key: cord-0875354-1taf0245
authors: Okuonghae, D.; Omame, A.
title: Analysis of a mathematical model for COVID-19 population dynamics in Lagos, Nigeria
date: 2020-06-20
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2020.110032
sha: 8aecd6cbafc99130f4872780b48c5edf25f9eec1
doc_id: 875354
cord_uid: 1taf0245

This work examines the impact of various non-pharmaceutical control measures (government and personal) on the population dynamics of the novel coronavirus disease 2019 (COVID-19) in Lagos, Nigeria, using an appropriately formulated mathematical model. Using the available data, since its first reported case on 16 March 2020, we seek to develop a predicative tool for the cumulative number of reported cases and the number of active cases in Lagos; we also estimate the basic reproduction number of the disease outbreak in the aforementioned State in Nigeria. Using numerical simulations, we show the effect of control measures, specifically the common social distancing, use of face mask and case detection (via contact tracing and subsequent testings) on the dynamics of COVID-19. We also provide forecasts for the cumulative number of reported cases and active cases for different levels of the control measures being implemented. Numerical simulations of the model show that if at least 55% of the population comply with the social distancing regulation with about 55% of the population effectively making use of face masks while in public, the disease will eventually die out in the population and that, if we can step up the case detection rate for symptomatic individuals to about 0.8 per day, with about 55% of the population complying with the social distancing regulations, it will lead to a great decrease in the incidence (and prevalence) of COVID-19.

The following are important findings of the article:

The impacts of various non-pharmaceutical control measures (government and personal) on the population dynamics of the novel coronavirus disease 2019 in Lagos, Nigeria, using an appropriately formulated mathematical model are presented (ii ) We provide forecasts for the cumulative number of reported cases and active cases for different levels of the control measures being implemented. (iii) Global sensitivity analysis is carried out to determine the parameters that strongly drive the dynamics of the model, when the infected populations and when the reproduction number are used as response functions, respectively

Numerical simulations of the model showed that (iv) if at least 55% of the population comply with the social distancing regulation with about 55% of the population effectively making use of face masks while in public, the disease will eventually die out in the population (v) if case detection rate for symptomatic individuals can be stepped up to about 0.8 per day, with about 55% of the population complying with the social distancing regulations, it will lead to a great decrease in the incidence (and prevalence) of COVID-19.

The world is presently battling with ongoing outbreaks of a coronavirus disease, namely, coronavirus disease 2019 (COVID-19) caused by the novel coronavirus, SARS-COV2, a highly virulent virus that has caused COVID-19 to be a fatal disease, a disease which targets the human respiratory system. The pandemic started with a cluster of patients being admitted in late December 2019 to hospitals with an initial diagnosis of pneumonia and the patients were linked to a seafood and wet animal market in Wuhan, Hubei Province, China [33] . By January 2, 2020, 41 admitted hospital patients had been confirmed of being infected with COVID-19 [33] . By January 22, 2020, a total of 571 cases of COVID-19 were reported in 25 provinces in China [23, 33] . On January 30, 2020, about 7734 cases have been confirmed in China, with 90 cases reported in about 13 countries [4, 33] including the United States, India, Canada, France, Germany and the United Arab Emirates. As of May 27, 2020, 5,656 ,615 cases of COVID-19 have been reported worldwide, with about 355,355 deaths [13] . COVID-19 is transmitted from human-to-human via direct contact with contaminated surfaces and through the inhalation of respiratory droplets from infected individuals [3] . Presently, there is no vaccine or antiviral treatments approved for the prevention or management of COVID-19 [38] . To effectively reduce the spread of COVID-19, governments have been implementing different control meaures such as imposing strict, mandatory lockdowns, and encouraging (and in some cases strictly enforcing) other measures such as individuals maintaining a minimum distance between themselves (social distancing), avoiding crowded events, imposing a maximum number on individuals in any gathering (religious and social) and the use of face masks while in public [10] . To further help in mitigating the spread of COVID-19, contact tracing of suspected infected cases have been stepped up in several countries and detected cases (asymptomatic and symptomatic) are quickly placed on isolation for prompt treatment [10] .

