key: cord-0984897-bqz6seha
authors: Abdul Razzaq, Oyoon; Alam Khan, Najeeb; Faizan, Muhammad; Ara, Asmat; Ullah, Saif
title: Behavioral response of population on transmissibility and saturation incidence of deadly pandemic through fractional order dynamical system
date: 2021-06-12
journal: Results Phys
DOI: 10.1016/j.rinp.2021.104438
sha: 4cc1e99d46637f50a03319fd624c22ec85057806
doc_id: 984897
cord_uid: bqz6seha

The world entered in another wave of the SARS-CoV-2 due to non-compliance of standard operating procedures appropriately, initiated by respective governments. Apparently, measures like using face masks and social distancing were not observed by populace that ultimately worsens the situation. The behavioral response of the population induces a change in the dynamical outcomes of the pandemic, which is documented in this paper for all intents and purposes. The innovative perception is executed through a compartmental model with the incorporation of fractional calculus and saturation incident rate. In the first instance, the epidemiological model is designed with proportional fractional definition considering the compartmental individuals of susceptible, social distancing, exposed, quarantined, infected, isolated and recovered populations. By virtue of proportional fractional derivative, effective dynamical outcomes of equilibrium states and basic reproduction number are successfully elaborated with memory effect. The expansion of this derivative greatly simplifies the model to integer order while remaining in the fractional context. Subsequently, the memory effects on the asymptotic profiles are demonstrated through various graphical plots and tabulated values. In addition, the inclusion of saturation incident rate further explains the transmissibility of infection for different behavior of susceptible individuals. Mathematically, the results are also validated through comparative analysis of values with the solutions attained from fractional fourth order Runge-Kutta method (FRK4).

Healthcare researchers believe on mathematical simulations to make prompt measures and policies, and save the community from deadly viral infections. These models use epidemic systems through the provided data from clinicians and health workers for surveillance of the transmissibility of a disease. Different mathematical functions are being incorporated in epidemic models in order to design a model that considerably articulates the realistic behavior of a disease. Literature depicts many systems of epidemic models, which propose different rates and functions to prevent and control the spread of the infectious disease [1] [2] [3] . An incredible number of models have been formulated, investigated and functional, which improve our understanding and extrapolative skill about a diversity of transmissible disease qualitatively and quantitatively [4] [5] [6] [7] .

Calculus of non-integer orders have an extended and thrilling history beginning with communication by two scientist Leibniz and de L'Hopital. Equations with fractional order derivatives are controlling tool for unfolding processes with non-locality and memory. Some new definitions of fractional derivatives have a conventional of unusual things that contains violations of Leibniz rule for the derivative of product function and chain rule that define the composition of function. Some of these new operators and results are just formal non-integer calculus of the known conventional theories, often without any justification and motivation. There are different types of non-integer derivatives that are suggested by Liouville, Caputo and Reisz, that described the procedures of non-locality and memory [8] . Moreover, some of the definitions provide useful expansion that reduces the computational complexity of fractional order differential equations. For instance, Laplace expansion of Caputo fractional derivative [9] , tempered fractional derivative [10] and proportional fractional derivative [11] etc., have added a great contribution in fractional calculus. The operators of fractional calculus have been broadly functional in many fields such as, engineering and sciences [11] [12] [13] [14] . SARS-CoV-2, familiarized as coronavirus disease (COVID '19) , being a communicable disease is a detriment to human-life in all respect. Carelessness in following the precautionary measures have caused the world to face another wave of this pandemic with more severity. People infected with the COVID'19 infection involve mild to moderate respiratory sickness and recuperate without needful special treatment in some cases, whereas some are at death's door. As mathematical demonstrations play important role in the area of epidemiology, many models have been structured in this regard [15] [16] [17] . Keeping in view the behavioral change and crowding effect of populace that became one of the major reason of another wave of COVID'19, we assimilate the Holling type II function [18] in dynamics of susceptible and infective population. The incorporation of this function in the model is epidemiologically effective to discuss this novel scenario of COVID '19 transmission. It elaborates the overall effects that occur due to the behavioral change of susceptible from the crowding effect of the infective individuals, vice versa. In addition to this property, it prevents the unboundedness of the contact rate by choosing suitable parameters [19] [20] [21] [22] [23] [24] . Therefore, the representation of incidence rate using Holling type II function is more reasonable than the bilinear incidence rate. Moreover, fractional order derivative is also considered to evaluate the historical effects in continuation with the future dynamics, innovatively. Here, we utilized proportional fractional order derivative to greatly simplify the fractional order system without losing the originality of the model [17, 21] . The governing epidemiological model comprises of system of seven nonlinear fractional order differential equations for the dynamics among susceptible (S), social distancing (P), exposed (E), quarantined (Q), infected (I), isolated (M) and recovered (R) populations. The dynamical effect on the system due to the existence of social distancing population is an epidemiological modernity. Growth of susceptible and individuals with social distancing are contemplated with a threshold through the perception of logistic growth function [23] [24] [25] [26] , whereas growth in other compartments is due to the transmission of infection. This function intends to explain the per capita growth of population by limited resources in the environment. The pictorial representation in Fig. 1 defines the transmission of the deadly pandemic around the individuals in the considered compartments. Moreover, the validity of the attained numerical solutions are also further verified through a comparative study. This study is carried out with help of fractional fourth order Runge-Kutta method [27] , which is a well-known numerical technique that can significantly generate numerical solutions with a choice of adequately small step size of iterations in fractional environment. Sequentially, the remaining sections of the paper comprise of; the detail deliberation of modelling in section 2, findings of basic reproduction number and equilibrium states are demonstrated in section 3, and the numerical discussions and conclusion are established in section 4 and 5, accordingly.

