Predicting Required COVID-19 Vaccine Coverage and its Impact in Sierra Leone Using Mathematical models

In this article, we predict the required vaccine coverage to eradicate the COVID-19 outbreak in Sierra Leone. We also, investigate the impact of facemask and vaccine coverage on the spread of the COVID-19 virus using a modified symptomatic-asymptomatic infection transmissions Susceptible-Latent-Infectious-Asymptomatic-Recovered (SLIAR) model. We derived an explicit formula for the basic reproduction number and used it to understand the dynamics of the disease when it is greater than unity. Numerically, we show that 58 per cent of the Sierra Leone national population required vaccination to eradicate the COVID-19 virus. Also, the SLIAR with vaccine model results reveal that the impact of using facemask is very challenging to understand and the vaccine coverage decrease the infected transmission rate but cannot completely eradicate the infection. materials. Authors can confirm that all relevant data sources are included in the article.


Introduction
The 2019 coronavirus (COVID- 19) outbreak has been global pandemic with over 140,000,000 cases and more than 3,000,000 deaths [1]. Because of its high rate of infection and mode of transmission many control strategies have been implemented like social distancing, use of face mask, handwashing, and vaccine.
Facemask is used because the virus of the COVID-19 can transmit through sneezing and coughing [2,3].
Regular handwashing because human frequently touches their face which is the fastest transition point of the virus [4]. Social distancing because the virus can spread between a distance less than 6.1ft [5]. The fear imposed by the COVID-19 is that people without symptom whether they can show symptoms or not, as long as they have exposed to the virus they can cause transmission [6][7][8][9]. That is, there is an asymptomatic form of transmission.
Although the recovered rate of the COVID-19 is encouraging, the deaths rate in most developed nations is alarming and because of that, the use of the vaccine has been the focus by many developed nations to eradicate the COVID-19 virus. Many vaccines have been used which developing countries like Sierra Leone is also a beneficiary. In February 2021, the People Republic of China donated 200,000 doses of vaccine [10] and another 96,000 doses of the AstraZeneca-Oxford COVID-19 vaccine shipped via the COVAX facility was donated on March 8, 2021, [11] help to fight the COVID-19 virus in Sierra Leone.
The targeted populace are frontline health workers (18 years and above), vulnerable including aged people, 30 years and above healthy individuals and people with underlying health conditions [11,12].
However, whilst many countries are preventing the third wave of the COVID-19, there is concern among public health officials that Sierra Leone is on the verge of a third wave of COVID-19 infections [12]. New cases are surging and have hit highs record since June 1, 2021 [12]. Like many African countries, citizens in Sierra Leone have been reluctant to take the COVID-19 vaccine. As of June 16, 2021, estimated vaccine total doses given is 91,789 and people fully vaccinated is 13,831 which is about 0.2 per cent of Sierra Leone populations [13]. Among the recent restriction by the Sierra Leone government as of June 17, 2021 are; age for vaccination has been revised further to thirty (30) years and educational institutions shall comply with mandatory face mask, hand-washing and social distance protocols [12]. As a researcher, our main concern is the fraction of the population that must be successfully vaccinated to eradicate the infectious agent so that the government policy which states that government ministries, departments and agencies will only be accessed by employees and members of the public with proof of at least one dose of COVID-19 vaccination [12] can be revisited.
A mathematical models has been an effective tool to understand disease dynamics [14]. The state compartments in the system are demonstrated using differential equations from which an expression of the basic reproduction number can be derived [14]. The basic reproduction number ( 0 ) is a value that determines on average how many individuals an infected person infect during the infectious period. If 0 > 1, it believes that the disease will persist and becomes endemic and if 0 < 1, the disease will die out and the system becomes disease-free [15]. After the 0 has been determined, the basic reproductive number under vaccination which is the number of secondary cases caused by one primary case introduced into a population in which a proportion have been vaccinated [16] can also be determined. Also, the critical vaccination proportion that will achieve the eradication of the virus, for which the basic reproductive number under vaccination is equal to unity [16] can be determined.
In this article, we modified a symptomatic-asymptomatic infection transmission Susceptible-Latent-Infectious-Asymptomatic-Recovered (SLIAR) model for Influenza outbreak with a vaccine (V), used of facemask and handwashing. The symptomatic-asymptomatic infection transmission SLIAR model is common in the literature of SARS-CoV-1 [17,18] due to its mode of transmission as it is done in COVID- 19. That is why scientist referred to the COVID-19 as SAR-CoV-2 [19,20] as they get to know more about the COVID-19 virus. We aim to use the mathematical SLIARV model to inspect the impact of facemask, and vaccine coverages using epidemic parameters in Sierra Leone. We also, aim to use Maximum Likelihood modelling [21] to estimate the 0 from Sierra Leone daily reported incidence data and use it to estimate the proportion of people to vaccinate in other to eradicate the SARS-CoV-2 in Sierra Leone. By considering the vaccine coverage period in Sierra Leone, the estimated 0 is 2.4 showing that 58 per cent of the national population required to vaccinate to eradicate the COVID-19 virus. Also, our mathematical model results shows that the impact of using facemask is very challenging to understand and the vaccine coverage decrease the infected transmission rate but cannot completely eradicate the infection.

