This is an openaccess article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work, first published in JMIR Public Health and Surveillance, is properly cited. The complete bibliographic information, a link to the original publication on http://publichealth.jmir.org, as well as this copyright and license information must be included.
With the sensitivity of the polymerase chain reaction test used to detect the presence of the virus in the human host, the worldwide health community has been able to record a large number of the recovered population.
The aim of this study was to evaluate the probability of reinfection in the recovered class and the model equations, which exhibits the diseasefree equilibrium state for the coronavirus disease.
The model differential equation was evaluated for the diseasefree equilibrium for the case of reinfection as well as the existence and stability criteria for the disease, using the model proportions. This evaluation shows that the criteria for a local or worldwide asymptotic stability with a basic reproductive number (
With a total of about 900,000 infected cases worldwide, numerical simulations for this study were carried out to complement the analytical results and investigate the effect that the implementation of quarantine and observation procedures has on the projection of further virus spread.
As shown by the results, the proportion of the infected population, in the absence of a curative vaccination, will continue to grow worldwide; meanwhile, the recovery rate will continue slowly, which means that the ratio of infection rate to recovery rate will determine the death rate that is recorded. Most significant for this study is the rate of reinfection by the recovered population, which will decline to zero over time as the virus is cleared clinically from the system of the recovered class.
The coronavirus disease (COVID19) pandemic has had a major impact on the global economy and on behavioral practices of people worldwide. Until its early detection in Wuhan, China in 2019, the virus was unknown to the scientific world, and the extent of its damage was unmeasurable. However, upon its outbreak, various research, including but not limited to Victor [
In the literature (Victor [
The SEIRUS model was used due to the resulting solutions that captured the relevant parameters for the exposed and untransmitable classes, which are not present in the SIR model as used by Nesteruk [
The resulting equations from the SEIRUS model are a system of coupled homogenous differential equations used to capture the susceptible rate, rate of exposure, infectious rate, and the rate of recovery. In addition, the equations capture the rate of reinfection, which is captured in the undetectable class that is clinically ascertained by the PCR testing approach for the recovered population.
Numerical experiments, with relevant simulation showing how the variation of the reproductive number (R_{0}) affects the number of infected individuals, were carried out as well as a projection for the rate of reinfection by the recovered class. Conscious effort to evaluate the new deterministic SEIRUS model was done to reduce the R_{0} to zero and possibly halt the spread of the disease, thereby leading to an endemic equilibrium and eradication of the disease in the future.
The worldwide COVID19 pandemic and the lack or inefficiency of purposeful and resultbased interventions are great calls for other empirical and scientific interventions that seek to review strategic models and recommendations of social and scientific research for disease control. Although previous studies have been tailored toward the epidemiology and the diseasefree equilibrium (DFE) where the R_{0} of the infectious population is at its bare minimum, this study seeks to evaluate the impact of a new endemic deterministic model on the endemic equilibrium while taking into consideration the possibility of the recovered population being undetectable and fit to be moved to the susceptible class, which will, therefore, imply zero secondary infection of the disease worldwide.
In summary, this study aims to use the new deterministic endemic SEIRUS compartmental model for COVID19 dynamics, which combines quarantine and observation procedures, and behavioral change and social distancing in the control and eradication of the disease in the most exposed subpopulations to predict the chances of reinfection by the recovered class.
As suggested in Victor [
The variables for the new deterministic endemic model.
Variable  Description 
Number of susceptible population at time 

Number of exposed population at time 

Number of infected population at time 

Number of infected population quarantined and expecting recovery at time 

Number of recovered adults satisfying undetectable criteria at time 
The parameters for the new deterministic endemic model.
Parameter  Description 

Natural death rate of the population 

Maximum death rate due to coronavirus disease (α≤α_{0}) 
α  Death rate of the infected population due to coronavirus disease 

Disease induced death rate of infected population not quarantined 

Disease induced death rate of infected receiving quarantine 

Maximum lifespan after infection ( 

Efficacy of quarantine (0≤ 

Rate of recovery 

Rate of transmission 

Proportion of infected population in quarantine per unit time (treatment rate) 

Proportion of population from susceptible to exposed/latent class 

Proportion of removed population still being observed and being moved to susceptible class 
Incidence rate or force of infection in the population 
The following assumptions, as suggested in Victor [
There is no emigration from the total population and there is no immigration into the population. A negligible proportion of individuals move in and out of the population at a given time.
Maturation (or maturity) is interpreted as the period between infection and symptom observation (days 114).
The susceptible population are first exposed to a latent class where they can be infected or not.
Some infected individuals move to the removed class when they are quarantined for observation procedures.
The recruitment from the S class into the E class is through contact with populations in the I class to the S class.
The recruitment into the R class from the I class is at a rate of σ.
The recruitment into the U class from the R class depends on the effectiveness of the quarantine and observation procedures at a rate of ρ.
Death is implicit in the model, and it occurs in all classes at a constant rate μ. However, there is an additional death rate in the I and R classes due to infection for both juvenile and adult subpopulations, denoted by φ and ϖ, respectively.
This study uses the deterministic endemic model where a susceptible class is a class that is yet to be infected but is open to infection as interactions with members of the
The following diagram [
The following equations are a system of coupled homogenous differential equations for projecting the detection rate of the presence of the virus in the clinically prescribed recovered population based on the assumptions and the flow diagram previously mentioned:
The incidence rate or force of infection at time
is given by:
The model equations in proportion according to Victor [
However,
Equations 1014 are the model equations in proportions, which define the prevalence of infection.
The DFE state of the endemic SEIRUS model is obtained by setting the lefthand sides of equations 1014 to zero while setting the disease components
0=
After substituting equation 16 into 15 we have:
We then take 15, where
0 =
Simplifying this further gives us:
In equation 18,
Therefore, the solution for the equations in 18 are given by:
Ignoring the native values of
In the event that patients recover from COVID19, it is assumed that they are disease free for at least 14 days after their last clinical test shows that they have clinically recovered from the virus. Hence, to study the behavior of the equations 1014 around the DFE state,
Hence, according to Gerald [
From equation 20:
Similarly from the Trace of the Jacobian matrix
Hence, since
The basic
To compute the basic reproductive number (
Here, 1≤
The inverse of
The next matrix will then be denoted by
We find the eigenvalues of
The characteristics polynomial is:
The characteristics equation is given as:
We solve the characteristics equation for the eigenvalues
The basic reproductive number (
According to the WHO [
A world map showing the number of cases for each country with a coronavirus disease case.
A cumulative case chart showing the number of cases of coronavirus disease.
The agestructured deterministic model in equations 1014 was solved numerically using the RungeKuttaFehllberg fourth to fifth order method and implemented using Maple Software (Maplesoft). The model equations were first transformed into proportions, thus, reducing the model equations to 10 differential equations. The parameters used in the implementation of the model are shown in
Hence from equation 26, the reproductive number
Estimated values of the parameters used in the numerical experiments.
Parameters  Values  Data source  Parameters  Values  Data source 
7.57 billion  WPR^{a} [ 

