key: cord-0734085-znis54sg authors: Din, Rahim ud; Algehyne, Ebrahem A. title: Mathematical Analysis of COVID-19 by Using SIR Model with Convex Incidence Rate date: 2021-02-19 journal: Results Phys DOI: 10.1016/j.rinp.2021.103970 sha: 673091c7698ae3a1492f69ce48d611a69798a84b doc_id: 734085 cord_uid: znis54sg Our this manuscript is about a new COVID-19 SIR model, which contain three classes; Susceptible S(t), Infected I(t) and Recovered R(t) with Convex incidence rate. Firstly, we present the consider model in differential equations form. Secondly, “ the disease-free and endemic equilibrium” is calculated for the model. Also the basic reproduction number [Formula: see text] is derived for the model. Furthermore, Global stability is calculated through constructing Lyapunov Function and Local Stability is found through Jacobian matrix. Numerical simulation are calculated through (NFDS) Nonstandard Finite Difference scheme. In numerical simulation, we testify our model using data from Pakistan. Simulation mean with change of time how S(t), I(t) and R(t), protection, exposure and death rates affect the people. It was reported in December 2019, that a new kind of virus named Corona had affected badly the Wuhan, a city in China. Firstly, the said virus and it's resultant outbreak hit the Wuhan city and later it affects almost the whole world. It took the lives of hundreds of thousands lives throughout the world. It is hard to find a single view about the origin of the origin of the said virus, for example , it is due to seafood market; migration of people from people one place to other places by transmission from animals to human or may be it is due to human to human interaction. Currently it has almost devastated everything around the world. Social life, health, economy, educational almost every field of the life have been affected badly. Researchers of the field of health, policy makers of the countries and health field are puzzled how to tackle with this deadly outbreak. They all have their own point of views observing the situations. They are trying hard to at least minimize the number of deaths due to this outbreak. People infected with this pandemic experiences a mild respiratory problems. Fever, dry cough, throat infection, and tiredness are the symptoms of this disease. People may have these symptoms; nasal infection, aches, and sore throat. Mathematical modeling is playing an important role in describing the epidemic of infectious diseases. The purpose of mathematical modeling is to represent different types of real world situation in the language of mathematics. A number of mathematical models are studies in the pervious literature [2, 8, 10] . Also SARS-CoV-2 is study by many researchers in current research literature [9, 11, 12, 14] . We will study SARS-CoV-2 by developing SEIR model later on in this work. Recently many authors have established numerious models for COVID-19 under different concept of fractional calculus. In this regards very useful models have been established, we refer some as [20] [21] [22] [23] [24] [25] . To find out the different dynamics of a disease and therefore to overcome it at an early stage, mathematical modeling plays an important role there. The area dedicated to the investigation of biological pandemic and also epidemic models for recent diseases SARS-CoV-2 of research. Numerous examples of mathematical models for this pandemic are found in the current study [16, 17] . To understand the stability theory, existence theory, and theory of reform SARS-CoV-2 [6, 13, 19] , can be model, and its outcome can be predicted. Plan of prevention is also possible. In addition, one can find a possible lockdown strategy. Especially impressed with the excellent features of the SEIR model using non-linear saturated incidence rates [6, 8, 15 ]. In this section of manuscript, we formulate our new model for N COV ID − 19 in the form of following system (1). We take whole population N(t) into three classes S(t), I(t) and R(t), which represent Susceptible, Infected and Recovered compartment in the form of differential equations given below (1), (1) For above system (1) Table 1 , we describe parameters used in system (1). In system (1), add all equations, implies Here N (t) represent whole population as N (t) = S(t) + I(t) + R(t). We get For the system (1), we suppose the existence of equilibrium. Disease free equilibrium is exist for some values of the variables used in (1), which is denoted by E 0 = (S 0 , 0, 0). In epidemiology there R 0 is most important parameter, which give us idea about how the disease is flow in the whole population. From R 0 , we look how the disease id spread in population and we can control it from this. The method of finding R 0 is below let X = (S(t), I(t)), then from system Hence We have From this, we get R 0 is To computes the basic reproduction number we obtained R 0 = 0.7831 from the parameters used in Table 2 [18], which show that the COVID-19 that occurred in Pakistan is well controlled by Pakistan government. We have the following theorem on the basis of (3). (1), called the endemic equilibrium." We reduced our model (1) for local stability. Furthermore, to obtained the result which show "disease free and endemic equilibrium". For system (1) . We reduced and get Subject to initial condition For local stability, we have the following theorem. Theorem 5.1. "If R 0 < 1, then the system (4) is locally asymptotically stable at the disease free Proof. At E 0 the jacobian matrix is given by The auxiliary equation of J 0 is given