In Nigeria, the first (index) case of COVID-19 was announced on 27 February, 2020 [27] . As of May 27, 2020 , Nigeria had reported 8,733 cases of COVID-19, with 254 deaths, 5,978 active cases (undergoing treatment) and 2,501 discharged cases [27] . As of that same date, the city of Lagos, Nigeria (the epicenter of the disease in the country) had reported 4,012 confirmed cases, with 47 deaths, 3,220 active cases and 745 discharged cases [27] . However, it is imperative to clearly state that just 48,544 samples were tested that led to the detection of the cases in the country of Nigeria as a whole (a country of over 200 million persons), as of May 27 , 2020 [27] . Hence, it is expected that, likely, several cases of COVID-19 remains undetected in the general population (whether in Nigeria or even in the city of Lagos, in particular), hereby rendering the announced figures as a gross underreporting of the actual burden of the disease in the country (Lagos, inclusive). Of course, it is expected that, with improved testing rates, more cases of COVID-19 will be detected and a clearer picture of the true state of the disease burden in Nigeria will become evident.

Mathematical models have long been used as tools in gaining insight into the dynamics of infectious diseases [12, 19, 28, 29, 31, 32, 35, 34, 36, 39] . Several mathematical models have already been formulated for the population dynamics of COVID-19 in several countries [1, 2, 14, 16, 18, 20, 21, 26, 37] . In this study, we will be presenting a mathematical model that investigates the impact of some control measures on the spread of covid-19 in a human population; the work will be looking critically at the situation in Lagos, Nigeria, hence we will be making use of data for the aforementioned city. The model, parameterized using the COVID-19 cumulative number of reported cases and the number of active cases for Lagos, will provide a realistic assessment of the burden of the disease for Lagos. Using the results from the quantitative study of the model, we will assess the impact of social distancing, use of face masks and case detection (via testing) on the disease burden in Lagos and make predictions for the cumulative number of reported cases and number of active cases as well as the likely peak time for the Lagos situation.

The total human population at time t, denoted by N h (t), is split into a mutually exclusive subpopulations of susceptible humans (S(t)), exposed humans (E(t)), asymptomatic infectious humans (A(t)), symptomatic infectious humans (I(t)), detected infectious humans via testing (and are isolated and in some form of hospitalization for prompt treatment) (I D (t)) and recovered humans (R(t)). Thus, N h (t) = S(t) + E(t) + A(t) + I(t) + I D (t) + R(t). We assume that those in the I D are completely isolated and do not come in contact with the general population.

The model for COVID-19 transmission dynamics in a population is given by the following system of deterministic non-linear differential equations in (2.1), with Table 1 describing the associated state variables and parameters in the model (2.1) while Figure 1 gives the flow diagram of model (2.1). 

Mathemarical models without demographic parameters (that is, birth and natural death) of the form (2.1) have been used extensively in studying the dynamics of epidemics [6, 11, 19, 38, 42, 43] . Demographic parameters including births and natural deaths can be exluded when investigating the dynamics of an epidemic that is occuring within a few weeks or months [11, 17, 38, 42, 43] . If we introduce parameters that represents social distancing and the use of face masks into the force of infection of the basic model (2.1), whereby a new parameter δ represents the proportion of the population that maintains the minimum distance required to prevent an infection (at least 1 meter apart), so that 0 ≤ δ ≤ 1, and another parameter ε represents the fraction of the entire population that effectively makes use of a face mask (where we assume that the face masks are with high efficacy), any time they are in public, so that 0 ≤ ε ≤ 1, the basic model (2.1) now becomes

It is imperative to state that, in the setting being considered in this work, the strict adoption of the use of face mask was being promoted well into the ongoing outbreaks in Nigeria, in particular the city of Lagos, after a five weeks total lockdown in the city was lifted. 

In this section, we seek to qualitatively study the dynamical properties of the COVID-19 model (2.2).

For the model (2.2) to be epidemiologically meaningful, it is important to show that all its state variables are non-negative for all time (t) > 0 and that D is, indeed, bounded. We claim the following: 

Let

We have, from the first equation of the system (2.2) that

can be re-written as

Similarly, it can be shown that: E > 0, A > 0, I > 0, I d > 0, R > 0. 

which can be re-written as

Thus:

Hence the region D attracts all solutions in R 6 + 3.2 Local asymptotic stability of the disease-free equilibrium (DFE) of the model (2.

2)

The COVID-19 model (2.2) has a DFE, obtained by setting the right-hand sides of the equations in the model (2.2) to zero, given by

The linear stability of the disease free equilibrium, ξ 0 can be established using the next generation operator method on the system (2.2). Using the notation in [40] , the matrix F (of new infections) and the matrix V (of the transfer of individuals between compartments) are respectively, given by

Hence, it follows from [40] that the basic reproduction number of the model (2.2), denoted by R c , is given by

We claim the following:

The DFE, ξ 0 , of the model (2.2) is locally asymptotically stable (LAS) if R c < 1, and unstable if R c > 1.

The local stability of the model (2.2) is analysed by the Jacobian matrix of the system (2.2) evaluated at the disease-free equilibrium ξ 0 , given by: 

The eigenvalues are given by λ 1 = −(γ i + d D ) and the solutions of the characteristic polynomial

Applying the Routh-Hurwitz criterion, the cubic equation (3.2) will have roots with negative real parts if and only if Φ 1 > 0, Φ 3 > 0 and Φ 1 Φ 2 > Φ 3 . Clearly, Φ 1 > 0 and φ 3 > 0 (if R c < 1). As a result, the disease-free equilibrium, ξ 0 is locally asymptotically stable if R c < 1.

The threshold quantity R c is the control reproduction number for the model (2.2). It represents the average number of secondary COVID-19 infections generated by a typical infectious individual (asymptomatic and symptomatic) in a completely susceptible population where control measures are present [40] . By Theorem 3.2, biologically speaking, COVID-19 can be eliminated from the population when R c < 1 if the initial sizes of the population of the model are in the region of attraction of the DFE. The first term of R c represents the average number of secondary infections generated by a typical asymptomatically infectious individual while the second term represents the average number of secondary infections generated by a typical symptomatically infectious individual. Consider the Lyapunov function (the linear Lyapunov function has been widely used to prove the GAS of DFE [15, 30] ):

The time derivative is given bẏ

Substituting the expressions for the derivatives,Ė,Ȧ andİ from (2.2), into the Lyapunov derivative,L, and carrying out some algebraic manipulations, we have noting that

Hence, L < 0 if and only if R c < 1, and L = 0 if and only if E = A = I = 0. Therefore, L is a Lyapunov function for the system (2.2). Thus, it follows by the La Salle's Invariance Principle [22] , that the DFE of the model (2.2) is globally asymptotically stable whenever R c < 1.

In this section, we shall carry out uncertainty and sensitivity analyses on the parameters of the model due to uncertainties which may arise in the estimation of some of the parameters of the model. We also carry out numerical simulations of the model, in order to assess the impact of various control strategies on the dynamics of the disease. The equations of the model (2.2) are solved numerically using the MATLAB ode45 solver which is based on the fourth-order Runge-Kutta method. The stability of the method is well established in [24] .

As a result of the uncertainties which may come up in parameter estimates used in the numerical simulations, a Latin Hypercube Sampling (LHS) [5] is implemented on the parameters of the model. For the sensitivity analysis, we carry out a Partial Rank Correlation Coefficient (PRCC) between values of the parameters in the response function and the values of the response function derived from the sensitivity analysis. 1,000 simulations of the model (2.2) were run. Using the reproduction number, R c , as the response function, it is observed in Table 2 that the top-ranked parameters that drive the dynamics of the model are transmission rate, β c , the case detection rates for asymptomatic and symptomatic infectious humans, θ and ψ, respectively, the modification parameter accounting for infectiousness of asymptomatic infectious humans, α. β c and α are positively correlated whereas θ and ψ are negatively correlated. The public health implication of this is that, COVID-19 can be controlled effectively in the population by reducing the transmission rate, achieved through preventive measures like strict social distancing regulations and compulsory face-masks wearing in public, and also reducing the infectiousness of asymptomatic humans through proper treatment. In addition, the disease burden can be significantly reduced in the population if efforts are put in place to step up detection rates for asymptomatic and symptotic infectious humans so as to isolate and offer them adequate treatment. Using the exposed class, E, as response function, the parameters that strongly influence the dynamics of the disease are the effective contact rate, β c , the case detection rate for symptomatic infected, ψ and the transition rate out of the exposed class, σ. Also, when the total number of asymptomatic infectious, A, is used as the response funtcion, the parameters that strongly dominate the dynamics of COVID-19 disease are the transmission rate, β c , the fraction of exposed individuals who progress to the asymptomatic stage, ν, the transition rate out of the exposed class, σ and the detection rate for asymptomatic infectious humans, θ. Using the population of symptomatic infectious humans, I, as the response function, the top-ranked parameters that drive the dynamics of the disease are the transmission rate, β c , the transition rate out of the exposed class, σ, the detection rate for asymptomatic humans, ψ and the fraction of exposed humans who progress to asymptomatic stage, ν. Moreover, using the population of detected infectious humans, I d , as the response function the five top-ranked parametrs that drive the dynamics of the diasease are the transmission rate, β c , the detection rates for asymptomatic and symptomatic infectious, ψ and θ, respectively, the transition rate out of exposed class, σ and the recovery rate for detected infectious humans, γ i .

We now apply our model (2.2) to study the dynamics of COVID-19 in the city of Lagos, Nigeria. We will be making use of the outbreak data released daily by the Nigeria Center for Disease Control (NCDC) [27] . Specifically, we will use the cumulative number of reported cases and the number of active cases for our data and model fitting since the reported cumulative number of deaths is relatively small to give a good fit for parameter estimation. We implement our model and conduct numerical simulations for an epidemic period starting from when the index case was announced in Lagos, i.e., March 16, 2020 to May 2, 2020, two days before a total city-wide lockdown was lifted ( and replaced with a partial lockdown with strict measures on enforcing social distancing and the use of face masks when in public).

The incubation period for COVID-19 is estimated to average 5.2 days [33] , with a range of 3 to 14 [8, 7] , so that we set γ i = 1/15. We also set the natural recovery rates 0.13978 per day, so that γ a = γ o = 0.13978 [38] , although the value 0.111 was used in [1] . The disease induced death rates was set to 0.015, following the work in [14] , so that d o = d D = 0.015. We estimate the following parameters in model (2.2) by fitting the model with the daily cumulative number of reported cases and number of active cases: the transmission rate (β c ), the case detection rate for the asymptomatic individuals (θ) and the case detection rate for the symptomatic individuals (ψ). We will also estimate the initial number of infected individuals (latent and infectious) as at the time the index case was reported for Lagos, on March 16, 2020 i.e., E(0), A(0) and I(0).

We have that the total population of Lagos is estimated to be 14,368,332 [41] . Hence, since the first case was announced in March 16, 2020 (the start period for all the simulations herein), we set our initial conditions to be S(0) = 14, 367, 982, I D (0) = 1 and R(0) = 0, taking into account the fact that, as at the date of the first reported infected case, it is expected that some undetected infected within the population (since wide spread population screening and testing for COVID-19 had not begun at the time). Therefore, as stated above, it is imperative to estimate the likely values of the other initial conditions, namely, E(0), A(0) and I(0). Table 3 gives the values of the parameters used in the simulations. 

The model fitting was performed using a genetic algorithm (GA) [25] for our function optimizer, implemented in MATLAB; the GA algorithm helps us find the correct basin of attraction, which provides the starting values (for the parameters being estimated) for use in the fmincon function in the Optimization Toolbox of MATLAB. Hence, we will be combining two optimization algorithms for data fitting, a GA algorithm and the fmincon algorithm in MATLAB to get a more accurate estimate. We implement our model fitting for an epidemic period starting from when the index case was announced in Lagos, i.e., March 16, 2020 to May 2, 2020 .

Most of the days captured in the model fitting for Lagos was for the period when there was a government imposed lockdown in the city (from March 30, 2020 to May 3, 2020) . Hence, the impact of the lockdown on the dynamics of the disease in Lagos will inherently be captured in the transmission rate of the disease (β c ).

Using the data available for the daily cumulative number of reported cases and number of active cases, we seek to estimate the values of β c , ψ, θ, and the initial conditions E(0), A(0) and I(0). It is imperative to state, that the reported number of COVID-19 cases in Lagos is from a population that has only had very small number of tests carried out, thereby resulting in a likely gross under-reporting of the actual disease burden in the population. Also, all fittings were done with δ = 0 and ε = 0, since during the period being fitted in this work, both measures (social distancing when in public and the use of face masks) were not strictly monitored or enforced until after a partial lockdown (which was a relaxing of the previous 'total' 24 hours lockdown, with debatable level of compliance from the entire population) was introduced starting from May 4, 2020. The values of other parameters of the model (2.2), used in all simulations herein, are as stated in Table 3 .

We observe that we had, approximately, the same fitted values for β c , ψ and θ, as well as the calculated disease reproduction number (R c ), when we used the cumulative number of reported cases, the number of active cases and when we combined both datasets for the model fitting. However, there were significant differences in the estimated values of E(0), A(0) and I(0). Table 4 gives the estimated values of of β c , ψ, θ, and the initial conditions E(0), A(0) and I(0), together with the calculated reproduction number (R c ), when we used the cumulative number of reported cases, the number of active cases and both sets of data for fitting the model; we make use of the following notations: Table 4 : Estimated parameters fitted using three different data sets.

As seen from Table 4 , the estimated parameters β c , ψ and θ, as well as the calculated disease reproduction number (R c ), are approximately the same (for the three different datasets used for the model fitting); however, there are significant differences in the estimated values of the initial values E(0), A(0) and I(0). It is very imperative to state that, as at the time when the first detected case was announced in Lagos (16 March, 2020) , it will be difficult to determine the exact number of individuals in the population who were already infected with COVID-19. Hence, we are not surprised that the model fitting carried out using the above 3 different data sets gave different values for the initial conditions (even though each set of estimated parameters and initial conditions resulted in good model fits with the actual data).

These differences in the initial conditions (E(0), A(0) and I(0)), expectedly, affected the values of the projections (future predictions) made with regards to the likely number of infected individuals (asymptomatic and symptomatic) who are yet to be detected (via testing), the cumulative number of reported cases and the number of active cases, as we shall see in the numerical simulation results in the following subsections.

In this section, simulations of the model when fitted using the daily cumulative number of reported cases for Lagos, from March 16, 2020 to May 2, 2020 are presented. It is appropriate to note, except otherwise stated, that all fittings were done with δ = 0 and ε = 0 as explained in Section 4.3. Figure  2 shows that our model fitted well with the Lagos data (daily cumulative number of reported cases). The projection in Figure 3 reveals that, without enforcing the social distancing regulation and the use of face masks, when in public places, with the present case detection rates (for asymptomatic and asymptomatic infectious individuals), the daily cumulative number of reported cases may get up to 460,000 in 300 days (from March 16, 2020). As seen in Figure 4 , an increase in the detection rate for symptomatic humans resulted in an increase in the number of active and an expected decrease in the number of undetected symptomatic cases, with the peak period varying between 130 and 150 days (counting from March 16, 2020). The simulation results depicted in Figure 5 (a) shows two pandemic peaks: the first occuring between 120 and 150 days (counting from March 16, 2020), when the social distancing regulations and face-mask usage are not enforced in the population and the second occuring after 200 days (by the end of September, 2020), when 20% of the population observe only face-mask usage as a preventive measure. However, Figure 5 (b) shows that if 20% of the population complied with only the social distancing regulation from the beginning (starting from March 16, 2020), then we observe an infection peak after 200 days (by the end of September, 2020). Moreso, if 20% complied with both the social distancing regulations and face-mask usage, then infection peak occurs between 330 and 360 days (counting from March 16, 2020), as depicted by Figure 5(b) . It is important to note that infection peaks are lower and takes longer time to attain when social distancing regulation compliance and face-mask usage are combined than when they are not. Figure 6 (a) shows that if at least 55% of the population comply with the social distancing regulation with about 55% of the population effectively making use of face masks while in public, the reproduction number of the disease can be brought below 1, which indicates that the disease will eventually die out in the population. Figure 6 (b) reveals that, if we can step up the case detection rate for symptomatic individuals to about 0.8 per day, with about 55% of the population complying with the social distancing regulations, we can also get the reproduction number of the disease below 1, with a great decrease in the incidence (and prevalence) of COVID-19 guaranteed. In addition, Figure 7 (a) reveals that if the detection rate for symptomatic humans can be up to 0.5 per day, then if at least 55% of the population comply with the social distancing regulation while in public, the reproduction number of the disease can be brought below 1, indicating that the disease will eventually die out in the population. Figure 7 (b) reveal that, if we can step up the case detection rate for symptomatic individuals to about 0.8 per day, with about 0.5 case detection rate for asymptomatic individuals, we can equally bring the reproduction number of the disease below unity, with a great decrease in the incidence (and prevalence) of COVID-19 certain. In this section, simulations of the model when fitted using the daily cumulative number of active cases for Lagos, from March 16, 2020 to May 2, 2020 are presented. Please note, except otherwise stated, all fittings were done with δ = 0 and ε = 0 as explained in Section 4.3. Figure 8 shows that our model fitted well with the Lagos data (daily cumulative number of active cases). The projection in Figure 9 , when the cumulative number of active cases are used to fit the model, reveals also that, without enforcing the social distancing regulation and the use of face masks, when in public places, and with the present case detection rates (for asymptomatic and asymptomatic infectious individuals), the daily cumulative number of active cases may get up to 160,000 by mid August, 2020. It is observed in Figure 10 , that an increase in the detection rate for symptomatic humans led to an increase in the number of reported cases and an expected decrease in the number of undetected symptomatic cases, with the peak period varying between 130 and 150 days (counting from March 16, 2020), before the curve flattens. The simulation results depicted in Figure 11 (a) shows two pandemic peaks: the first occuring between 120 and 150 days (counting from March 16, 2020), when the social distancing regulations and face-mask usage are not enforced in the population and the second occuring after 200 days (by the end of September, 2020), when 20% of the population observe only face-mask usage as a control measure. However, Figure 11 (b) shows that if 20% of the population complied with only the social distancing regulation from the beginning (starting from March 16, 2020), then we observe an infection peak after 200 days (by the end of September, 2020). Moreso, if 20% complied with both the social distancing regulations and face-mask usage, then a lower infection peak occurs between 330 and 360 days (counting from March 16, 2020), as depicted by Figure 11 (b). It is interesting to note here, that the peak figures vary in comparison to when the cumulative number of reported cases are used to estimate the parameters and initial conditions. Figure 12 (a) reveals that if face-masks are effectively used by at least 55% of the population and at least 55% of the population observe strict social distancing rule in public then the disease will be effectively eliminated from the population. Figure 12(b) shows that, if we can step up the case detection rate for asymptomatic individuals to about 0.8 per day, with about 55% of the population complying with the social distancing regulations, we can also get the reproduction number of the disease below 1, with a great decrease in the incidence (and prevalence) of COVID-19 guaranteed. Figure 13 (a) reveals that, if we can step up the case detection rate for symptomatic individuals to about 0.9 per day, with about 50% of the population complying with the social distancing regulations, we can also get the reproduction number of the disease below unity. In addition, Figure 13 (a) reveals that if the detection rate for symptomatic humans can be up to 0.5 per day, then if at least 55% of the population comply with the social distancing regulation while in public, the reproduction number of the disease can be brought below 1, indicating that the disease will eventually die out in the population. Figure 13 (b) reveal that, if we can step up the case detection rate for symptomatic individuals to about 0.8 per day, with about 0.5 case detection rate for asymptomatic individuals, we can equally bring the reproduction number of the disease below unity, with a great decrease in the incidence (and prevalence) of COVID-19 certain. In this section, simulations of the model when fitted using the daily cumulative number of reported cases and the number of active cases for Lagos, from March 16, 2020 to May 2, 2020 are presented. It is imperative to note, except otherwise stated, that all fittings were done with δ = 0 and ε = 0 as explained in Section 4.3. As shown in Figures 14, our model fit well to the Lagos data, when the cumulative number of reported cases and the number of active cases are simulataneously used to fit the model. The cumulative number of reported cases and active cases are projected to attain their peak figures of 240,000 and 48,000, respectively, by mid July, 2020, when preventive measures like the social distancing rule and face-mask wearing in public are not strictly enforced ( Figure 15 ). It is observed from Figure 16 that cumulative number of reported and active cases increase steadily, while the undetected asymptomatic and symptomatic infectious cases reduce, as efforts are intensified to increase detection of new cases (via testing). Simulations of the total number of infected humans at different detection rates for asymptomatic and symptomatic infected are depicted in Figure 16 . It is observed that as detection rate for symptomatic infected is stepped up, more confirmed cases are identified and isolated for adequate treatment. In Figure 17 , different infection peaks are observed for reported and active COVID-19 cases, between 100 and 200 days (from March 16, 2020). The simulation results depicted in Figure 18 (a) shows two pandemic peaks: the first occuring after 150 days (by mid August, 2020), when the social distancing regulations and face-mask usage are not enforced in the population and the second occuring after 180 days (by mid September, 2020), when 20% of the population observe only face-mask usage as a preventive measure. However, Figure 18 (b) shows that if 20% of the population complied with only the social distancing regulation from the beginning (starting from March 16, 2020), then we observe an infection peak between 180 and 200 days (counting from March 16, 2020). Moreso, if 20% complied with both the social distancing regulations and face-mask usage, then infection peak occurs between 220 and 300 days (counting from March 16, 2020), as depicted by Figure 18(b) . It is observed that Infection peaks are lower than the peaks reported when social distancing regulations are not complied with. It is also imperative to note, that the peak figures vary in comparison to when the cumulative number of reported or active cases are used to estimate the parameters and initial conditions of the model. Figure 19 shows that if 60% of the population comply with the social distancing regulations and effectively use face-masks in public and detection rates stepped up to 0.9 per day, then we can get the reproduction number below unity and the burden of the disease brought very low. Similar conclusions are reached for the plots in Figure 20 . In this section, we want to demonstrate the fact that, while different data fitting runs, using the optimization routines implemented in this work, the significant difference in the values of the initial conditions (E(0), A(0) and I(0)) gave markedly different projections for the number of cumulative reported cases and the number of active cases, even when the different runs gave good model fits to the data being used. Please note, except otherwise stated, all fittings were done with δ = 0 and ε = 0 as explained in Section 4.3. Table 5 shows the values of the estimated parameters when the model was fitted using the cumulative number of reported cases. These estimates were from the six 'best fits' after multiple runs of both the GA and the fmincon optimization routines; all estimates resulted in a very good match between the data and the numbers from the simulated model. As seen from Figure 21 , as these parameter values (the estimated values in Table 5 ) are used to make future projections on the cumulative number of reported cases for 300 days, starting from March 16, 2020, we see the significant differences in the projections, which is due to the differences in the values of the estimated (unknown) initial number of exposed individuals (E(0)) and infectious individuals (asymptomatic and symptomatic, A(0) and I(0), respectively) in the population at the time the first reported case was announced on the start day of the simulation results herein.

