Copyright © M. Khalil et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unre- stricted use, distribution, and reproduction in any medium, provided the original work is properly cited. SPC Journal of Environmental Sciences, 1 (1) (2019) 1-11 SPC Journal of Environmental Sciences Website: www.sciencepubco.com/index.php/JES Research paper Fractional order models of infectious diseases: a review M. Khalil 1 *, M. Said 1 , H. Osman 1 , B. Ahmed 2 , D. Ahmed 3 , N. Younis 4 , B. Maher 3 , M. Osama 3 , M. Ashmawy 3 1 Department of mathematics, Faculty of engineering, Modern sciences and arts University(MSA), Egypt 2 Department of electrical and computer engineering, Faculty of engineering, University of Victoria, Canada 3 Department of electrical systems engineering, Faculty of engineering, Modern sciences and arts University(MSA), Egypt 4 Department of English and American studies, University of Vienna, Austria *Corresponding author E-mail: mkibrahim@msa.eun.eg Abstract The aim of this paper is to present a succinct review on fractional order models of infectious diseases. Fractional order derivative is a potential tool which gives a better understanding of the impact of memory on spread of infectious diseases. This paper reviews different infectious diseases models with constant, variable or complex fractional order. Fractional order models with time delay are presented in this paper as well. We argue that, such models are essential for decision makers in health organizations. Keywords: Constant/Variable Fractional Order Models-Models with Complex Fractional Order -Fractional Order Models with Time Delay-Infectious Diseases Models with Memory. 1. Introduction Infectious diseases and epidemics have become one of the crucial global issues as they cause of death and disability not only in developing countries, but also worldwide. Every year, HIV, TB and malaria cause 10% of all deaths [37], [28]. Over the last few decades, the number of infectious diseases outbreaks has increased dramatically, resulting in economic crises and millions of disability and deaths [48], [61]. For example, West Africa suffered up to $32 billion loss by 2015 and more than 11,000 deaths during Ebola outbreak [60], [66]. Infectious diseases are spreading around the globe faster than ever before, and new diseases are emerging at a high rate [20]. Speed modern transportation has helped spread of some communicable diseases. International travelling and commerce drive the rapid, global distribution of microbial pathogens and the organisms that harbor them [45]. Recently, the Zika virus is now spreading explosively in the world through modern transportations. There is no, no background immunity in the population, or vaccine currently available [70]. Outbreaks provide an opportunity to collect and analyze initial data. These gathered data are essential to predict the behavior of diseases and to adjust control strategies. But sometimes data collection is impossible due pathological limitations [37], [40], [54]. Also, testing spread of infectious diseases in human societies are unethical or needs big budgets [39], [40], [54]. So, mathematical models of infectious diseases should be used to give a better understanding of spread of diseases spread in human communities [36] and to predict crucial data that should be collected. It is worth mentioning that, mathematical modeling of diseases extends to ancient history. One of the earliest mathematical models in epidemiology was presented by Daniel Bernoulli (1700–1782) [50], [22]. He predicted the impact of immunity with smallpox disease that made the idea of eradication feasible [50], [22]. Nowadays, modern mathematical models of infectious diseases play an in- creasingly significant role to evaluate the potential impact of eradication and control programs in reducing morbidity and mortality [23], [25], [62], [46]. Such enormous models are significant to link between clinical data for selected subpopulations and population-level used [30]. Mathematical modelling can achieve a better understanding of the indirect protection provided by immunization [29]. For example, the spread of influenza virus in USA were simulated in 2009 from gathered data from different areas in [29]. Gathered data from the H1N1 epidemic have been used to approve SEIR mathematical model. Also, an estimate for the vaccination coverage needed to block the spread of infectious diseases can be obtained from such models after estimating its parameters from available medical and epidemiological data [29]. Mathematical models enable researchers to understand the development of drug resistance throughout therapy [47]. Although many of these models have been proposed in literature, it has been restricted to integer order models [17]. However, integer order systems do not convey any information about prior states [4-11]. Understanding the concept of memory in biological systems can be very essential to predict the future of infectious diseases outbreaks and to control infectious diseases. Studying immunological memory is es- sential to develop vaccines. Memory and learning process in vector and host are critical in vector-borne disease transmission like Malaria and dengue fever [56]. Learning behavior and memory in vectors like mosquitoes are important in vector borne diseases transmission [56]. Mosquitoes’ experience is considered as the ability for mosquitoes to accurately identify their hosts. Also memory and learning behavior is significant in immunological memory which is defined as the potential of the immune system to respond more effectively to threats that have been encountered previously [38]. Fractional order differential equations can be potential flexible tools for modelling epidemiological and biological systems related with memory. Adding fractional-order parameter enhances the system potential as it adds a new degree of freedom which leads the model to more space [17], [19]. So, the fractional order is supposed to be the memory index [21]. However, if the fractional order value tends to unity, the system