The epidemic models have been undergone with different mathematical modifications to design more realistic models. Since these equations deal with the study of the diseases on human population, therefore the basic biological postulations are retained. Here, we adapted the model in which a host population (N) is divided into seven compartments categorized as: susceptible (S), individuals following social distancing (P), individuals exposed to infected (E), exposed individuals quarantined (Q), infected individuals (I), infected individuals isolated (M) and individuals recovered from quarantine and isolation (R), per day. So the total population of the considered region can be expressed as, 

All individuals are equally possible to catch infection except those who are following social distancing even in susceptible. The growth of susceptible is defined with a maximum surviving capability in the available resources. The transmission process of disease is considered with the Holling-type II function, which describes the infection transmission rate together with the inhibitory effect. This term commonly known as saturation incidence function in epidemiology that intends to define the variation in infection force due to the crowding effect of infective. The exposed to infection with mild symptoms might get recovered after completing incubation period in quarantine. Beside, some may get infected that can transmit the disease to the others and either get recovered after an isolation period or die due to the infection. Moreover, the individuals once recovered are considered to gain immunity that they cannot be infected again and also if they join the individuals who are following social distancing measures. Different rate of treatments for exposed, quarantine and infected individuals is availed during the respective degree of infectiousness. Linked with the historical variations in the dynamical changes, the future interpretations is contemplated through the memory effect of proportional fractional order derivative. Based on aforementioned assumptions, the fractional order system of seven equations is thus designed as,

With initial condition,

where, δ i ⩾0 for i = 1,2,...,7. All the variables and parameters presented in system (2) are defined in the Table 1 . Special case of proportional fractional derivative defined in [11] is a lucrative operator that greatly simplifies the fractional order differential equations into integer order. The proportional fractional derivative of any continuous function h(t) expands to,

where, g 0 (α, t) ∕ = 0 for α ∈ (0, 1], with lim α→0 + g 0 (α, t) = 0 and lim α→1 − g 0 (α, t) = 1. Moreover,

Thus, using the expansion (5) in model (2), we get the expression as,

With initial conditions same as defined in Eq. (3). Some essential theorems for the system (6), are discussed as follows:

+ be the set of all possible solutions of the system (6), such that

then it is uniformly bounded subset of R 7 + .

Proof. Since,

Expanding proportional fractional derivative, we get the expression as: 

using system (6) with some manipulation, Eq. (8) becomes,

for 0 < α⩽1, where d N represent the total proportion of natural deaths i. e.,

As, 0 < S(t) kS ⩽1and 0 < P(t) kP ⩽1, the inequality (9) reduces to

After some manipulation, as t→∞ , we attain the boundedness relation as,

Theorem 2.. Let Λ(G(t) ) : R 7 + →R 7 + be the matrix of right hand side of system (6) with Λ(G(t) ) and ∂Λ(G(t) ) ∂G(t) being continuous, such that

Then, for given initial conditions satisfying (2), there exists a unique, bounded and non-negative solution of the system (6) for all t ∈ [0, ∞).

Proof. Proving the uniqueness of the solution, the system (6) can be represented as:

where,

and

which can be further compressed as:

such that

Taking norm on both side of Eq. (16),

Furthermore, using the positivity of initial conditions (2) and the first differential equation of system (6) , to prove the non-negativity of the solutions as:

Hence, the non-negativity of solution of S(t) is proved. Correspondingly, in the similar manner, it can be concluded that the solutions of remaining equations of system (6) may also be non-negative with positivity of initial conditions.

