Abstract
COVID-19 is the short name of the coronavirus disease discovered in Wuhan, China, in 2019. In the context of Tanzania, we develop a mathematical model in this work that compares lockdown and quarantine. Again, we provide evidence in favor of local and global stability, with the basic reproduction number, , determined to be at the diagnostic test rates . In comparison to the lockdown, it has been discovered that isolating (or quarantining) affected individuals is the most effective way to stop the spread of COVID-19. Additionally, it is advised that governments in Tanzania and other African countries permit their citizens to go about their daily lives as long as they take the necessary precautions, such as donning face masks, washing their hands, and avoiding crowded gatherings in case of a recurrence of any form of COVID-19.
1. Introduction
The coronavirus disease of 2019 is known by the abbreviation COVID-19, and it was identified in patients who had just visited a seafood and wet market in Wuhan, China. Although having comparable symptoms and a similar route of transmission to SARS-CoV and the Middle East Respiratory Syndrome, it has been determined to be the most lethal pandemic disease in human history [1]. On 11th March 2020, the World Health Organization (WHO) declared the COVID-19 disease as a pandemic disease due to its higher rate of spreading [2]. The coronavirus disease is mainly transmitted through inhaling or sneezing the droplets of the infected human; the droplets containing the virus are heavy in air; thus, they quickly fall onto surfaces [3]. The incubation period ranges from 2 to 14 days where the common symptoms of the COVID-19 disease include high fever, cough, fatigue, and sore throat; breathlessness, diarrhea, headache and nausea, and fatigue are earlier symptoms which appear on many patients before they develop fever [4, 5]. The zoonotic origin of the coronavirus disease is still not clear, but it is likely that bats are the primary reservoir for the virus [1]. It has been identified that out of 34 children with COVID-19 examined, there were no deaths identified for these children indicating that the coronavirus disease has little effect on children and young people; again many studies show that the adults with more than 60 years are highly affected by the virus, and a large number of deaths for these age groups have been observed in many countries [4]. Basing on the report of WHO and Johns Hopkins University, up until 5th October 2020, there were million deaths reported worldwide due to COVID-19 disease in which deaths are from Africa [6]. Early detection, diagnosis, treatment, and quarantine have been recommended as control measures to reduce secondary infection spread from person to person. The MRNA vaccine candidate and BNT 162b2 were discovered to be effective in 28 days following the first dosage; according to the World Health Organization (WHO) reports, 10 more COVID-19 vaccines have been confirmed to be effective [7].
The first COVID-19 case in Tanzania was identified on 16th March 2020 in a Belgian woman who had traveled to Tanzania via Kilimanjaro International Airport (KIA) [2]. After the occurrence of the first case, the government of Tanzania through the prime minister Kasim Majaliwa decided to take some control measures by announcing the closure of all schools and public and social gatherings as well as colleges and universities. Again, the government decided to introduce the quarantine for 14 days for all incoming travelers at their own costs [2]. Although there were about coronavirus cases with deaths and recoveries by May 2020 [8], the late president of the United Republic of Tanzania, Hon. Dr. John Pombe Magufuli, on 21st May 2020 announced the reopening of all colleges, schools, and sports and allowing all tourists to use international flights by 1st June 2020 [2]. At the time of writing this paper, there is scientific and biological evidence of reappearance of new COVID-19 disease patients in Tanzania, and the government of Tanzania under the President Hon. Samia Suluhu Hassan has decided to distribute the COVID-19 vaccine among her citizens [9].
Nearly all nations, especially those in east Africa, have been adversely affected by the COVID-19 disease politically, socially, and economically. In Tanzania, the COVID-19 epidemic has had a significant negative impact on a number of industries, including tourism, trade, finance, education, health, and agriculture. Also, it has been found that more than of Tanzanians workers especially in private sectors were in danger of losing their jobs and some of them lost their jobs while the other workers’ salaries were reduced due to the COVID-19 pandemic [8]. Once more, it has been discovered that COVID-19 has caused the deaths of numerous people in east African countries who had HIV, tuberculosis, malaria, and noncommunicable disorders like blood pressure (BP) and diabetes [10]. Despite that, Tanzania had no total lockdown (stay home), and the number of deaths due to COVID-19 was fewer compared to other east African countries like Rwanda, Kenya and, South Sudan where there were lockdowns [8].
