FPGA Realizations of Chaotic Epidemic and Disease Models Including Covid-19

The spread of epidemics and diseases is known to exhibit chaotic dynamics; a fact confirmed by many developed mathematical models. However, to the best of our knowledge, no attempt to realize any of these chaotic models in analog or digital electronic form has been reported in the literature. In this work, we report on the efficient FPGA implementations of three different virus spreading models and one disease progress model. In particular, the Ebola, Influenza, and COVID-19 virus spreading models in addition to a Cancer disease progress model are first numerically analyzed for parameter sensitivity via bifurcation diagrams. Subsequently and despite the large number of parameters and large number of multiplication (or division) operations, these models are efficiently implemented on FPGA platforms using fixed-point architectures. Detailed FPGA design process, hardware architecture and timing analysis are provided for three of the studied models (Ebola, Influenza, and Cancer) on an Altera Cyclone IV EP4CE115F29C7 FPGA chip. All models are also implemented on a high performance Xilinx Artix-7 XC7A100TCSG324 FPGA for comparison of the needed hardware resources. Experimental results showing real-time control of the chaotic dynamics are presented.


I. INTRODUCTION
Deterministic chaos is a common behavior in continuoustime dynamical systems of differential equations with nonlinear terms, which exhibit aperiodicity, ergodicity and sensitivity to initial conditions [1]. These properties of chaotic systems are needed in many applications such as modeling of robots [2], motion control [3], Random Number Generation and encryption applications [4]. This demand on chaotic systems in various applications has encouraged the exploration of different methods for their analog and digital hardware realizations [5]- [7]. Meanwhile, mathematical models of biological systems have been associated with chaotic behavior long before the emergence of chaos theory and dates back to the logistic equation model of population The associate editor coordinating the review of this manuscript and approving it for publication was Ludovico Minati . growth [8]. Epidemics and infectious diseases modeling is a relatively difficult problem, since their dynamics vary largely from one outbreak to another. For such emerging and reemerging diseases, the causes and transfer processes are often poorly known and understood. Susceptible, infected, recovered and possibly also exposed classes of a given population need to be considered. Many models have been developed for viral infectious diseases such as Influenza and Ebola [9]- [14] and nearly all of them show chaotic dynamics even after vaccination is administrated [15]. The emergence of chaotic behavior can be attributed mostly to predator-prey and competition dynamics [16]- [18] as well as nonlinear interactions between cell populations such as in cancer models [19]- [22] and Parkinson's disease [23], [24].
Recently, following the COVID-19 epidemic crisis, more attention has been directed towards epidemics and infectious diseases modeling and the study of their state behavior VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ against different factors [25]. These models usually contain many parameters and are sensitive to variations in their values. Therefore the implementation of these model should be attempted with caution. In this regards it is known that fixed-point implementation of chaotic models is generally more reliable and reproducible than floating-point ones [26], [27]. In fixed-precision fixed-point addition, there is no mantissa alignment and hence, no rounding errors in addition operations. A fixed-point representation was used in [27] to implement modified versions of the product integration rules; which are used to solve differential equations. Moreover, fixed-point operations can be performed efficiently in any Hardware Description Language (HDL) and realized on FPGA modules providing the advantages of re-programmability, reduced hardware cost, high speed, noise immunity and reliability. Many chaotic systems have already been digitally realized and tested on FPGA platforms [28]- [30] and also using Field Programmable Analog Arrays (FPAAs) [31]. One major advantage of FPGA realizations is that they facilitate experimenting with various chaos control schemes and real-time chaotic time-series prediction algorithms [32].
In this work we show the feasibility of implementing complex chaotic epidemic models using fixed-point architectures on FPGAs on two different FPGA platforms to explore the utilization of resources. We present the realization process of three virus spreading models and one disease progress model. Sensitivity of the chaotic dynamics to parameter variation is explored through continuous bifurcation diagrams prior to implementation. Difficulty arises from several aspects: (i) all the considered models have many parameters (17 parameters in the case of the Ebola model for example) and (ii) the sensitivity of chaotic behavior with respect to all these parameters needs to be carefully investigated before implementation. (iii) All models require several multiplication operations (7 multiplications and 1 division in the Cancer model for example). While it is possible for chaotic systems implemented on FPGAs for the purposes of encryption applications to be deigned such that they contain very few parameters and a limited number of multiplication/division operations [29], this is clearly not the case in epidemic models. It is worth noting that an analog circuit realization of a tumor growth model was recently presented in [33]. To the best of our knowledge, no FPGA realization of an epidemic or disease model on an FPGA platform has yet been reported in the literature. This article shows the feasibility of implementing these complex models efficiently on FPGAs for real-time simulation, control and prediction purposes.
This article is organized as follows: section II presents the investigated models and explores their chaotic dynamics. Section III presents the first proposed FPGA design procedure based on an Altera Cyclone IV EP4CE115F29C7 FPGA chip. A second implementation based on a Xilinx Artix-7 XC7A100TCSG324 chip is also reported. Section IV shows the FPGA resource utilization facts and experimental results.