In this section, the equilibria of the system (6) along with the basic reproduction number and its dynamical properties are discussed in particular. Let the constant solutions of system (6) be χ 0 , χ 1 and χ 2 as trivial, disease free and endemic equilibria, respectively. Suppose, T → be the sub-model of system (6), expressed as:

Such that the vector T → includes the E(t), Q(t), I(t) and M(t) equations

, containing all other outputs and inputs of these compartments, respectively. Since, no new infected individual enter the Q(t) and I(t) compartments, rather they transit from the E(t) compartment into Q(t) and then I(t) compartments, therefore, the second and third element of Z

Taking Jacobian matrix of Eq. (18) at disease free equilibrium,

Using Eq. (19) and the eigenvalues with the largest magnitude of the next generation matrix K = ZX − 1 , the required basic reproduction number is attained as,

Furthermore, again constructing the Jacobian matrix J, but for the complete system (6), the corresponding characteristic equation for the eigenvalues λ is then expressed as,

of system (6), is asymptotically unstable, for all and 0 < α⩽1.

Proof. The eigenvalues of J at this equilibrium point are obtained as,

Theorem 4.. Let χ 1 ∈ R 7 + , be the disease free equilibrium solution of system (6) , then it is asymptotically stable if R 0 < 1 and unstable when R 0 > 1.

Solving J for χ 1 ∈ R 7 + , we get the negative real eigenvalues,

with the transcendental equation,

where,

) < 4b 2 k p r S and r S > − 1 − b − d S + α,so it is clear that λ i < 0, for i = 1, 2, 3, 4 and for all 0 < α⩽1. Moreover, from Routh-Hurwitz condition [28, 29] for cubic polynomial, if B 2 > 0,B 1 > 0 and B 0 (1 − R 0 ) > 0, then T(λ) > 0 for all non-negative real eigenvalues. Therefore, it can be clearly seen that B 2 > 0,B 1 > 0 and B 0 > 0, the only thing left to be proved is Proof. Solving J for the equilibrium point χ 2 (S * , P * , E * , Q * , I * , M * , R * ) ∈ R 7 + , we get the negative real eigenvalues,

with the transcendental equation,

where,

and r S > − 1 − b − d S + α, so it is clear that λ i < 0, for i = 1, 2, 3, and for all 0 < α⩽1. From Routh-Hurwitz condition [28, 29] for fourth order polynomial, if C i > 0, for i = 0,1,2,3, then T(λ) > 0 for all non-negative real eigenvalues. Therefore, it can be clearly seen that

which yields Eq. (23) to have a non-negative real part, thus it becomes unstable.

Numerical investigations are carried out to validate the aforementioned analytical discussions of the model (6) . Some fitted values of the parameters are considered for this purpose, which are mentioned in Table 2 as sets 1 and 2 along with the initial conditions,

These computations are conducted on Mathematica 12.0 software to reduce the workforce and produce well-defined graphical representations of the governing model. In addition, a comparative analysis is also added in the deliberation to study the effectiveness of the expansion of proportional fractional derivative.

The solutions attained by the virtue of special case of the expansion of proportional fractional derivative, are justified through a comparative analysis with the generated solutions of FRK4 algorithm. In [27] , FRK4 algorithm is exemplified through programming codes of Mathematica software, which made it an easy user interface. Here, after re-designing the programming code according to the governing model and parameter values, in both sets of Table 2 

One of the most important threshold of epidemiology, R 0 , is studied through the Tables 3 and 4 . These tables contain the computed values of R 0 for the model (6) with the parameters of Table 2 Table 1 , are mentioned in Table 3 . It can be seen clearly, that basic reproduction number remains less than 1 as α approaches 1 for these values of parameters and validates the situation of disease free equilibrium points. So, the dynamical history also outlines the same behavior that the population remains disease free until theR 0 < 1. Unlike this, Table 4 defines the existence of disease in the population for the parameters of 

) set 2 at different values of α. Throughout the memory till α approaches to 1, the basic reproduction number remains greater than 1. Consequently, it verifies the aforementioned analytical result, which states the stability of endemic equilibrium point whenR 0 > 1.

Another effective parameter in model (6) is the fractional index of the proportional fractional derivative. It enables to sketch the historical dynamical behavior of the individuals in each compartment, before the occurrence of a complete change in the system. It can be further illustrated through the attained graphs for model (6) 

t ∈ [0, 10]using the set 1. Evidently, the curves represent the stability of disease free equilibrium points, as it can be easily read that the populations of E(t), Q(t), I(t), M(t) and R(t) become extinct after some time. On the other hand, susceptible and those individuals who are take precautionary measures will survive and thus disease will die out.

α ∈ [0.1, 1] and t ∈ [0, 10]using the set 2. The curves represent the existence of individuals in all compartments that validate the stability of endemic equilibrium points and consequently, it indicate the persistence of diseases in the population. The historical behavior of these discussions are also depicted in figures through grayscale curves. The gradual rise of the curves from grayscale to a dark color explain the memory effect in the asymptotic profiles of S(t), P(t), E(t), Q(t), I(t), M(t) and R(t). Hence, as value of α approaches 1, the level of the curves moves to the higher level, which clearly illustrate the realistic change in the compartments.