Many researchers have so far proposed prevention measures and diagnostic methods in their work on the COVID-19 virus’s propagation [3]. In the work of Ming et al. in [11], they used the modified Susceptible-Infectious-Recovered (SIR) model to project the actual number of infected people and the specific burdens on isolation wards and intensive care units (ICU); they found out that the estimated burdens on the health care system should be largely reduced if at least efficacy of public health intervention is achieved. In February 2020, Singhai in his work [4] suggested that there can be a possible outbreak of other viruses and pathogens of zoonotic origin; thus, apart from curbing the coronavirus disease outbreak, efforts should be made to devise comprehensive measures to prevent future outbreaks of zoonotic origin.
Based on the SEIR model proposed by Prem et al. [12] using Wuhan, China, as a case study, their results showed that the rapid transmission of the coronavirus disease had been reduced as the result of a complete lockdown for about two months up to the end of April 2020. Zab et al. [3] formulated the Susceptible-Exposed-Infected-Isolated and Recovered model for COVID-19 in which they found that human-to-human contact is a potential cause of outbreak of the disease; thus, they suggested that the isolation of the infected human is a necessary approach to reduce the spread of the disease. Furthermore, Mumbu and Hugo [2] formulated the Susceptible-Masked-Unmasked-Exposed-Infected-Hosptalized-Recovered (SMUEIHR) model to analyze the significance of wearing masks and hospitalizing the detected infections as preventive and supportive measures in reducing the transmission rate of COVID-19 in Tanzania.
In order to analyze the COVID-19 transmission rate, we develop a mathematical model incorporating home stay (lockdown) and quarantine elements using Tanzania as a case study where there was a partial lockdown. The effectiveness of home stay and quarantine in Tanzania is then contrasted.
2. Model Formulation
In this section, the Susceptible-Exposed at Home-Exposed Quarantined-Serious Infected at Home-Mild Infected at Home-Quarantined Mild Infected-Quarantined Severe Infected, Hospitalized-Recovered model has been formulated to analyze the transmission of the COVID-19 disease. The total human population at any time is defined by
The susceptible individuals are recruited at the rate . The transmission rate at home and quarantine are, respectively, defined by
The susceptible person can contract the coronavirus by coming into contact with the droplets of the infected person or by breathing the droplets of the infected person at a rate of . The exposed human has COVID-19 but does not show the serious symptoms of the disease, and it has been assumed that all exposed people diagnosed were infected. Again, there is no death due to disease for exposed humans; however, natural death can happen at any stage at rate . The exposed humans become severe and mild infectious at rate . It is also assumed that COVID-19 is transmitted when the human is at the infectious stage, and is the stage when the symptoms show, and the person can be severe if his/her immunity is weak due to some other diseases like HIV or cancer or because of age. The mild infectious COVID-19 patients become severe at the rate and the severe infected patient becomes hospitalized at the rate . The parameter represents the recovery rate after taking medications or using natural herbs. The rest of the model parameters and variables are described in Tables 1 and 2. We present the dynamics of the disease by using the compartmental model in Figure 1 where it has been divided into home stay (lockdown) and quarantine compartments.
The system of differential equations are derived from Figure 1 as follows:
2.1. Positivity and Boundedness of the Model
In this subsection, we show that system (3) is epidemiologically and mathematically well-defined in the positive invariant region:
Theorem 1. There exists a positive invariant region where the solution set is contained and bounded.
Proof. If we have the solution set , with the positive initial conditions , we define the total human population at any time by The derivative of is taken with respect to time to obtain It follows that Thus, we have the differential equation This implies that Therefore, the solution of system (3) exists and is contained in the invariant region :
3. Model Equilibria and Stability Analysis
In this section, the model equilibria, basic reproduction , and stabilities at disease-free and endemic equilibrium are derived.
3.1. Model Equilibria
The disease-free equilibrium is calculated in the absence of COVID-19 in the population while the endemic equilibrium is when there is COVID-19 within the human population. Thus, the disease-free equilibrium is given by and endemic equilibrium point () is As in [13], the model’s parameters by themselves are unable to mathematically express the endemic equilibrium point. Hence, the following equations relate the elements of the endemic equilibrium point :
3.2. Basic Reproduction Number,
The basic reproduction number, , is the number of secondary infections caused by one infected human in a completely susceptible population; it plays a great role on the threshold value for the dynamics of the system and the disease [14]. When , the number of infected people declines to zero implying that the disease dies out from the population; however, if , the disease remains in the population and the disease is said to be endemic in the population [14]. The next-generation matrix approach as done by P. van den Driessche and Watmough [14–17] is used to derive the basic reproduction number. From the compartmental diagram 1, we let and be the column matrices from the new infection and transition compartments, respectively, where
We further denote and where .
The basic reproduction number, , is equal to the spectral radius of the next-generation matrix . Thus, then,
Therefore, the basic reproduction number, , is given by
Epidemiologically, the obtained represents the average number of new COVID-19 infections generated by a single secondary infected Tanzanian in a susceptible population.
3.3. Local and Global Stability Analysis
The disease-free equilibrium (DFE) and endemic equilibrium (EE) denoted by and , respectively, are proven to be both locally and globally asymptotically stable in Theorems 2 and 3 as follows:
Theorem 2. If , the DFE is locally asymptotically stable in the region defined by equation (4), and it is unstable if .
Proof. The DFE is locally stable only if the Jacobian matrix has all eigenvalues negative; this is possible if and only if the trace of and the determinant of . By linearizing system (3) at , the Jacobian matrix is obtained as follows: where , , and . The trace of matrix is negative. We now use the matrix properties and matrix reduction techniques as in [18] to verify that its determinant is greater than zero. The eigenvalue of the matrix has negative real parts; thus, the Jacobian matrix is reduced to where , , and . The matrix has a negative real eigenvalue . Using the same procedures and reduction techniques, the matrix is reduced to the matrix below: It can observed that the eigenvalue in the matrix has negative real parts. The matrix , therefore, was reduced to a : The trace of matrix is clearly negative; thus, the determinant is The determinant of matrix is greater than zero only if . Therefore, the DFE is locally stable if .
Theorem 3. If the disease-free equilibrium (DFE) defined by is globally asymptotically stable; otherwise, it is unstable.
We use the following definitions and Theorem 4 in [14] to prove Theorem 3.
Definition 4. A scalar function such that is called radially unbounded if as .
Definition 5. Let be a continuous scalar function such that ; the function is positively definite on the entire space if for .
Theorem 6. If a function is globally positively definite and radially unbounded and its time derivative is globally negative, i.e., for all , then the equilibrium is globally stable.
Proof. We define the Lyapunov function by where are constants to be determined. We have considered the infected compartments only because if the infected compartments are globally stable, then the DFE is globally stable and the disease dies out from the population. The function is clearly radially unbounded and globally positively definite. It is enough to show that for all . Differentiating with respect to time gives System (3) is then substituted into equation (24) following simplifications. Thus, By assuming , , we have Let and , and after further simplifications, we get Since all variables and parameters are nonnegative, it follows that The epidemiological implication of Theorem 6 is that the COVID-19 disease dies out from the population at any size of the subpopulations if
Theorem 7. The endemic equilibrium point is globally asymptotically stable in the region in equation (4) if ; otherwise, it is unstable.
Proof. We use the Lyapunov function technique as used in [2]. The Lyapunov function is defined at the endemic equilibrium by
where represents the human population classes and is the human population at endemic equilibrium .
We show that for all . By substituting the classes and the endemic equilibrium point into equation (29), we obtain
The derivative of with respect to time gives
From equation (6) and , we have
Thus,
Therefore, ; if and only if , which imply that the endemic equilibrium point is globally asymptotically stable in the region . Epidemiologically, this indicates that COVID-19 will persist in the human populations in Tanzania for a long time.
4. Numerical Results and Discussion
4.1. Numerical Simulation
In this section, the numerical results were obtained using the MATLAB programming language by using the set of values of parameters obtained from several literatures as cited in Table 3. Some of the parameters have been assumed for the sake of illustrations under the given reasonable proportions. Some initial datasets were taken from the reports of the Tanzanian Ministry of Health, Community Development, Gender, Elderly and Children as explained in [2] where the following initial conditions were considered: the total susceptible population is assumed to be 55,000,000, the mild infected humans at their homes , the severe infected humans at their homes , the mild infected humans under quarantine , and the severe infected humans under quarantine . Home-exposed humans and the quarantined-exposed Tanzanians ; also, the hospitalized humans are taken to be 297 and the recovered humans are supposed to be 183. According to Mumbu et al., the recruiting rate is mathematically possible because Tanzania has a 55-year life expectancy [2]. The rest of the parameters are summarized in Table 3.
The numerical results in Figures 2, 3, 4, and 5 show that the number of infected humans (both mild and severe) due to lockdown is greater than the COVID-19 cases due to quarantine at different diagnostics test rates.
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4.2. Results of the Numerical Analysis
In this subsection, we present and analyze the numerical simulation results obtained using the MATLAB programming language by using the set of values of parameters in Table 3.
The home stay (lockdown) was partially implemented in Tanzania; rather, the country implemented the approach of isolating COVID-19-infected patients (quarantine) where the infected humans were isolated for 14 days for treatments.
Figure 2 shows the number of COVID-19 infection cases for both severe and mild situations when the diagnostic rates and the basic reproduction number for both home stay and quarantine aspects; it is observed that the number of infections due to home stay is greater than the COVID-19 cases due to quarantine. Again, Figures 3 and 4 show that when and , respectively, the number of infected people due to home stay (lockdown) is greater than the number of infected people due to quarantine. Furthermore, the results from Figures 5 and 6 depicted that the number of exposed cases due to home stay is higher than exposed quarantined people at different diagnostic test rates. The value of the obtained indicates that the number of infected individuals in both scenarios can be decreased to zero by raising the diagnostic test rates if the diagnostic test rates are . In Tanzania and other African countries where a major portion of the population depends on everyday activities for survival, the results of these figures suggest that isolating infected individuals is the best way to stop human-to-human contact and slow the spread of COVID-19. For instance, the number of COVID-19 cases and deaths in the two eastern countries of Kenya and Rwanda, where there was lockdown (home stay), was higher than the number of cases and deaths reported in Tanzania [8].
The findings from other articles are consistent with the results from this paper. For instance, research by Zeb et al. [3] and van Zandvoort et a. [24] shows that the COVID-19 transmission can be stopped by keeping infected people apart and maintaining a moderate physical distance between them. Therefore, based on the numerical findings, it is advised that the governments of Tanzania and other African nations permit their citizens to go about their daily lives as long as they take the necessary precautions, such as donning face masks, washing their hands, and avoiding person-to-person contact by forbidding pointless gatherings while the infected patients are isolated for treatment.
5. Conclusion
Based on the quantitative findings of this study, it can be deduced that close contact between people has the potential to spread the COVID-19 pandemic. As a result, isolation (quarantine) of infected individuals may be the best strategy for reducing COVID-19 transmission in Tanzania and other African countries, as opposed to lockdown. Thus, it is recommended that Tanzanian and other African governments allow their citizens to go about their daily lives while taking the necessary precautions, such as wearing face masks, washing their hands, segregation of infected individuals in order to receive medical care, and preventing person-to-person contact by outlawing meaningless gatherings.
Data Availability
The secondary data supporting this research are from previously reported studies and datasets, which have been cited. The processed data are available in the references cited.
Conflicts of Interest
The authors declare that they do not have any conflict of interest.