The model
In this article, we consider a varying population SLIARV model since the SARS-CoV-2 has been going on for more than a year and there is no restriction of movement in and out of Sierra Leone during the time of writing. The model is demonstrated using the differential equation given as . The total population is = + + + + + , Λ is the rate of birth/migration into the S population, is the symptomatic infective transmission coefficient and the infectiveness due to an asymptomatic person is reduced by the proportion . The rate of facemask and handwashing are denoted as 1 and 2 respectively, whereas and are the natural death rate and proportion of vaccinated individual respectively. The incubation period rate is measure by and the probability of being symptomatic infective is measure by .
Death related to the disease is measure by the rate , the recovered rate from populations of , and are respectively denoted as 1 , 2 , and 3 . For simplicity equations (1) to (6) is converted to proportional form given as Where = ⁄ , = ⁄ , = ⁄ , = , ⁄ = ⁄ , and equation (6) is excluded for further analysis since it does not influence the other equations. It easy to see that if 1 = 2 = = 0, equations (7) to (11) will be in the same form as the influenza model in [18]. However, it is not difficult to see that the feasible region of the model (2) Γ = {( , , , , ) is positive invariant and attracting sets that attracts all solutions of model (2) with nonnegative initial condition. Also, it is not difficult to see that model (2) has disease-free equilibrium (DFE), 0 = ( 0 = Λ ( + ) ⁄ , 0, 0, 0, 0) and using the next-generation matrix method [15]A we get .
It is well known that the disease will die out whenever 0 < 1 or continue to persist in the population whenever 0 > 1. For a perfect vaccine that confers life-long protection, the basic reproductive number under vaccination ( 0 ) can be determined as (1 − ) 0 , and the vaccination proportion that will achieve eradication ( ) is determine by (1 − 1 0 ⁄ ) [16].

