Modelling the Transmission Dynamics of COVID-19 Incorporating public Enlightenment Campaign

: A mathematical model to study the transmission dynamics of COVID-19 incorporating public enlightenment campaign as control is presented. The effective reproduction number ( 𝑅 𝑐 ) was computed using the next generation method. Using the Lyapunov method, the global stability of the disease-free equilibrium was found to be globally asymptotically stable whenever 𝑅 𝑐 ≤ 1 . Sensitivity analysis was conducted on the effective reproduction number in order to determine parameters of the model that are most sensitive and targeted by way of intervention strategies. Numerical simulations of the COVID-19 model shows that if 90% of both treatment and public enlightenment campaign is achieved, the pandemic will be greatly controlled and subsequently eradicated in the population.

Corona virus disease COVID-19 is an infectious disease which was first identified amid an outbreak of respiratory illness caused in Wuhan City, China (Cennimo, 2021). The corona virus has been responsible for 270,327,277 cases and 5,316,017 deaths globally, European Center for Disease Prevention and Control (ECDC), (2021). In Nigeria, it is on record that 217, 481 cases have been confirmed while 2,981 deaths recorded so far (Nigeria Centre for Disease Control (NCDC), (2021) and World Health Organization, (WHO), 2021). The COVID-19 virus spreads primarily through droplets of saliva or discharge from the nose when an infected person coughs or sneezes (WHO, 2020). Thus, the disease spreads from one person to another as a result of close contact with an infected person. Symptoms of COVID-19 may range from mild symptoms to severe illness. Symptoms may appear within 2-14 days after exposure to the virus and these include: fever, chills, cough, shortness of breath or difficulty in breathing, sore throat, congestion or runny nose, fatigue, headache, muscle pain, loss of taste or smell, nausea, vomiting, and diarrhea (Felman, 2021). A test can detect the infection, even if there are no symptoms (CDC, 2021). Over the last two years, numerous models have been developed in order to understand the spread and control of COVID-19, see for example, Gweryina et al., (2021), William et al., (2021) and Iboi et al., (2021) just to mention a few have all developed mathematical models to study the transmission dynamics, control and prevention of COVID-19 pandemic. In all the papers referenced in this work, none considered public enlightenment campaign as one of the control strategies in curbing further spread of COVID-19. Public enlightenment campaign can help play a key role in educating the public about COVID-19 pandemic stressing the available control/prevention strategies among other steps that could help curb the spread of the pandemic. This could be achieved through the organization of workshops, television adverts and radio jingles to help enlighten the public about COVID-19. In this paper, a mathematical model to study the transmission dynamics of COVID-19 incorporating public enlightenment campaign as a control strategy is presented. This research is a modification of the work by Olaniyi et al., (2020) and based on the following assumptions: all successfully treated individuals become susceptible after treatment from the isolation centre, proper COVID-19 protocols are fully observed during burial rites of the dead, hence no need for the dead compartment, the present study incorporates the parameters of birth, death, and control strategies like the public enlightenment campaign, which was not captured in the works of Olaniyi et al., (2020), we neglect immigration of individuals into the country in order to curtain the spread of the pandemic, this study did not incorporate recovery class since individuals who are properly treated from the hospital moved to their various houses (become susceptible). Individuals who are either symptomatic (I) or Isolated (H) can die due to COVID-19 infections.