II. SELECTED MODELS
In this section, we describe the four models that are considered in this work. The details of these models and their respective derivation assumptions can be found in [11], [13], [25] and [34].

A. EBOLA MODEL
In [13], a 4-D model for the spread of the Ebola virus in West Africa between 2013 and 2016 was proposed. This deterministic model suggests that societal and environmental conditions were conducive to the propagation of Ebola. Once the epidemic had broken out, its propagation was driven by a few predominant processes. The spread model based on recorded time series data of x 1 and its higher order derivatives y 1,2,3 optimally fit the following model where f (·) is a nonlinear function with 14 terms given by This model can be easily discretized using an Euler method with a suitable step size h. Figure 1 . Prior to considering an implementation of this system, the type of response obtained at different values of the parameters and the sensitivity to parameter variations needs to be studied. This was done by considering variations in narrow steps of one parameter at a time while fixing the other parameters using bifurcation diagrams. A bifurcation diagram reveals periodic windows and examines the robustness of the chaotic behavior versus parameter variations. For this system, the bifurcation diagram versus a chosen parameter was generated through plotting the value of the state x 1 , every time it reaches a local maximum after discarding two thirds of the total number of points. Following extensive simulations of the system (2), we found that some parameters correspond to chaotic behavior only in very narrow ranges of their values, such as b 3 . Other parameters can generate chaos for wider ranges such as b 6 . In addition, some parameter variations drift the model into quasi-periodic state (donut-shaped attractor) such as when a 2 > 1.5 and a 3 < 1.

B. INFLUENZA MODEL
The influenza model considered here was proposed in [11] for avian influenza in a seabird colony and is given bẏ This model is obviously a non-autonomous periodically forced model where I and R respectively correspond to the infected and recovered individuals. p may be interpreted as the recruitment rate of infectious individuals, β is the transmission rate constant, α is the recovery rate constant and ω is the natural death rate constant. The driving forcing function p(t) oscillates with a period of 1 year corresponding to the annual breeding season. It is also straight forward to discretize this model using Euler's backward method and the simulation results in this case are shown in Fig. 1(b) for (β, α, ω, p, p 1 ) = (0.1, 100, 0.05, 10 3 , 0.2) and using a step size h = 10 −3 . The effect of the parameter variations on the state variables in this system was also carefully examined using bifurcation diagrams. The best two parameters that result in the widest range of chaotic behavior were found to be β and p 1 and the corresponding bifurcation diagrams are plotted in Fig. 2 versus I max .

C. COVID-19 MODEL
A recent model of the COVID-19 pandemic was proposed in [25] taking into consideration data from China, Japan, South Korea and Italy. This model is given bẏ where C(t) is the daily number of new cases, S(t) is the daily additional severe cases and D(t) is the daily number of new deaths. The model contains three multiplier-type nonlinearities (DS, CD, CS) in addition to two quadratic terms. Simulation results of this model after being discretized with an Euler method are shown in Fig. 3 with (a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 , a 9 , a 10 , a 11 , a 12 Fig. 1(b).
where x 1 is the number of tumor cells, x 2 is the number of healthy host cells, and x 3 is the number of effector immune cells. a 1,..,5 and r 1,2 are constants obtained by rescaling the original system model, which are described in [34], to obtain VOLUME 9, 2021  Simulation results of this model after being discretized with an Euler method are shown in Fig. 4 with  (a 1 , a 2 , a 3 , a 4 , a 5 , r 1 , r 2 ) = (1, 2.5, 1.5, 0.2, 0.5, 0.6, 4.5).
For this system, the study of the effects of the parameter variations on the state variables using bifurcation diagrams revealed interestingly that it exhibits a period doubling route to chaos against r 2 and reverse bifurcation against both a 3 and a 4 , as shown in Fig. 5.

