Hopf and backward bifurcations induced by immune effectors in a cancer oncolytic virotherapy dynamics
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Hopf and backward bifurcations induced by immune effectors in a cancer oncolytic virotherapy dynamics Martial Kabong Nono1 · Elie Bertrand Megam Ngouonkadi1,2 · Samuel Bowong3 · Hilaire Bertrand Fotsin1 Received: 7 May 2020 / Revised: 24 August 2020 / Accepted: 27 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Oncolytic virotherapy is emerging as a promising new method for cancer treatment. In this paper, we consider a mathematical model for treatment of cancer by using oncolytic virotherapy in the presence of immune effectors. We provide a theoretical analysis of the model and derive the basic reproduction number R0 which determines the extinction and the presence of oncolytic viruses during therapy. The existence of equilibria and their stability are investigated. More precisely, we show that, depending of the values of the parameters, there exits a quantity ε so that, when ε < 1, the tumor can be eliminated in the body. However, we also show that, if R0 < 1 and ε > 1, the infection-free equilibrium is stable and the model is shown to exhibit the phenomenon of backward bifurcation (where a stable infection-free equilibrium coexist with one or more stable endemic equilibria when the associated basic reproduction ratio is less than unity). Furthermore the model presents a Hopf bifurcation which is supercritical, from which birth of oscillation occurs. Numerical simulations support our theoretical results. Keywords Cancer models · Oncolytic virotherapy · Immune effectors · Backward bifurcation · Hopf bifurcation
1 Introduction Cancer remains one of the most dangerous illnesses which causes many deaths worldwide. It is predicted to be the increasingly major cause of mortality in coming years. According to the World Cancer Report 2018, compiled by the International Agency for Research on Cancer (IARC), there were about 18.1 million new cancer cases and 9.6 million cancer deaths [1]. One in 5 men and one in 6 women worldwide develop cancer during their lifetime, while one in 8 men and one in 11 women die from the disease. Worldwide, the total number of people who are alive within 5 years of a cancer diagnosis, called the 5-year prevalence, is estimated
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Martial Kabong Nono [email protected]
1
Unité de Recherche de Matiére Condensée, d’Electronique et de Traitement du Signal (URMACETS), Department of Physics, Faculty of Sciences, University of Dschang, P.O. Box 067, Dschang, Cameroon
2
Department of Electrical and Electronics Engineering, College of Technology, University of Buea, P.O. Box 63, Buea, Cameroon
3
Department of Mathematics and Computer Science, Faculty of Sciences, University of Douala, P.O. Box 24157, Douala, Cameroon
to be 43.8 million. If these trends are seen globally in the future, the burden of cancer will increase to 27.5 million new cases each year by 2040 [2]. Cancer is a group of diseases resulting from the uncontrolled growth and spread of abnormal cells in the body. If the spread is not controlled, it can results in death. Cancer occurs when
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