








key: cord-299312-asc120pn
authors: Khoshnaw, Sarbaz H.A.; Shahzad, Muhammad; Ali, Mehboob; Sultan, Faisal
title: A Quantitative and Qualitative Analysis of the COVID–19 Pandemic Model
date: 2020-05-25
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2020.109932
sha: 
doc_id: 299312
cord_uid: asc120pn

Global efforts around the world are focused on to discuss several health care strategies for minimizing the impact of the new coronavirus (COVID-19) on the community. As it is clear that this virus becomes a public health threat and spreading easily among individuals. Mathematical models with computational simulations are effective tools that help global efforts to estimate key transmission parameters and further improvements for controlling this disease. This is an infectious disease and can be modeled as a system of non-linear differential equations with reaction rates. This work reviews and develops some suggested models for the COVID-19 that can address important questions about global health care and suggest important notes. Then, we suggest an updated model that includes a system of differential equations with transmission parameters. Some key computational simulations and sensitivity analysis are investigated. Also, the local sensitivities for each model state concerning the model parameters are computed using three different techniques: non-normalizations, half normalizations, and full normalizations. Results based on the computational simulations show that the model dynamics are significantly changed for different key model parameters. Interestingly, we identify that transition rates between asymptomatic infected with both reported and unreported symptomatic infected individuals are very sensitive parameters concerning model variables in spreading this disease. This helps international efforts to reduce the number of infected individuals from the disease and to prevent the propagation of new coronavirus more widely on the community. Another novelty of this paper is the identification of the critical model parameters, which makes it easy to be used by biologists with less knowledge of mathematical modeling and also facilitates the improvement of the model for future development theoretically and practically.

This work reviews and develops some suggested models for the COVID-19 that can 19 address important questions about global health care and suggest important notes. Then, we 20 suggest an updated model that includes a system of differential equations with transmission 21 parameters. Some key computational simulations and sensitivity analysis are investigated. 22

Also, the local sensitivities for each model state concerning the model parameters are 23 computed using three different techniques: non-normalizations, half normalizations, and 24 full normalizations. 25

Results based on the computational simulations show that the model dynamics are 26 significantly changed for different key model parameters. Interestingly, we identify that 27 transition rates between asymptomatic infected with both reported and unreported 28 symptomatic infected individuals are very sensitive parameters concerning model variables 29

in spreading this disease. This helps international efforts to reduce the number of infected 30 individuals from the disease and to prevent the propagation of new coronavirus more 31 widely on the community. Another novelty of this paper is the identification of the critical 32 model parameters, which makes it easy to be used by biologists with less knowledge of 33 mathematical modeling and also facilitates the improvement of the model for future 34 development theoretically and practically. 

The idea of chemical kinetic theory is an important approach for understanding and 126

representing the biological process in terms of model equations. The important assumptions 127 to build such models are model states, parameters, and equations. This is because it helps 128 the investigation of mathematical modeling effectively and easily [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] infected, shows symptoms of COVID-19, and detected by the government, either 185 from a rapid test or from voluntary action to report to the hospital. We assume that 186 all individuals in this class will get a specific treatment and supervision, whether it's 187 through monitored isolation or treatment in the hospital. 188 5. Recovered group (R). This group present individual who get recovered from 189 COVID-19, and had a temporal immunity. 190

The transmission diagram which illustrates the interaction between each group described in 191 Figure 1 . 

where 208 209 Furthermore, the model's constant parameters and initial states with their definitions are 210 described in Table 1 . Using equations (2)-(4), the model dynamics are described by the 211

following system of non-linear ordinary differential equations 212

The model initial populations are expressed in the following equation 213 Co  263  mp  264  utat  265  ion  266  al  267  sim  268  ulat  269  ions  270  for  271  the  272  mo  273  del  274  stat  275  es  276 giv 277 en 278 in system (9) parameter has an effective role in the dynamics of . 419

3. The transition rate has also affected asymptomatic infected people, reported 420 symptomatic infected people, and unreported symptomatic infected people. It 421

can be seen that the model dynamics for such states become more flat when the 422 value of is increased. This is an important key element for controlling this 423

disease. practically discussed and investigated in this area. have improvements in interventions and healthcare programs. 510

The used model in this paper has further improved based on the computational results using 511 MATLAB for different initial populations and parameters. Some main results can help in 512 understanding the suggested model more widely and effectively. By using computational 513 simulation, we identify some key critical parameters that have a great role in spreading this 514 virus among the model classes. One of the identified key parameters is the transmission rate 515 between asymptomatic infected and reported symptomatic individuals. This is an important 516 finding in the understanding of the COVID-19 and how this virus spreads more quickly. 517 Some other critical model parameters have investigated in this paper. For example, the 518 transmission parameter between asymptomatic infected and unreported symptomatic 519

individuals has a great impact on the dynamics of the model states. Besides these findings 520 provide additional information about estimations and predictions for the number of infected 521 individuals. Accordingly, our results in identifying key parameters are broadly consistent 522

with clinical and biological findings. 523

Remaining issues are subject to sensitivity analysis. This is also an important issue that can 524 be further studied. We have applied the idea of local sensitivity to calculate the sensitivity 525 of each model state concerning model parameters for the updated model of the COVID-19. 526

Three different techniques are investigated which are non-normalizations, half 527 normalizations, and full normalizations. These provide us an important step forward to 528 identify critical model elements. By using local sensitivity approaches we concluded 529 that almost all model states are sensitive to the critical model parameters { } . This 530 becomes a great step forward and helps international attempts regarding the COVID-19 531 pandemic outbreak. This may help to reduce the number of infected individuals from the 532 disease and to prevent the coronavirus more widely in the community. It can be concluded 533 that the identified factors can be controlled to reduce the number of infected individuals. 534

Overall, our results demonstrate a strong effect of the key critical parameters on the 535 spreading COVID-19. 536 Therefore, based on the effect of each involved parameters over the model states, more 537

suggestions and interventions can be proposed for controlling the COVID-19 disease. That 538 will be useful for any interventions and vaccination programs. Accordingly, the healthcare 539 communities should pay more attention to the quarantine places for controlling this disease 540 more effectively. It can be strongly suggested that anyone in the quarantine places should 541 be separated from the others and should use only their separate equipment, bedroom, and 542 toilet to prevent the transmission of the virus through the touching of shared surfaces. 543

Another suggestion is that reducing the contact between asymptomatic-symptomatic groups 544

and susceptible groups, this is effectively minimizing the number of infected people. It 545 seems necessary to plan a certain strategy to put the asymptomatic infected individuals on 546 quarantine places sooner rather than later. Future research on identifying key critical 547 elements might extend the explanations of the new COVID-19 more widely. It will be 548 important that future research investigates more suggested transmissions between the 549 model groups. For example, the model will further improve by adding two transmission 550 paths, one of them is between unreported symptomatic infected and reported symptomatic 551

infected, the other one is between asymptomatic infected and recovered individuals. 552 553

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