The Corona (COVID-19) epidemic has triggered interest in many fields of technology, medicine, science, and politics. Most of the mathematical research in this area focused on analyzing the dynamics of the spread of the virus. In this article, after a review of some current methodologies, a non-linear system of differential equations is developed to model the spread of COVID-19. In order to consider a wide spectrum of scenarios, we propose a susceptible-exposed-infected-quarantined-recovered (SEIQRS)-model which was analyzed to determine threshold conditions for its stability, and the number of infected cases that is an infected person will transmit on a virus to, reproduction number R0 is calculated. It is established that the disease-free state is globally asymptotically stable when the reproduction number is less than unity and unstable if its value is more than one. The model is tested against real data taken from the Ministry of Health in Jordan covering three time periods between March and September 2020 wherein two infection peaks occurred in the country. Simulations show consistency and accurate spread predictions within the optimistic range and the proposed model is distinguished by its applicability to aspects including recurrent infections, asymptomatic carriers over several timespans as well as the aforementioned waves of infection.

The spread of COVID-19 zoonotic corona virus from December 2019, which occurred by crossing species from animals to humans similar to another two previous outbreaks, namely:

The scientific research community has also participated strongly in the search for a cure/vaccine by medical and microbiology communities meanwhile statisticians, mathematicians and data scientists have embarked on analyzing the spread of the pandemic and produced much work and analytics in this area. Human behavior specialists and data scientists developed models, descriptions and analyses to enable national and international organizations to monitor, understand and track the patterns of spread of the virus to minimize impact. Nonetheless, as late as August 2020, there remains much to be understood about human-to-human transmission patterns as well as spectrum of infection of this virus [

The Center for Disease Control [

The endemic dynamics of disease outbreaks in large cities as well as persistent dynamics thereof in smaller communities necessitated the development of mathematical models to represent and capture the outbreaks of diseases such as measles. Mathematical models allow us to utilize mathematical notation to represent the behavior of a system in a precise way such that we can study and analyze the epidemic dynamics of diseases as well as the likely effects of preventative measures and vaccinations. Model accuracy supports our ability to predict pathogen future trends whereas model transparency provides better understanding of the disease being studied/analyzed and quite often, the two parameters work in contrast to each other. Model weaknesses stem from: i) unknown disease attributes, ii) unknowable disease attributes or, iii) missed behavior/information that was inadvertently dropped from the model design, e.g., not picked upon early enough or was not reported.

The classification of infectious diseases and the types of mathematical models that have aimed to represent or analyze infection dynamics are explained well by [

In [

The mathematical modeling and simulation proposed in this article highlights the significance of this Quarantine class. Another important distinction of the proposed model is the supposition that if the entire population is inherently susceptible and converting into the Recovered category it is still no guarantee of immunity. The basis on which a Quarantine class was added is imputed to the nature of the data and this class's importance in elucidating the actual precautions being taken in Jordan; adding the Quarantine category is rather common when modeling infectious disease as in [

The proposed, SEIQRS model encompasses several other parameters including the incubation period which is reflected in the constant rates of the model such as the immunity rate of the disease. Constant rates have a strong impact on the credibility of the model in terms of outbreak value and time, all constant rates have been chosen to best emulate the actual conditions in Jordan; it is common that several constant rate values fit the model felicitously however give inconsistent predictions. Optimization of the constant rate values is applied using mathematical algorithms that ensure convergence of the global minimal at effective speed. This simulator will predict the trends of the rates of Susceptible (S), Latent (E), Infectious (I), Quarantined (Q) and Recovered (R) groups. Overtime prevailing the peak value of the infectious population, the number of days for the peak of the infectious population and the number of days where the entire susceptible class population shifts into the recovered class simultaneously (this value highlights the mortality rate). In the proposed model, immunity is not conferred; the derivation and structure of the proposed model are described in the next section. In Section 3 we analyze the model, and in Section 4 we introduce numerical methods and experiments conducted using this model. Finally, in Section 5 we conclude the proposed work.

The main parameters of the proposed model represent Susceptible (S), Latent (E), Infectious (I), Quarantined (Q) and Recovered (R) class of cases. Numerical methods are applied in order to solve the resulting set of ordinary differential equations (ODEs). The dynamic behavior is as follows: Susceptible nodes first go through a latent period (and are said to become exposed). Next, Infected persons may be asymptomatic and recover without being noticed; this is represented by moving from class I to class R. The remainder of the infected persons who show symptoms will go to Quarantine class. Since the acquired immunity is not permanent, the recovered persons may still return to the susceptible class.

Model Parameters Description: The total population size N(t) = S(t) + E(t)+ I(t)+ Q(t)+ R(t) may depend on the time variable. Here S(t), E(t), I(t), Q(t), R(t) denote the cardinality of S, E, I, Q, R compartments at time t, respectively. The per capita contact rate β, is the average number of effective contacts with other nodes per time unit. The number of new infections is denoted by βSI. b is the recruitment rate of susceptible nodes to the population. d is the per capita natural mortality rate, γ is the rate for nodes leaving the exposed compartment E for infective compartment I, δ is the rate of leaving the infective compartment I for quarantine compartment. μ is the disease related death rate in the infected compartments

A population of nodes of size N(t) is partitioned sub-classes: Susceptible, Exposed (Infected but not yet Infectious), Infectious, Quarantined, and Recovered, with sizes denoted by S(t), E(t), I(t), Q(t), R(t) respectively. The SEIQRS model is:

on the closed, positive invariant set

Let

Using the main model

hence, a standard theorem [

Reproduction number and equilibrium points: The proposed SEIQRS is virus-free at _{0} can be established using the next generation operator method on system

It follows then that the basic reproduction number, denoted by _{0}, is given by^{−1}) is the spectral radius of ^{−1}.

Suppose

_{0} ≤ 1, the unique virus-free equilibrium _{0} is locally asymptotically stable in the model

The Jacobian matrix at free equilibrium is

without loss of generality, we suppose

Clearly _{1}, _{2}, _{3} < 0 (negative), also _{4} < 0 when _{0} ≤ 1, we have

therefore,

According to the stability theory in [

_{0} < 1, and is unstable if _{0} > 1.

^{′} = ^{′} = 0 if and only if

Because _{0} ≤ 1, it can be shows that

Also,

Moreover, _{0} is globally asymptotically stable.

The second equilibrium point of the steady state of system _{0} > 1, as shown in

Hence, the unique epidemic equilibrium

with

and

In this section we applied the analysis using Matlab [

From

_{cost}(_{cost}(

In this article, we proposed the SEIQRS model to describe the dynamic behavior of COVID-19 infection rates in humans. The model analysis was described and the local and global stability of this model has been proven. Despite the simplicity of the model, it considers the most likely scenarios for the spread of COVID-19. The model is broad in its scope and caters for several parameters including varying timespans and infection waves; it represents a powerful instrument that can provide strategic support for decision makers in healthcare and related areas of policy and administration.