Harmonic Analysis via an Integral Equation: An Application to Dengue Transmission

A collection of oscillatory basis functions generated via an integral equation is investigated here. This is a new approach in the harmonic analysis as we are able to interpret phenomena with damping and amplifying oscillations other than classical Fourierlike periodic waves. The proposed technique is tested with a data set of dengue incidence, where different types of influences prevail. An intermediate transform supported by the Laplace transform is available. It facilitates parameter estimation and strengthens the extraction of hidden influencing accumulations. This mechanistic work can be extended as a tool in signal processing that encounters oscillatory and accumulated effects.


Introduction
Harmonic analysis is central to many applications in signal processing. Its applicability does not restrict to physical waves, showing potential applications in many phenomena from biology to finance [1]. Superpositioning of basic waves to represent a wave or a function is the key mechanism in harmonic analysis. Fourier analysis is a frequently used method in signal processing, where sinusoidal waves perform as basis functions [2]. It is the baseline in many comparison studies as well. For instance, Broadbent and Maksik utilized the Walsh transform in analyzing rectangular waves [3].
In addition, statistical evaluations such as correlation analysis can also be adopted in studying signals, particularly time series. However, some details in multiple periodicities or unrecognized variables would not be extracted [4]. Thus, we opted to build our work advancing upon the Fourier series. Furthermore, there are techniques of accelerating convergence in the Fourier series, for instance, incorporating derivative information at the endpoints [5]. However, our approach allows changing harmonic patterns rather than restricting them into periodic waves.
As attributed to the Fourier series, periodic basis functions may not underlie every application. Therefore, variants with damping and amplifying oscillations would be subsequent improvements. Here, in this paper, we propose such work via the Volterra integral equation (1), which has been implemented to model population growth [6].
The model equation (1) describes the population nðtÞ according to the survival incorporated by a function f ðtÞ as age increases. It caters to an accumulated effect on the population as each surviving individual contributes to newborns. We alter its ideology into a context of epidemiology to mimic the incidence of infectious diseases. Then, an accumulated process of already infected individuals governs further incidence. Other external influences can also be incorporated.
We introduce a harmonic process carrying both sinusoidal and exponential effects to the solutions of (1). It endows with damping and amplifying oscillations other than to classical periodic waves. Next, we plan to extract influencing function f ðtÞ to understand the process behind each wave. This activity is further motivated by the difference kernel kf ðt − τÞ in (1), which leads to potential solutions for nðtÞ via Laplace transforms. We validate this new harmonic analysis for dengue incidence with a comparison of the classical technique of discrete Fourier series.

Preliminaries and Alterations to the Integral Model.
In the exact terminology of the model equation (1), nðtÞ is the total population at time t with an initial value n 0 . Here, f ðtÞ, the survival function, represents the fraction of people surviving to age t. Hence, f ðtÞ depicts the age-dependent potential of human survival, which is the main routine of the model. The parameter k represents the births per individual, and the argument t − τ in the kernel kf ðt − τÞ catches a lagging effect due to the time of birth.
In our move towards dengue incidence, f ðtÞ acts as an influencing function that tolerates ultimate transmission dynamics. For instance, when nðtÞ detects the incidence, then f ðtÞ can represent an influencing factor such as rainfall patterns. Later, we aim to extract hidden influencing functions. Disease-specific interpretations can be aligned with the influences by vector mosquitoes, weather patterns, and biological or behavioral factors of human hosts [7]. Definition 1. We define nðtÞ to be a measure on the number of infected cases (incidence) at time t influenced by an influencing function f ðtÞ.
Our aim is to use nðtÞ directly to formulate basis functions that approximate dengue incidence data.
The manipulation of f ðtÞ is twofold as articulating a sinusoidal pattern as a base and allowing damping and amplifying variants through exponential effects. Here, parameter a determines the wavelength or frequency and parameter b determines any shift due to initial requirements in the absence of an exponential effect. The role of the parameter λ is also twofold as standing for scaling requirements and partially responsible for the final linear combination of basic waves.

