Contents lists available at ScienceDirect Medical Hypotheses journal homepage: www.elsevier.com/locate/mehy A simple mathematical model for relapsing-remitting multiple sclerosis (RRMS) M.F. Elettreby a,b, ⁎ , E. Ahmed b a Mathematics Department, Faculty of Science, King Khalid University, Abha 9004, Saudi Arabia b Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt ARTICLE INFO Keywords: Multiple sclerosis Relapse Recurrence Autoimmune diseases Complex system ABSTRACT Multiple sclerosis is one of the most common autoimmune-related neurological disorders affecting the central nervous system. One of its main properties is dissemination of lesions in time and space. Here, we propose a simple mathematical model to describe the stages of disease progression with recurrence. We tried to find the steady states and evaluated their stability to identify the necessary conditions to produce the recurrence. Also, we described the analytical results with some numerical simulations. Introduction Multiple sclerosis (MS) is an autoimmune-related neurological dis- ease involving the central nervous system (CNS), primarily damaging oligodendrocytes and consequently disturbing the neuronal function in the brain and spine. MS is known to be an unpredictable disease. It is a long-lasting disease that affects the brain, optic nerves, and spinal cord; symptoms may cause gait disturbance, impaired vision, and more mild other neurological symptoms. The type and severity of symptoms differ from one patient to another, from a mild case that needs no treatment to a severe case with disturbed daily activities who requires relapse- preventive therapies [1–4]. The cause of MS is still inconclusive. Several factors have been proposed to explain the mechanism of the disease, such as genetic, epigenetic, immune, viral or environmental. Most of the patients have their clinical onset between the ages of 20–50 years, and the disease is about three times more likely to be seen in females than in males. At present, we cannot predict the clinical type of relapses, the severity of neurological disability, and the relapse frequency in each MS patient based on the patient’s background, laboratory data, or past clinical history [5–8]. The accumulation of neurological impairment in MS is mainly comprised of the following two components: repeated attacks (relapses) and indolent progressive brain atrophy (progressive forms). Mostly based on the combinations of these two components, the clinical course in MS patients are usually divided into the following three subtypes: primary progressive multiple sclerosis (PPMS), relapsing-remitting multiple sclerosis (RRMS), and secondary progressive multiple sclerosis (SPMS) [9]. Difference between RRMS and PPMS is that RRMS shows clinical relapses and cerebral atrophy occurring randomly [10,11], whereas PPMS predominantly shows a progressive cerebral atrophy from the early stage of the disease without apparent clinical relapses [12–14]. There are several disease-modifying drugs (DMDs) that are effec- tively suppress the relapse frequencies in MS patients, such as inter- feron, fingolimod, and natalizumab. However, these DMDs are not ef- fective for all MS patients. Besides, although DMDs are effective for preventing relapses, most of them have not been shown to successfully suppress the progressive brain atrophy process irrelevant of relapses. Further researches to elucidate the pathophysiological mechanism of MS are waited to invent further therapeutic approaches. Building a reliable mathematical model to reproduce the clinical manifestation in MS is one of the possible approaches. In this report, we present a new mathematical model that can suc- cessfully reproduce the clinical manifestation of MS only with three simple simultaneous differential equations. In the following Section “The model of multiple sclerosis”, we propose the simple mathematical model with three ordinary differential equations. In the proposed model, we modify the theories presented in the previous studies [18,19]. In Section “The stability analysis of the MS model”, we drive the steady states and their stability conditions. In Section “Numerical simulations”, we ensure our theoretical results by some numerical si- mulations. In the last Section “Summary and conclusions”, we sum- marize and conclude our results. https://doi.org/10.1016/j.mehy.2019.109478 Received 21 October 2019; Received in revised form 5 November 2019; Accepted 8 November 2019 ⁎ Corresponding author. E-mail address: mohfathy@mans.edu.eg (M.F. Elettreby). Medical Hypotheses 135 (2020) 109478 0306-9877/ © 2019 Elsevier Ltd. All rights reserved. T