method proposed above to the real data of reported cases of influenza in Japan from 2006 to 2015. The data is available in the website of the National Institute of Infectious Diseases, Japan (see [20]). Let the unit time step be a week.
The actual number of reported cases per sentinel and prediction results for influenza in Japan from week 1 to week 12 in 2015
Week (t)  Number of reported  U(t)  Prediction  Result 

cases per sentinel  
1  21.46  –  –  – 
2  33.28  21.46  Increase  Correct 
3  37  33.28  Increase  Correct 
4  39.42  37  Increase  Incorrect 
5  29.11  39.42  Decrease  Correct 
6  19.03  29.11  Decrease  Correct 
7  12.15  19.03  Decrease  Correct 
8  8.26  12.15  Decrease  Correct 
9  5.88  8.26  Decrease  Correct 
10  4.32  5.88  Decrease  Correct 
11  3.99  4.32  Decrease  Correct 
12  3.85  –  –  – 
Accuracy  0.90(=9/10) 
In Table 1, it is easy to see that the control U(t) is equal to the number of reported cases at the previous week t−1 as derived in (13). For each time step t∈ [ 2,11], the number of reported cases is compared to U(t) and if it is greater than U(t), then the prediction is “Increase” and if it is less than U(t), then the prediction is “Decrease”. In this case, the number of correct predictions is 9 and the total number of predictions is 10. Hence, the accuracy of the prediction is 9/10=0.90.
The accuracy of our prediction for influenza in Japan from 2006 to 2015
Year  Number of correct  Number of incorrect  Accuracy 

predictions  predictions  
2006  44  6  0.88 
2007  42  8  0.84 
2008  45  5  0.90 
2009  37  13  0.74 
2010  35  15  0.70 
2011  42  8  0.84 
2012  38  12  0.76 
2013  41  9  0.82 
2014  41  9  0.82 
2015  39  11  0.78 
Total  404  96  0.81 
As in the previous example, the prediction is not performed for the first week (t=1) and the last week (t=52). Therefore, the total number of predictions per year is 50. The accuracy of total predictions is 0.81 and relatively high.
Comparison
The accuracy of the alternative prediction based on ARIMA(p,d,q) models for influenza in Japan from 2006 to 2015 (total 500 predictions)
(p,d,q)  Number of correct  Number of incorrect  Accuracy 

predictions  predictions  
(0,0,1)  224  276  0.45 
(0,1,0)  244  256  0.49 
(1,0,0)  156  344  0.31 
(0,1,1)  382  118  0.76 
(1,0,1)  152  348  0.30 
(1,1,0)  393  107  0.79 
(1,1,1)  355  145  0.71 
From Table 3, we see that (p,d,q)=(1,1,0) is the best choice in this case. Nonetheless, its accuracy is lower than that of our prediction method (0.79<0.81).
Estimation of the number of reported cases
Discussion
Although our prediction method only need the data from each of the past 2 weeks (including the current week), the accuracy of it (0.81) was higher than that of the bestfitted ARIMA model ARIMA(1,1,0) (0.79) which is based on the data from each of the past 10 weeks. Moreover, our estimation method (14) for the number of newly infected population was better than ARIMA(1,1,0) in the sense of mean square error. From these results, we conjecture that focusing only on the data in the current and previous weeks can lead to a good prediction for the spread of influenza in Japan.
Conclusions
In this study, based on the backstepping method in engineering, we developed a new prediction method for the increase or decrease of newly infected population. Under the assumption that the period of infectiousness is same for all infected individuals (that is, the recovery rate is given by the Dirac delta function multiplied by a sufficiently large positive constant), the method was simplified to the comparison of the number of reported cases at the current and previous time steps. In spite of its simplicity, its accuracy was relatively high (0.81) for the spread of influenza in Japan from 2006 to 2015. Furthermore, the simple estimation method (14) based on the linear law was proposed and its accuracy was better in the sense of mean square error than that of the bestfitted ARIMA model ARIMA(1,1,0), which is based on the data from each of the past 10 weeks. From these results, we conjectured that focusing only on the data in the current and previous weeks can lead to a good prediction for the spread of influenza in Japan.
As future tasks, not limited to influenza in Japan, our prediction method would be applied to various infectious diseases in various countries. The assumption that the period of infectiousness is same for all infected individuals might have to be modified for each case. Our model (1) was based on the SIR epidemic model in which the newly infected individuals immediately have the infectiousness without latency. To consider more realistic situations, we might have to start the discussion from some different models such as the SEIR and SIRS epidemic models.
Declarations
Acknowledgements
The authors would like to thank anonymous reviewers for their helpful comments and suggestions to the previous version of the manuscript. TK was supported by GrantinAid for Young Scientists (B), No.15K17585 of Japan Society for the Promotion of Science. HS was supported by GrantinAid for Scientific Research (C), No.15K04999 of Japan Society for the Promotion of Science. TK and HS were supported by the program of the Japan Initiative for Global Research Network on Infectious Diseases (JGRID); from Japan Agency for Medical Research and Development, AMED.
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Authors’ Affiliations
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