# Improving Healthcare Facilities in Remote Areas Using Cutting-Edge Technologies
## 4 Mathematical Modeling
It is only possible when IoT sensors are working correctly so that patients' vitals can be collected in a timely manner, IoT data are transferred to the cloud using 5G, and previous status and vitals are available. \[\widetilde{\vartheta}=max_{\vartheta}\rho\Big{(}I\Big{|}\vartheta,\vartheta^{ \Downarrow}\Big{)} \tag{10}\]  
\[=max_{\vartheta}\rho(g|I)max_{\vartheta}\rho\Big{(}I\Big{|}\vartheta^{ \Downarrow}\Big{)} \tag{11}\]  
According to Equations (10) and (11), vital signs must be correct, which is only possible if the remote patient follows the clinical guidelines and the IoT sensors are working correctly to capture vitals. Furthermore, the previous vitals collected through IoT sensors are also available and correct. Based on Equations (4)-(11), our objective function can be modeled as  
\[\Theta\mathbb{F}=\max f(\mathcal{T}|[d_{i}])\rhd\max\left(\widetilde{\mathrm{S }},\widetilde{\vartheta},\widetilde{\Gamma},\widetilde{\vartheta}\right) \Longleftrightarrow\Omega(\mathrm{I},\mathbb{C},\mathbb{F},\mathrm{A}) \tag{12}\]  
\[\begin{array}{c}max_{\vartheta}\rho\Big{(}S\Big{|}\vartheta,S^{\Downarrow} \Big{)}\vspace{-0.1 cm}\boxplus max_{\vartheta}\rho\Big{(}\vartheta,S, \vartheta^{\Downarrow},\Gamma\Big{)}\vspace{-0.1 cm}\boxplus max_{\Gamma}\rho \Big{(}\mathrm{I}\Big{|}\vartheta,\mathbb{F},\mathbb{C},S,\vartheta^{ \Downarrow}\Big{)}\\ \boxplus max_{\vartheta}\rho\Big{(}I\Big{|}\vartheta,\vartheta^{\Downarrow} \Big{)}\Omega(\mathrm{I},\  
\(\widetilde{\mathsf{S}},\widetilde{\boldsymbol{\vartheta}},\widetilde{\Gamma}\), and \(\widetilde{\boldsymbol{\vartheta}}\) respectively. Then, the comfort index \(C_{i}\) will be calculated using the formula mentioned in Equation (19):  
\[C_{i}=\tau_{1}-\left(\frac{e_{1}}{S}\right)^{2}+\tau_{2}-\left(\frac{e_{2}}{ \widetilde{\boldsymbol{\vartheta}}}\right)^{2}+\tau_{3}-\left(\frac{e_{3}}{ \widetilde{\Gamma}}\right)^{2}+\tau_{4}-\left(\frac{e_{4}}{\widetilde{ \boldsymbol{\vartheta}}}\right)^{2} \tag{19}\]  
where \(\tau_{1}+\tau_{2}+\tau_{3}+\tau_{4}=1\) and \(e_{1},e_{2},e_{3}\), and \(e_{4}\) represent differences between actual values and estimated values. Let \(T(n_{1}),T(n_{2}),T(n_{3}),\ldots.\) be the execution time taken to transfer vitals between various components of the proposed methodology and \(P(n_{1}),P(n_{2}),P(n_{3}),\ldots.\) be the probability that a particular transfer will be done successfully. Then, the average case time complexity of the proposed model will be calculated using the formula given in Equation (20):  
\[T(n_{1})P(n_{1})+T(n_{2})P(n_{2})+T(n_{3})P(n_{3}),\ldots.T(n_{n})P(n_{n}) \tag{20}\]  
The suggested framework's throughput varies in various places. The quantity of data transmitted successfully from one location to another in a certain time period is measured in bits per second (bps). When data from various sources are stored in the cloud, the storage throughput is estimated based on the quantity of data that can be received and written to the storage medium. Thus, the throughput of the proposed model will be calculated using the formula mentioned below:  
\[d_{i}=\left\{\vartheta,\Gamma,S,\vartheta\right\} \tag{2}\]0  
where N refers to the number of requests and T refers to the time in which certain transmission or storage requests are received, while S refers to the number of storage requests in a particular amount of time. In the proposed system, quick information transmission is required since healthcare data are particularly sensitive and require fast delivery and reaction. As a result, the delay should be as short as possible. The proposed framework's delay will be determined using the methodology shown below. \[d_{i}=\left\{\vartheta,\Gamma,S,\vartheta\right\} \tag{2}\]1  
where P is the packet size, T is the time taken to transmit a packet, D is the distance between the sender and the receiver, and S is the speed of the link. Now, we evaluate the proposed model using a case study. The description of the case study is as follows: