Introduction: About 50 million people worldwide have epilepsy, a condition that can occur at any age. Some patients with a history of recurrent seizures have significant cognitive and functional impairment, requiring life-long pharmacology treatment or temporal lobectomy for pharmaco-resistant epilepsy. On one extreme, epilepsy can be understood from the electromagnetic interactions around the cellular membrane. Alternatively, seizure activity is similar to pandemic behavior for which mathematical models exist, such as the Kermack-McKendrick equations. This work focuses on the suitability and impact of epilepsy modeled by Kermack-McKendrick equations, as well as the underlying atomic interactions around the cellular membrane in neurons with seizure activity.
Methods: Appodictical mathematical derivations from differential calculus in analogy to pandemic behavior are used to apply the Kermack-McKendrick equations to predict seizure electrical activity in neurons. We modeled the cellular membrane as an infinite uniformly charged plate to operate with integral calculus properties to find the electric field that initiates and perpetuates the seizure activity.
Results: Seizure onsets follow Thom's mathematical theory of catastrophe, while a chaotic Lorentzian conversion to a non-Markovian process state represents seizure propagation.
The likelihood of neurons becoming part, enhancing the reverberant circuit that occurs during a seizure, could be modeled by Kermack-McKendrick equations.
Conclusion : Mathematical models play a crucial role in our understanding of disease behavior. The Kermack-McKendrick equations, when used to model seizure activity in patients with epilepsy, have the potential to enlighten us about the effectiveness of current epilepsy drugs and guide us in developing better anti-epileptic medications.
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