Model
Solution. Now, we present the solution of the model equation (1) and its direction to the proposed application in dengue incidence. Since Lf f ðtÞg = λðððs − cÞ sin b + a cos bÞ/ððs − cÞ 2 + a 2 ÞÞ, we arrive at If a 2 − kλa cos b − ððkλ sin bÞ/2Þ 2 > 0, then we reach the solution in Case 1 and if a 2 − kλa cos b − ððkλ sin bÞ/2Þ 2 < 0, then we have Case 2.
Next, Proposition 4 presents the inverse process of obtaining f ðtÞ that ensures the analytical existence of f ðtÞ for a given nðtÞ.
Definition 5. We structure the superposition upon nðtÞ as Nð tÞ = ∑ ∞ i=1 n i ðtÞ, where n i ðtÞ = n 0i λ i e β i t ðP i cos μ i t + Q i sin μ i t Þ is the basis function corresponding to the influencing function f i ðtÞ = λ i e c i t sin ða i t + b i Þ for each case of i = 1, 2, ⋯. We call this NðtÞ as an overall transmission potential.
Approximations in finite context can be obtained by N m ðtÞ = ∑ m i=1 n i ðtÞ, where m number of harmonics are used. This finally leads to approximate incidence data in the least square sense.
iωt is taken to approximate data in a discrete sense. First, we determine the Fourier harmonics F i ðtÞ = A i cos iωt + B i sin iωt,i = 1, 2, ⋯, m, and the intercept term A 0 (i.e., F 0 ðtÞ) using the MATLAB curve fitting toolbox that minimizes ∑ p j=1 ðd j − T m ðt j ÞÞ 2 . Here, d j refers to the data value at a point t j for j = 1, 2, ⋯, p.
Next, we estimate initial guesses for each set of parameters of f i ðtÞ infused in n i ðtÞ. These estimations are obtained by aligning n i ðtÞ with sequential harmonics F i ðtÞ separately. It is fulfilled by a trial and error method with a reasonable tolerance (ε) on the squared deviation between each n i ðtÞ and its Fourier counterpart F i ðtÞ. Here, near-periodic behaviors are attributed into n i ðtÞ while an exponential effect finally plays the role of fine-tuning the overall fit.
The following result guarantees that no complete restriction is employed on the exponential effect of f i ðtÞ while forcing the removal of the exponential effect in the space of n i ðtÞ.

Journal of Applied Mathematics
Proof. The result is straightforward as we have β = c + ððλk sin bÞ/2Þ.
Next, we initialize the parameters as described above and minimize ∑ p j=1 ðd j − ðA 0 + N m ðt j ÞÞÞ 2 that represents the deviation between data and superposition output A 0 + N m ðt j Þ. For that, we implement a MATLAB fminsearch tool, which uses the Nelder-Mead simplex algorithm [11]. By using the resultant parameter values, we can determine all n i ðtÞ and f i ðtÞ.

Outlook of Convergence.
It is obvious that both the Fourier fit and the model fit yield better approximations of data when the number of series increases. For a given number of series, our model fit may reach data much closer than that of the Fourier fit by the effect of e β i t in n i ðtÞ. It is assisted by the boundedness of t and the possibility of any real value for β i . Note that the terms involving t in the Fourier fit are sine and cosine terms allowing only oscillations tolerated by respective coefficients. However, in the model fit, one may see the adaptability of coefficients as per the effect of e β i t . Options on β i are indirectly subjected to the condition on μ i 2 (Proposition 3). Notwithstanding, sophisticated searching abilities in the software may reduce its burden in computational trials.