III. FPGA IMPLEMENTATIONS
A. USING ALTERA CYCLONE IV Figure 6 depicts the FPGA design process applied to implement three of the studied models (Ebola, Influenza  and Cancer). This process consists of two parallel phases, namely the Verilog Design and the Design Verification phases. The two phases run in parallel to ensure that the hardware implementation meets all the specifications of the system at different key points during the process. The design steps allow for an iterative mechanism.

1) FIXED-POINT ARCHITECTURE
Floating-point units typically need more than one clock cycle to produce an output and consequently, synchronizing  fixed-point results is much easier and usually produces a simpler and faster hardware design [35]. In fixed-point architectures, memory and bus widths are smaller, contributing significantly to a lower cost and power consumption [36]. That is why a fixed-point architecture is used to implement the hardware components of the three models. The fixed-point signed notation of (1.m.n) is used to represent the number of bits allocated to the integer part m and the fraction part n of the number. The number of bits was determined by taking into account the maximum possible value of all state variables in the chaotic models. Both the Ebola and Cancer models use an internal architecture of 64-bit signed fixed-point whereas, the Influenza model requires more precision in the fractional part and hence a 256-bit signed fixed-point architecture is used. A typical histogram of the distribution of values for the four-state variables in the Ebola model is shown in Fig. 7. The more spread the values are, the less the required precision to discriminate among successive values.

2) HARDWARE
Here the details of implementing the Cancer model (as an example) are discussed. This model is the most complex from an implementation perspective. Figure 8(a) shows the architecture of the top-level module of the Cancer model. One of the main components of this module is the phase-locked loop which is responsible for generating the clock frequency required to run the model blocks. This clock frequency is fed to the control unit which is responsible for controlling the number of iterations through a state machine. The Cancer_top module, shown in Fig. 8(b), is responsible for generating the values of the next state from the existing state values in the model. The selection logic enables the user to route any two-state variables to a digital-to-analog converter for external observation on an oscilloscope. Figure 9 shows the internal architecture of the combinational cancer module particularly the arithmetic and shift operations needed. The arithmetic operations are implemented using IP cores from Intel. To optimize the multiplication operation for example by a constant equal to 2.5, we first shift by 2 bits, which is a multiplication of 4, then we add the result to the original value resulting in a multiplication by 5. Finally, we shift the product 1 bit to the right in order to divide by 2. This enables us to compute a multiplication by 2.5 without the need to use the resource-consuming multiplication operation. The dotted lines in the figure are the state VOLUME 9, 2021 FIGURE 10. General block diagram for a 3D dynamical system, where a three combinational circuits are used to compute x, y and z.  registers used to store the values of the three state variables of the model. These state registers enable synchronize the inputs and outputs with the clock cycles.

3) TIMING ANALYSIS
Timing analysis is performed using the Timing Quest Timing Analyzer tool in Intel Quartus Prime software to find the maximum frequency at which the realized models can operate. The minimum input and output delays are set to 2ns, whereas the maximum input and output delays are set to 3ns. The maximum frequencies at which we can operate our three modules are found to be 17.02MHz, 10.86MHz, and 1.62MHz for the Ebola, Influenza, and Cancer models, respectively. The reduction in operating frequency in the Influenza model is mainly attributed to the necessary higher precision when compared to the Ebola model (256-bits vs. 64-bits). The drastic reduction in frequency in the Cancer model mainly results from the complex division operation needed to createẋ 3 . The PLL in all three designs was used to provide the clocks at maximum frequencies.
B. USING XILINX ARTIX-7 Figure 10 shows the general block diagram used where a 3D chaotic system is considered with outputs x, y and z stored in three registers. Three combinational circuits are used to compute the numerical solution for x, y and z. Taking the Covid 19 chaotic system as an example for this design procedure, Fig. 11 displays the three combinational circuits required to calculate the numerical solution of x, y, z which correspond respectively to C, S, and D. Different arithmetic blocks are used to compute the numerical solution including adders, subtractors, and multipliers. 32-bit fixed points are used for the implementation each with 4-bits for the integer part and 28-bits for the fractional part. For the Cancer model, Fig. 12 presents the three combinational circuits needed to compute the numerical solution of x, y, and z which represent x 1 , x 2 and x 3 for this model respectively. The term 1 x n +1 is computed based on a linear approximation method. The linear binomial coefficients are generated based on MATLAB curve fitting. Since the range of 1 + x is [1:2], the linear approximation is applied for an input interval [1:2]. Four uniform segments  are used for the approximation. Here 64 bits fixed point are used each with 16-bits for the integer part and 48-bits for the fractional part. The outputs are truncated to 12-bits and hence the three registers x, y, and z use 12 bits.

IV. RESULTS
The first implementation described in section III-A was based on the DE2-115 development board equipped with Altera Cyclone IV EP4CE115F29C7 FPGA device. This board includes many input/output (I/O) peripherals including switches and a High-Speed Mezzanine Card (HSMC) connector. A Texas Instruments high-speed Digital to Analog Converter (TI DAC5672) was connected to the HSMC connector on the DE2-115 board. The TI DAC5672 is a 14-bit dual-channel DAC, hence the main outputs of the three models needed to be scaled down to 14 bits in order to be displayed on the oscilloscope (RIGOL DS4012). Figure 13 shows the final experimental setup while the FPGA is running the Ebola model code. Table 1 summarizes the FPGA resource utilization for each of the three implemented models with this FPGA.
For the second implementation described in section III-B the Xilinx XC7A100TCSG324 FPGA was used. Table 2 presents a summary of the resources needed to implement the Covid-19 and Cancer models. The Covid 19 model is the most resource demanding of all models. Figure 14 shows the oscilloscope output from all epidemic models using this FPGA.

V. CONCLUSION
We reported on the feasibility of implementing four complex chaotic models for virus spreading (Ebola, Influenza and Covid 19) and disease progress (Cancer) on two different FPGA platforms using fixed-point architectures. Although these models require a large number of multiplier blocks and have many parameters, their efficient realization on FPGAs is possible. Future work on applying and investigating the robustness of chaos control techniques to these epidemic models in real-time is ongoing.  Dr. Aloul received a number of awards including the Global Engineering Deans Council (GEDC) Airbus Engineering Diversity Award, the Sheikh Khalifa Award for Higher Education, AUS Excellence in Teaching Award, the Abdul Hameed Shoman Award for Young Arab Researchers, and the Sheikh Rashid's Award for Outstanding Scientific Achievement. He is a regular invited speaker and panelist across a number of international conferences related to Cyber Security, Technology, Innovation, and Education. He is a Certified Information Systems Security Professional (CISSP).
A. SAGAHYROON (Senior Member, IEEE) received the B.Sc. degree in electrical engineering from the University of Khartoum, the M.Sc. degree in electrical engineering from Northwestern University, Evanston, IL, USA, and the Ph.D. degree from The University of Arizona, Tucson, AZ, USA. From 1993 to 1999, he has been a member of the Department of Computer Science and Engineering, Northern Arizona University. In 1999, he joined the Department of Math and Computer Science, California State University. In 2003, he joined the Department of Computer Science and Engineering, American University of Sharjah, where he served as the Department Head for seven years. He is currently a Professor of computer science and engineering and the Associate Dean of undergraduate affair with the American University of Sharjah. He has many publications in international conferences and journals. His research interests include innovative applications of emerging technology in the medical field, power consumption of portable electronics, and FPGAs based design. He was an Invited Technical Reviewer for National Science Foundation Programs, and served in technical program committees of many international conferences.