Colour Code in Figure 21 β Table 5 : Estimated parameters when the model (2.2) was fitted using the cumulative number of reported cases. These estimates were from the six 'best fits'. Table 6 shows the values of the estimated parameters when the model was fitted using the number of active cases. These estimates were from the five 'best fits' after multiple runs of both the GA and the fmincon optimization routines; all estimates resulted in a very good match between the data and the numbers from the simulated model. These parameter values (the estimated values in Table 6 ) were used to make future projections on the number of active cases for 300 days, starting from March 16, 2020. Again, as seen from Figure 22 , just as was observed when we performed data fitting using the cumulative number of reported cases (Table 5 and Figure 21) , we see the significant differences in the projections, which is due to the differences in the values of the estimated (unknown) initial number of exposed individuals (E(0)) and infectious individuals (asymptomatic and symptomatic, A(0) and I(0), Figure 21 β Table 6 : Estimated parameters when the model (2.2) was fitted using the number of active cases. These estimates were from the five 'best fits'. Table 7 shows the values of the estimated parameters when the model was fitted using the cumulative number of reported cases and the number of active cases. These estimates were from the seven 2) was fitted using the cumulative number of reported cases. The dark, red circles (at the beginning of the simulation) represents the actual data. These estimates were from the six 'best fits'. See Table 5   0  50  100  150  200  250 The dark, red circles (at the beginning of the simulation) represents the actual data. These estimates were from the five 'best fits'.