Results
3.1. Curve Fitting. In Figure 1, Fourier fit T m ðtÞ and model fit A 0 + N m ðtÞ are illustrated. We add the intercept term A 0 into N m ðtÞ to compromise with actual data. At the trial and error stage, the tolerance ε < 10 3 . We start with m = 5 and finish with m = 8, which is the maximum number of harmonics allowed in the MATLAB Fourier fit.
Graphs in Figure 1 visualize the reliability of the fitted curves for different m, and Table 1 contains the sum of squared deviations. It shows that A 0 + N m ðtÞ fits well compared to its corresponding T m ðtÞ for each m. Therefore, the basis function n i ðtÞ can further be used as a reliable intermediate tool to extract f i ðtÞ.
One can observe that the model fit and the Fourier fit mainly differ at the extremes of the data set. It is an optimistic rectification over Fourier harmonics to achieve a better fit via the model (1). This hints the effect of e β i t in n i ðtÞ as discussed in Section 2.4. Figure 2 further illustrates such an effect at an extreme. Here, the red curve stands for cos t + 0:5 sin t and the blue curve e 0:1t ðcos t + 0:5 sin tÞ contains an exponential effect. Observe that the additional fluctuations of the blue curve at the latter stage allow for reaching attributes in data, which cannot be extracted only via sinusoidal curves. Thus, the convergence of series at extremes is more equipped with exponential orders.

Basis Functions.
Since improvements are expected upon near-periodic waves, it is worthwhile to see the deviations in trigonometric terms. Figure 3 depicts the relevant cases in each harmonic for m = 5. We illustrate sine terms (a 1 -a 5 ) and cosine terms (b 1 -b 5 ) in the Fourier fit and their counterparts in the model fit, excluding the exponential effect. Besides, harmonics are also presented without A 0 (Figure 3, c 1 -c 5 ). The values of iω and μ are indicated to see the variations.
Here, different oscillatory behaviors are visible as damping where β < 0 (c 1 , c 3 , c 4 , and c 5 ) and amplifying where β > 0 (c 2 ). The strength of the damping or amplifying nature can be seen via the size of β. Thus, comparatively large jβj forces the curves to deviate from the classical periodic nature. Oscillations around zero preserve the same argument as in Fourier harmonics, which indicate changes over an intercept value A 0 . In the final model fit, parameter λ is partially responsible for combining basic patterns of the type e βt ðP cos μt + Q sin μtÞ. Originally, λ determines the amplitude (scaling) requirements of influencing pattern e ct sin ðat + bÞ. Besides, parameter n 0 stands for further compromise of the amplitude while representing the initial requirements of the model (1). These parameter values for the above case in Figure 3 are listed in Table 2. The values of n 0 λ are also displayed to see the outcome of combining e βt ðP cos μt + Q sin μtÞ type waves in the overall transmission potential. Thus, our approach establishes a mechanistic way of seeing different possibilities of oscillatory harmonics.   According to the size of n 0 λ, n 2 and n 5 contain the lowest impact and the highest impact, respectively, for the overall transmission. Such observations allow for seeing the quality (via the positive/negative sign of n 0 λ) and the quantity (via the size of n 0 λ) of contributions made by each harmonic structure e βt ðP cos μt + Q sin μtÞ.
Analogous to Fourier analysis, one can design a power spectrum by summing the size of coefficients in each exponentially weighted sinusoidal: e βt cos μt and e βt sin μt. Table 3 contains those coefficient values and the power taken as jn 0 λPj + jn 0 λQj. Here, also n 2 and n 5 contain the lowest power and the highest power, respectively.
Decomposing a data series into basic sinusoidal waves is the main routine of Fourier analysis. Similarly, our approach via model (1) accompanies a structure with exponentially weighted sinusoidals. A noted fact is that in influencing functions also, we can observe a similar context of exponentially weighted sinusoidal.

Influencing Functions.
The influencing functions f i ðtÞ of the above case in Figure 3 are shown in Figure 4. As described earlier, these functions incorporate accumulation effects of influences on disease transmission. Note that if no accumulation is enforced (i.e., k = 0), then the behavior of the basis      Here, also, we observe the quality (via the positive/negative sign of k) and the quantity (via the size of k) of contributions made by the accumulation Ð t 0 f ðt − τÞnðτÞdτ in model equation (1).

Interpretation Prospects.
Dengue is a mosquito-borne viral disease, which is caused by four virus serotypes (DENV-1, DENV-2, DENV-3, and DENV-4). Long-term exposure to one serotype generates immunity to that particular type, but only partial cross-protection is evident for other serotypes in the long run [12,13]. On the other hand, vector mosquitoes, Aedes aegypti and Aedes albopictus, are highly adapted to the urban environment [14,15]. Therefore, a collective risk may occur in different combinations of virus types and adapted mosquitoes. This leaves control measures unstable and inconsistent, which can be identified by amplifying oscillations in basis functions (e.g., n 2 ).
After a set of individuals are infected with dengue, their transmission capability fluctuates in different ways. As damping oscillations suggest, that capability may feature in two ways as aligning with rainfall patterns and aligning with awareness on disease control. Since awareness increases in rainy seasons, that combined effect is subject to oscillatory behavior. The concerned Colombo municipal area is vulner-able to a peak in rainfall in every six months due to monsoon and intermonsoon effects [16]. Therefore, as seen in (c 4 ) and (c 5 ), we may expect a peak in transmission around every six months. Meanwhile, as shown in (c 1 ), n 1 captures a higher incidence in the middle period of the data series, suggesting the capability of incorporating long-term influences.
The basis function n 3 in (c 3 ) does not indicate strong peaks but suggests a quicker saturation to a particular transmission level. This is understood via improved mosquito control and health care irrespective of rainfall effects. More specific interpretations can be drawn when we consider more harmonics in T m and correspondingly in N m . On the other hand, one can keep the desired information while removing any noisy or extraneous waves when we have more harmonics [1].

Discussion and Conclusion
A mechanism for extracting oscillatory behaviors with damping, amplifying, or periodic nature is discussed in this work. It is contrasting to the classical harmonic analysis by Fourier series, where only periodic waves are considered. The integral equation (1) allows the dependent variable to incorporate accumulation over a time period. Incidence of infectious diseases such as dengue can be modeled similarly as in (1). The whole work amalgamates decomposing data into basic oscillatory waves and infusing accumulation into those waves.
The difference kernel in (1) leads to the easier implementation of convolution property in the Laplace transform. We consider oscillatory basis functions according to the choice of influencing functions (Proposition 3). Our approach for parameter estimation aligns with near-periodic harmonics. Proposition 6 supports the existence of all oscillatory types for influencing functions though we impose near-periodic patterns for corresponding basis functions. In a theoretical sense, Proposition 4 guarantees a transform between basis functions and influencing functions. The Laplace transform entitles with numerous classes of problems, including differential equations, frequency analysis, and circuit analysis [17]. Our work also shows the importance of its intermediate role in bridging a model solution to harmonic analysis.
Overall, transmission potential NðtÞ assimilates dengue incidence better than its Fourier counterpart, as shown in Table 1. Subject to trialing with Fourier waves for initial guesses, we tend to fine-tune the overall behavior via an exponential effect in each basis function. Different combinations of influences for dengue transmissions can be interpreted via resultant harmonics. In particular, damping oscillations may extract the combined influence of rainfall and awareness on disease control. The amplifying oscillations would show the variations in transmission due to different virus types and adapted mosquitoes. Extraction of responsible influencing functions, along with their contribution to accumulation, leads to more disease-specific interpretations. Cumulative effects are not restricted to one disease or one phenomenon. As White et al. [18] presented, for a general health impact, one must go beyond disease-specific models. Thus, a work   Journal of Applied Mathematics similar to here can be extended to investigate accumulation cum harmonic effects. The model-based harmonic analysis proposed here can be extended for applications outside epidemiology too. Any phenomena having meaningful oscillatory basic waves influenced by accumulation effects are the potential candidates for that.

Data Availability
Data can be retrieved via Epidemiology Unit, Ministry of Health, Sri Lanka, Available at: http://www.epid.gov.lk.

Conflicts of Interest
The authors declare that they have no conflicts of interest.