MATERIALS AND METHOD
The total population at time , denoted by is subdivided into five (5) compartments of the susceptible individuals , exposed individuals , symptomatic individuals , asymptomatic individuals and isolated individuals .
Let Λ be the recruitment number into the susceptible into the population, be the recovery rate and be the rate of transmission of the COVID-19 pandemic. is the natural death rate associated to all the compartments. represents the rate at which humans exposed to COVID-19 progress from the exposed state to either the symptomatic or asymptomatic compartments after the incubation period of 14 days. The parameter 1 represents the rate of individuals with symptoms after the incubation period. As a result, the rate of the individuals that do not show symptoms after incubation period is(1 − 1 ).
The parameters ℎ 1 and ℎ 2 represent the rates at which individuals are isolated in both the symptomatic and asymptomatic compartments respectively. COVID-19 is known to have caused several deaths; the parameter represents the rate at which people die due to COVID-19 pandemic. The above description of variables and parameters are summarized on Tables 1 and 2 respectively.
While the flow diagram used in the formulation of the COVID-19 model is presented in figure 1. With the descriptions of the variables and parameters on Tables 1 and 2, assumptions and the flow diagram in Figure  1, the following set of non-linear ordinary differential equations were derived: where, Basic Properties of the Model: For Model (1) − (5) to be epidemiologically meaningful, it is important to prove that all its state variables are positive for all time( ).

Positivity and boundedness of solutions: Consider the biologically feasible region
It can be shown that the set ℳ is a positive invariant set.
Lemma 1. The region ℳ is positively invariant for the system (1) − (5) Proof. The total human population is given by (7) and its rate of change is Since the right-hand side of (2.8) is bounded by Λ − , a standard comparison theorem as outlined in the works of Iboi and Okounghae (2016) was applied here.
Thus, ℳ is positively invariant. In this region, the model (1) − (5) can be considered as being epidemiologically meaningful and mathematically well posed. Effective transmission coefficient Rate of disease progression from exposed class 1 Proportion of exposed with symptoms after the incubation period 1 − 1 Proportion of exposed without symptoms after the incubation period ℎ 1 Isolation rate for symptomatic class ℎ 2 Isolation rate for asymptomatic class after confirmation γ Recovery rate from isolation class COVID-19 induced mortality rate Natural death rate Where ( −1 )is the spectral radius of the matrix −1 . Equation (13) is the effective reproduction number of the COVID-19 model (1)-(5). By definition, it is the average number of secondary COVID-19 cases generated by an infected COVID-19 individual in a completely susceptible population.
The results below follows from Theorem 2 in van den Driessche and Watmough (2002). Lemma 2. The disease-free equilibrium 0 is locally asymptotically stable when < 1 and unstable when > 1. Proof: Here, we follow a similar approach as outlined in the works of Iboi and Okounghae (2016) to establish the global stability of the disease-free-equilibrium.
Having established the proof for local and global stability of the disease-free equilibrium state, it is extremely important to establish the bounds for both the COVID-19 transmission rate and the public enlightenment campaign.
From (13), we obtain, This suggests that for COVID-19 to be put under control in the population, the infection rate  should not exceed the value given by the right-hand side of (15) while equation (16) suggests that for public enlightenment campaign about COVID-19 to be effective, the campaign rate should target a value greater that the right hand of (16).
Sensitivity Analysis: Sensitivity analysis is conducted on the effective reproduction number using the normalized forward sensitivity index in order to determine the parameters that are most sensitive and that should be targeted by way of intervention strategies.
Mathematically, the normalized forward sensitivity index of a parameter ( ) that depends on differentiable ( ) (Williams et al, 2020) is expressed as: The results obtained from the analysis are presented in Table 3. The table shows that parameters with positive index will increase the endemicity of the pandemic while those with negative index will reduce the endemicity of the pandemic.   Figure 2 shows that strengthening public enlightenment campaign at 90% level among the susceptible population will greatly reduce the number of persons who will be exposed or infected with COVID-19 pandemic. Figure 3 reveals that, administering proper treatment at 90% level will lead to the prompt recovery of individuals in the isolation centres. Conclusion: A mathematical model to study the transmission dynamics of COVID-19 incorporating public enlightenment campaign is presented. The model was found to be locally asymptotically stable when < 1 and globally asymptotically stable whenever ≤ 1. A bound for COVID-19 transmission rate and that of public enlightenment campaign was obtained. Finally, numerical simulations conducted on the model reveal that, 90% of treatment and public enlightenment campaign must be attained for COVID-19 to be put under control in the population.