0.000005^{b}  Assumed  
845,292  WHO^{c} [ 

0.0000007  JHU^{d} [ 

1.0000  Estimation 

14 days  WHO [ 

1.0000  Estimation 

0.5^{b}  Assumed  
0.00002  WHO [ 

0.000095  JHU [ 

0.000095  JHU [ 

0.00002  WHO [ 

0.000095  JHU [ 

0.28404^{e}  Estimated  
μ  0.000001  WPR [ 

0.00567^{b}  Assumed 

0.000011  Nesteruk [ 

0.000095  JHU [ 
N/A^{f}  N/A  N/A  0.00000  Assumed 
^{a}WPR: World Population Review.
^{b}Assumed: Hypothetical data used for research purposes.
^{c}WHO: World Health Organization.
^{d}JHU: Johns Hopkins University.
^{e}Assumed: Based on Victor [
^{f}Not applicable.
Chart of recovered and infectious compartments for coronavirus disease.
Chart of the rate of reinfection of the recovered compartment from coronavirus disease.
The analysis clearly shows that the secondary infection rate satisfies the local and worldwide stability criteria and the DFE for an endemic situation. Unlike the respiratory syncytial virus, which causes a significant respiratory disease often in those 5 years or younger, COVID19 is estimated to burden more than 10,000 people worldwide. Although the stability analysis shows that there is no chances of secondary reinfection by the recovered class, the rate of the infectious will continue to rise asymptotically over a long period of time and there after begin to slide in a normal trajectory if no vaccine is available. Batista [
According to Victor [
However, with the various make shift treatments, social distancing measures, and quarantine strategies being adopted, the recovery rate will keep rising slowly but steadily over a long period of time. Therefore, as the recovery rate continues to grow steadily, the number of recovered patients who have been clinically declared free of the virus by the PCR test are also declared uninfectious as long as the virus is completely cleared from their system, and the rate of detection will vanish, making the rate of secondary infection
There is a need for a dedicated effort from individual populations, governments, health organizations, policy makers, and stakeholders. The world is hardly rid of COVID19, and further spread is eminent; the rate of infection will continue to increase despite the increased rate of recovery until a curative vaccine is developed.
With the worldwide health sector in a bid to tackle COVID19, this study gives encouragement to the policy makers and public health care sectors, as there is zero secondary reinfections by the recovered population. Therefore, the policy makers and public health sectors can enhance contact tracking, tracing, and testing to improve the isolation and quarantine of the infected and exposed classes. In addition, the health sector could use COVID19 antibodies from the samples of the recovered class to develop effective vaccines for the virus. However, since the hypothesis of zero reinfections has not been clinically proven, further observations should be carried out on the recovered class in clusters to study the progression of the exposed with the reexposed subpopulations to see, by clinical examination, the possibilities of reinfection and, thereby, promote the use of these antibodies for vaccine creation.
This study was limited by the variability of data available at the time of developing this paper. Meanwhile, from the statistics, the infected cases and fatalities were projected to increase geometrically. Therefore, the findings of this study are based on sample data taken at the time of the study.
In addition, with the SEIRUS model and the discovery that the
coronavirus disease
diseasefree equilibrium
polymerase chain reaction
reproductive number
susceptibleexposedinfectiousremovedundetectablesusceptible
susceptibleinfectiousremoved
World Health Organization
AV would like to acknowledge the various authors such as Nesteruk, Ming, Huang, and Zhang whose work motivated AV’s involvement in writing about the coronavirus disease. AV would also like to acknowledge the Johns Hopkins University and World Population Review for making their data available and accessible for use during this study. AV would like to acknowledge the Jomo Kenyatta University of Agriculture and Technology for allowing AV to use part of his work time to develop this paper.
The data used for this study was made available from John Hopkins University for the total number of infected patients and recovered patients for the entire population. In addition, the data for the worldwide population used to extract the susceptible class was made available from the World Population Review, and the material and methods were adopted from Victor and Oduwole from their work on disease control.
The author’s contributions include but were not limited to developing and using the novel SEIRUS model for COVID19 tracking, using the SEIRUS model to predict the probability of reinfection of COVID19 worldwide, confirming that there is no secondary spread of the virus after recovery without vaccine with a
None declared.