by We have The Routh-Hurtwriz criteria is satisfied as a 1 > 0, a 2 > 0, a 3 > 0 and a 1 r 2 − r 3 > 0 if R 0 < 1. which show the system (1) is locally asymptotically stable at E 0 . Furthermore, at E * the system (4) is locally asymptotically stability analogous to R 0 > 1. We are going to prove it in the next theorem. (4) is locally asymptotically stable." Proof. For system (4) jacobian matrix is After some operations on matrix J 1 , we get We calculate trace and determinant of M 1 and The determinant of J 1 > 0. The real part at E * (t) "endemic equilibrium" of model (4) has negative. Thus, with condition R 0 > 1, we have that the endemic equilibrium E * of system (4) is locally asymptotically stable. Here, we present Global stability for the system (1). For "global stability of disease-free and endemic equilibrium", we constructed a function known as Lyapunov function in the following theorem. Theorem 6.1. "If R 0 < 1 then disease free equilibrium of the system (4) is globally asymptotically stable. Otherwise unstable." Proof. To prove this, we construct a Lyapunov function as following such that c 1 , c 2 , c 3 > 0 are constants. With respect to time t taking derivative of (8) with, we have We get Let assume c 1 = c 2 = c 3 = 1, we get finally Hence "globally asymptotically stable" for system (1) with R 0 < 1 has reached. Further, We are going to prove a theorem for "global stability of the endemic equilibrium" of model (1). Theorem 6.2. "The endemic equilibrium E * of model (1) is stable globally asymptotically if R 0 > Proof. By constructing Lyapunov function, we prove the above result ω = (µ + β)(S(t) − S * (t)) + (µ + β)I(t). Taking derivative with respect to time (9), we get Putting the values from (1) dω dt = (µ+β)(b−k(1−αS(t)I(t))−αkβS(t)I(t)−µS(t))+(µ+β)(k(1−αS(t)I(t))+αkβS(t)I(t)−(µ+d 0 +γ)I * (t)). After some arrangement we get dω dt = −(µ + β)(µS(t) + (µ + d 0 + γ)I * (t)) < 0. Thus dω dt < 0, the "endemic equilibrium" E * of the model (1) is " globally asymptotically stable", show that R 0 > 1. Numerical value Susceptible compartment 220 in millions In this part of our manuscript, we calculated numerical simulation for model (1) with values used on table 2. We take data from 1 February 2020 to 20th September corresponding to different compartments involve in the system (1) from Pakistan. Here, we use (NSFD) Non-standard Finite Difference scheme [13, 15, 19 ] to rewrite the system is Which is decomposed in Nonstandard Finite Difference scheme as Just like above equation (11), we can write the system (1) in Non-Standard Finite Difference Scheme as concerned simulation was performed for taking the protection parameter α = β = 0.009. Now by deceasing the protection and isolation rate further up to α = 0.0009, β = 0.0009. We plot the results in the given figures 4-6. We see that the infection rate became slow on reducing the protection and isolation rate. Therefore the recovery is also become slow. From these simulation we observed that protection and isolation rate play significant roles in controlling the infection from further spreading in the community. Conclusion of the numerical results shows the projection of model (1) . The output derived from the NCOVID-19 display convex incidence rate. The current manuscript declared the high contiguous rate from infected population to susceptible population. To overcome the pandemic the migration should be strictly prohibited for the sake of saving humanity. Also the immigration of exposed population to infected community increased the infection. Isolation of infected one is the best option to secured the healthy community. It is necessary to judge the spread and model with various parameters for proper supervision. The proper treatment of this pandemic is to keep infected away from healthy people. High internal defense system aids to get healthy soon while the low internal defense system need more attention. This is the only solution to overcome recent outbreak within a short period. The current discussion demonstrate the quick transfer of NCOVID-19. The COVID-19 shared the same properties like SARS having mortality rate of 2 percent. There is no vaccine available in the current time but to isolate was the best option. Also social distancing is the best way to control this deadly various. Competing interest There exist no conflict of interest regarding this work. Funding There is no source to support this article financially. Availability of Data This is not applicable in this research work. Authors Contributions Both authors played their role as: • Conceptualization: Both authors have developed the concept of the topic. • Data curation The first author performed this process and the second author shaped it. • Formal analysis The first authors wrote it while the second author review it. • Funding acquisition Currently it is not available. • Investigation The first author investigated the model while the second draft it. • Methodology The methodology was point out by the second while first authors work on it. • Project administration This project is administrating by the second author. • Resources Not required. • Software Matlab 2015 software was used. • Supervision Both authors are supervising this project. • Validation This is the duty of the both authors. • Visualization The data visualization has performed by the first author. • Writing -original draftThe first author has written the draft. • Writing -review editing Both authors have reviewed the last version. 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