# Predictive Model of Lyme Disease Epidemic Process Using Machine Learning Approach
## 3 Materials and Methods  
### Model of Lyme Disease Epidemic Process  
To develop the neural network model of the Lyme disease epidemic process, a multi-layered neural network was used and the backpropagation algorithm or generalized delta rule was used for its learning. Training consists of adapting the parameters of all layers, so that the difference between the output signal of the network and the external training signal is on average minimal. From this, it follows that the learning algorithm is essentially an extremum search procedure for a specially designed target error function. A neural network is a distributed parallel processor consisting of elementary information processing units that accumulate experimental knowledge and provide it for further processing. The neural network is similar to the brain. Knowledge enters the neural network from the environment and is used in the learning process. Connections between neurons, called synaptic weights, are used to accumulate knowledge. Changing synaptic weights is a traditional method for tuning neural networks. This approach is very close to the theory of linear adaptive filters. However, neural networks can change their topology. A neural network's learning process is considered an adaptation of parameters for solving a given problem by optimizing the accepted quality criterion. The learning process is permanent, and over time the network improves its characteristics, gradually approaching the optimal solution of the task. The type and nature of training are determined by the amount of a priori and current information about the environment and the objective function, which characterizes the degree of compliance of the neural network with the task it is solving. Information about the external environment is given in the form of a training sample of images, and by processing them the network extracts the information necessary to obtain the desired solution. The neural network architecture developed for the simulation is shown in Figure 2.  
Each input shape represents \((n_{0}\times 1)\) vector \(x=(x_{1},\ldots,x_{i},\ldots,x_{n_{0}})^{T}\), output shape \((n_{3}\times 1)\) vector \(y=(y_{1},\ldots,y_{i},\ldots,y_{n_{3}})^{T}\) and learning shape \((n_{3}\times 1)\) vector \(d=(d_{1},\ldots,d_{j},\ldots,d_{n_{3}})^{T}\). In the process of learning, it is necessary to ensure minimal inconsistency between the current output values \(y_{j}(k)\) and the desired signals \(d_{j}(k)\) for all \(j=1,2,\,\ldots,n_{3}\) and \(k\). As an error option, a local quality criterion is used. \[E(k)=\frac{1}{2}\sum_{j=1}^{n_{3}}(d_{j}(k)-y_{j}((k))^{2}=\frac{1}{2}\sum_{j= 1}^{n_{3}}e_{j}^{2}(k)=\sum_{j=1}^{n_{3}}E_{j}(k), \tag{1}\]  
or global target function  
\[E^{k}=\sum_{k}E(k)=\frac{1}{2}\sum_{k}\sum_{j}(d_{j}(k)-y_{j}((k))^{2}=\frac{1 }{2}\sum_{k}\sum_{j}(e_{j}^{2}(k)). \tag{2}\]  
Also, the approach used in the model for finding the minimum of the adopted objective function is mainly associated with the objective function (1). It consists in sequentially adjusting the weights as the input images arrive (often in random order) one after another in real-time. Moreover, for each pair of shapes \(x\), \(d\), the weights \(w_{ji}^{[s]}\) (\(s=1,2,\,3\)) change by an amount \(\Delta w_{ji}^{[s]}\) proportional to the antigradient of the objective function \(E(k)\), i.e.,  
\[w_{ji}^{[s]}(k+1)-w_{ji}^{[s]}(k)=\Delta w_{ji}^{[s]}(k)=-\eta(k)\frac{\partial E (k)}{w_{ji}^{[s]}}. \tag{3}\]  
Figure 2: Neural network architecture for simulation the dynamics of the spread of the incidence of Lyme disease. If the step coefficient \(\eta(k)\) is sufficiently small, then this procedure minimizes the global objective function (2). Note also that the recurrent tuning procedure (3) corresponds in continuous time to a differential equation of the type:  
\[\frac{dw_{ji}^{[s]}}{dt}=-\eta(k)\frac{\partial E(t)}{\partial w_{ji}^{[s]}}, \eta>0. \tag{4}\]  
Let us consider the real-time learning algorithm associated with the minimization of the local function \(E(k)\) at each step.