Model (6) also comprises the Holling Type II function, which plays an important role as saturation incidence function. It basically measures the dynamics of disease transmission in a crowded population of infectious individuals. The two major parameters that are involved in this function are β and a, which illustrate the infection force and proportion of infectious individual who did not come to contact with susceptible, respectively. Subsequently, the effects of these parameters are portrayed in Figs. 18 and 19 for t ∈ [0, 30]. In these figures, the graphs are plotted forE(t) at α = 0.99 and for different cases of β and a, while other parameters are kept same as defined in Table 2 . Fig. 18 depicts different behavior of exposed individuals for these parameters for the disease free Fig. 3 . Comparative analysis of endemic equilibrium point χ 2 with FRK4 method [27] and PF for set 2 of Table 2 , h = 0.001and α = 0.998 at different values of t. Table 3 Basic reproduction number R 0 and disease free equilibrium point χ 1 for set 1 at different values of α. Table 4 Basic reproduction number R 0 and endemic equilibrium point χ 2 for set 2 at different values of α. α state. It can be clearly observed that with low infection force i.e.,β = 1 population of exposed individuals is lesser and disease dies out earlier when high proportion of infective are not in contact with susceptible i.e., at a = 2. Consequently, it may be the result of the social distancing that infectious are not in contact with population to transmit infection. However, when distancing measures are not followed all infective individuals are in contact with susceptible i.e., a = 0, which increases the population of exposed individuals and disease takes larger time to die out. The same graph also plots the curves of E(t)for the case when infection force is high i.e.,β = 2. Evidently, infection extinct faster for high infection force only if a larger proportion of infective is not in contact of susceptible i.e., at a = 2. As a result, plot of exposed individuals show lower level of population curve on this figure. Moreover, Fig. 19 illustrate different behavior of exposed individuals for these parameters for the endemic state. For the case when infection force is high i.e., β = 2 and only a small proportion of infective is not in contact of susceptible i.e., a = 1, more individuals expose towards the disease and thus asymptotically high profile of E(t)is attained, which remains constant throughout the time interval. Nonetheless, it remains high but less than the previous case if infection force is high and a larger proportion of infective is not in contact of susceptible i.e., at a = 2. Conversely, the helpful effectiveness of these parameters is also found in this figure, where the disease dies out from the population when infection force is low i.e., β = 1 and a larger proportion of infective is not in contact of susceptible i.e., at a = 2. In one-way or other, these parameters describe the effectiveness of social distancing to kill the transmission of the deadly pandemic from the population in a realistic manner.

Consequently, from all analytical discussions and numerical illustrations through graphs and tables we may conclude effective investigations from the whole study as follows.

The governing compartmental model was designed with special simplified case of proportional fractional derivative, where its expansion reduced the fractional order system to an integer order system of differential equations. From the comparative analysis of the obtained solutions with the solutions of FRK4 method, the effectiveness of the expansion of proportional fractional derivative is validated. This expansion enables to involve the fractional index in the findings of each epidemic threshold, therefore each can be studied with the memory effect. Knowing the historical behavior of any dynamical model of any disease is beneficial to control the infection before its outbreak.

The designed model for the susceptible S(t), individuals following social distancing P(t), individuals exposed to infected E(t), exposed individuals quarantined Q(t), infected individuals I(t), infected individuals isolated M(t) and individuals recovered from quarantine and isolation R(t), produced the following constructive outcomes:

• The parameters such as infection force, rate of exposed, rate of infection carrier after quarantine greatly control the basic reproduction number. • Decrease in these parameters makes the basic reproduction number less than one, thus a disease free environment is attained. • Increase in these parameters makes the basic reproduction number greater than one, hence to control the deadly pandemic becomes a challenge. • Saturation incidence rate is effective analysis in examining the inhibition effect of saturated population. • Saturation incidence function enables to produce realistic result of transmissibility process of the disease as compare to other linear function. • The parameters involved in this function deliberate different scenarios of the population at different level of infection force and contact proportion of infective. • This function concludes that if the infection force is high if the susceptible are not exposed to the infection through the social distancing, survival of disease decreases and eventually dies out from the population.

Epidemiological models play a vital role in managing the deadly diseases before its outbreak. Innovatively, designing such models is of great concern for health care researchers. In future, we aim to bring latest assumptions and techniques to make the study of such paradigm of equations effective and beneficial for health care system. 

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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The author declares that there is no conflict of interests regarding the publication of this paper. There are no funders to report for this submission.