'best fits' after multiple runs of both the GA and the fmincon optimization routines; all estimates resulted in a very good match between the data and the numbers from the simulated model. These parameter values (the estimated values in Table 7 ) were used to make future projections on the number of active cases for 300 days, starting from March 16, 2020. Again, as seen from Figure 23 , just as was observed in the two previous cases above (in this subsection), we see the significant differences in the projections, which is due to the differences in the values of the estimated (unknown) initial number of exposed individuals (E(0)) and infectious individuals (asymptomatic and symptomatic, A(0) and I(0), Figure 21 β c θ ψ E(0) A(0) I(0 Table 7 : Estimated parameters when the model (2.2) was fitted using the cumulative number of reported cases and the number of active cases. These estimates were from the seven 'best fits'.

It is important to state that we picked the 'best fit' with the lowest figures (peaks and cumulative values) for the numerical simulations and parameter estimations used in 2) was fitted using the cumulative number of reported cases and the number of active cases. The dark, red circles (at the beginning of the simulation) represents the actual data. These estimates were from the seven 'best fits'.

Based on the model fitting that was carried out in this work, with three sets of data, we observe an approximately similar estimates for the β c , ψ, θ and the control reproduction number R c but some significant variations for the initial values of individuals in the E, A and I classes. Figures 2, 8 and 14 showed that our model fits well to the data, irrespective of the data set used. However, we observed that the significant difference in the values of the initial conditions (E(0), A(0) and I(0)) gave markedly different projections for the number of cumulative reported cases and the number of active cases, even when the different runs gave good model fits to the data being used (as shown in Figures 3, 9 and 15 ). This strongly reveals that our lack of knowledge and understanding of how long community transmission may have been going on in the population (as at the time the first (index) case was announced on March 16, 2020) could hinder our knowledge of the actual burden of COVID-19 in Lagos. Hence, very strict measures must be taken to identify other cases as quickly as possible, through aggressive screening and testing of the population, especially for asymptomatic cases, and strict enforcement of the other control measures.

This work has revealed the impact of different non-pharmaceutical control measures on the population dynamics of the novel coronavirus disease 2019 in Lagos, Nigeria, using an appropriately formulated mathematical model. Using the available data, since its first reported case on 16 March 2020, we developed a predicative tool for the cumulative number of reported cases and the number of active cases in Lagos; we also estimated the basic reproduction number of the disease outbreak in the aforementioned State in Nigeria. Using numerical simulations, we showed the effect of control measures, specifically the common social distancing, use of face mask and case detection (via contact tracing and subsequent testings) on the dynamics of COVID-19. We also provided forecasts for the cumulative number of reported cases and active cases for different levels of the control measures being implemented. Numerical simulations of the model show that if at least 55% of the population comply with the social distancing regulation with about 55% of the population effectively making use of face masks while in public, the disease will eventually die out in the population and that, if we can step up the case detection rate for symptomatic individuals to about 0.8 per day, with about 55% of the population complying with the social distancing regulations, it will lead to a great decrease in the incidence (and prevalence) of COVID-19.

Therefore, to curtail the spread of COVID-19 at the community level, this study recommends, as a matter of urgency, very strict measures to be taken by policy makers and those in authority to identify new cases, through aggressive screening and testing of the population and strict enforcement of the use of face-masks and the social distancing regulations.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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