Numerical simulations
In this section, we use R-software to run simulations for our SLIAR-V model and the values of the parameters are taken from epidemiological data for Sierra Leone [26]. We also assume the rate of moving from an asymptomatic state to recovery rate to be 14 days since it is the same observed period use for quarantine individual. Table 1 gives the value of the parameter used in the numerical simulation, Furthermore, we now focus our analysis on the impact of the control measured ( , 1 , 2 ) considered in the SLIAR vaccine model. Firstly, we simulate the SLIAR-V model without any adjustment of the values of the parameters in Table 1 to observe the behaviour of the individual population trajectory, especially the visibly infectious compartment. It is observed in Figure.1A that the symptomatic infected trajectory increases with increasing time despite the use of facemask, handwashing and vaccine parameters.
Also, it observes that when and trajectories increase the susceptible decreases, indicating their negative effect to the susceptible populations if they are not controlled.  Figure 1B show the impact of increasing the vaccination rate constant by 0.7 %. It is well known that when vaccine coverage is large and immunity is lifelong, the prevalence of infection decreases. This is observed in Figure 1B as the vaccination rate increases. The trajectories peaks of the symptomatic infected proportion decrease but it does not reach zero even at 95% coverage. Moreover, even though the vaccine coverage increases equally with facemask and handwashing coverages (Figure 2A), the symptomatic infected proportion trajectories peaks decrease but do not reaches zero. This shows that large vaccine coverage reduces the infection spread, but does not lead to complete eradication of the disease as demonstrated in [28]. Thus, the strategy of controlling COVID-19 by vaccination needs to be improved in terms of vaccinating the younger ages of an individual to tackle the infection.  Table 1 and increase the magnitude of the facemask coverage by 0.7%.
In Figure 2B, it is observed that despite the 0.7% increment of the facemask coverage the same trajectories are produced and it is difficult to give an insightful characterization of the epidemic curve. This shows that there is a problem of identifying the importance of using facemask with the symptomaticasymptomatic infections SLIAR vaccine model.

Discussion
This paper aims to extend a basic symptomatic-asymptomatic infection SLIAR model without control of Influenzas propagation [17,18] to model the used of facemask, handwashing and vaccination as a control strategy for COVID-19 pandemic. We investigate the effects of the facemask and vaccination rate on the spread of the coronavirus infection in Sierra Leone. We have derived an explicit formula for the basic reproduction number ( 0 ), which has been the key parameter in our model. Using the method of Jacobian matrix, we have established the local stability results of the disease-free equilibrium points, the state where the infection die out in the system [14]. Surely, we have shown that the DFE is locally asymptotically stable if 0 < 1 and unstable otherwise. For this reason, where 0 > 1, there exists a unique endemic equilibrium, which is locally asymptotically stable. Numerical simulations are conducted to illustrate that the infection maintain in the population with time when 0 is great than unity (see Figure 1A).
In terms of calculating 0 , we show that the 0 estimated from the SLIAR vaccine model is equivalent to the current estimated 0 for Sierra Leone COVID-19 outbreak using Maximum Likelihood modelling [21]. It shows that from January 1, 2021, to June 20, 2021, the estimated 0 is 2.4, indicating that an infectious person can infect an average of two persons in his/her infectious period. Approximately, our estimated 0 is in agreement with reproduction rate of Our World in Data [22]. Because the transmission rate is alarming in Sierra Leone, we estimated the critical vaccination proportion of people to achieve the eradication of the virus as 58%, which is far above the vaccine program focus targeted population in Sierra Leone. According to the World Population Prospects 2019, Volume II [24], there were 39% of the country population between the age groups 25 years to 65 years and above. Therefore, targeting 18 years and above health professionals which majority are above aged 25 years, and the aged 30 years, for the vaccine coverage in Sierra Leone will impose a serious challenge to eradicate the COVID-19 disease outbreak.
Hence, the policy of vaccinating just portion of the susceptible population needs to be revisited by including young ages of an individual to tackle the COVID-19 in Sierra Leone. The predicting vaccine coverage in this article can only be reach if the targeted population can be reduce to at least age 15 years using the World Population Prospects 2019, Volume II data [24] Using the SLIAR vaccine model the vaccination coverage only involves those in the susceptible population. In terms of controlling the spread of COVID-19 by facemask, we show that the importance of using facemask cannot be identified, as the same trajectory produce each time the coverage proportion increases. By controlling the spread of the COVID-19 virus by vaccination, we also show that even though vaccination reduces the infection spread, it does not completely eradicate the COVID-19 in the community when the coverage is high which is in agreement with [28].

Declarations
Availability of data materials. Authors can confirm that all relevant data sources are included in the article.