Modelling the impact of detection on COVID-19 transmission dynamics in Ghana

ABSTRACT Recently, due to the global increase in new cases of infections, the World Health Organisation declared COVID-19 disease a pandemic. We present a deterministic model to investigate the impact of detection of infected individuals on the transmission dynamics of COVID-19. The model is extended to capture the role of immigration, governmental interventions and public perception of risk regarding the number of critical cases and deaths. The model is fitted to the currently available data on cumulative COVID-19 cases in Ghana and trends of outcomes quantitatively estimated. Results suggest that intervention in the form of lockdown lengthens the period of reaching the peak of infection, thereby giving time for policymaking and management of pandemic. Simulation results suggest that detection of exposed individuals has great potential to reduce daily detected cases and flatten the cumulative curve during the early stages of the pandemic. Thus, more effort should be targeted towards increasing contact tracing and detection of those suspected to be exposed to COVID-19 in order to curtail the spread of the disease. Results from this study would be useful in informing government policy direction and management regarding the control of COVID-19 in Ghana and other countries. ABSTRACT


Introduction
The novel coronavirus COVID-19, also named the severe acute respiratory syndrome coronavirus , is one of the many coronaviruses known to exist (Ndairou et al., 2020). COVID-19 is believed to have originated from Wuhan in China in early December 2019 (World Health Organization et al.). It is also believed to have been found in seafood, and the earliest patients of COVID-19 reported access to seafood in Wuhan. As of 25 May 2020, COVID-19 had affected 213 countries globally with 5; 165; 481 confirmed cases and 336; 430 deaths recorded (Worldometer, 2020). Africa accounted for 107; 747 confirmed cases of the recorded global cases with 3; 257 deaths and 42; 924 recoveries. South Africa has been the most hit with 22; 583 confirmed cases and 429 deaths as of 25 May 2020 (Worldometer, 2020).
Most of the first reported cases in Africa have been attributed mainly to international travel. This has resulted in most governments enforcing several intervention measures meant to curb the further spread of COVID-19. These measures included lockdowns, testing and screening individuals, recommending social distancing and wearing of face masks and use of hand sanitizers. On 12 March 2020, Ghana recorded its first two COVID-19 cases and noted a rapid increase in the daily reported number of cases. In response, the government of Ghana implemented a 21-day lockdown that started on 30 March 2020 and ended on 19 April 2020. The Ghananian government banned all public gatherings, closed all schools, universities and borders. It also encouraged strict hygiene practices on businesses that were allowed to operate. Furthermore, the government set up a COVID-19 response committee responsible for enforcing measures to reduce the spread of infection, contain the spread of infection within the country, provide enough care to those infected and mitigate the impact of the virus on the social and economic life (Danquah & Schotte, 2020).
Mathematical modelling provides a framework for finding solutions to emerging and re-emerging viruses such as COVID-19. For instance, Ivorra et al. (Ivorra et al., 2020) developed a θ-SEIHRD mathematical model describing COVID-19 dynamics in China. The model took into account undetected COVID-19 cases and results from this study established that enhancing detection of COVID-19 cases in China could be beneficial in ascertaining the number of hospital beds required for managing critical cases (Ivorra et al., 2020). Chen et al. (Chen & Yu, 2020) developed a second derivative model for assessing the detection rate of COVID-19 during a certain 2-month period. It was established that as soon as an outbreak of a disease or infection is detected it is important to closely monitor and assess the responses of individuals to the interventions put in place so as to reduce the spread of infection. Lin et al. (Lin et al.) developed a preliminary conceptual model taking into account the public perception of the disease together with governmental action effects to assess the spread of COVID-19 disease in Wuhan City of China.
In this study, we extend the model developed for the COVID-19 outbreak in Wuhan by Lin et al. (Lin et al.) to incorporate the class of detected and quarantined infected individuals so as to capture the disease dynamics in Ghana. The study seeks to investigate the impact of undetected cases on the COVID-19 disease spread in Ghana since low coverage of testing has been linked to the continued spread of the disease (Kim et al.). We thus capture the impact of varying detection rates on COVID-19 dynamics and propose a six-state SEIQRD compartmental model for the spread of COVID-19 in Ghana. Results from this study will be of great help in the fight against COVID-19 in Ghana as well as other countries.
The paper is arranged as follows; in Section 2, we formulate and establish the basic properties of the model. The model is analysed for stability in Section 3. In Section 4, we carry out some numerical simulations. Parameter estimation and numerical results are also presented in this section. The paper is concluded in Section 5.

Model formulation
The model developed in this paper comprises infection dynamics amongst humans only. Thus, the human population comprises of the following distinct compartments: SðtÞ, EðtÞ, IðtÞ, QðtÞ, RðtÞ and DðtÞ. The class SðtÞ represents individuals susceptible to COVID-19 infection, EðtÞ represents the exposed individuals, IðtÞ represents the infected individuals, QðtÞ represents the detected and quarantined individuals, RðtÞ represents recovered individuals and DðtÞ represents deceased individuals. Upon being infected susceptible individuals join the class of exposed individuals at a rate given by (2:1) The parameter η accounts for the relative infectivity of individuals in class QðtÞ as compared to individuals in class IðtÞ. Assuming that the infectivity rate of individuals in IðtÞ is higher than that of individuals in QðtÞ, it follows that 0 < η < 1. This is due to the fact that these individuals reduce their contacts either through selfisolation measures or due to hospitalization these individuals will have contact only with healthcare personnel who in most cases have protective gear. Exposed individuals join the class I of infectives after completing their incubation for an average period of γ À 1 days. We consider the class IðtÞ to include pre-symptomatic undetected individuals, asymptomatic undetected individuals and symptomatic (mild and severe) undetected individuals. Once individuals are in class IðtÞ, they can either recover naturally at a rate given by ρ 1 to join the class RðtÞ, experience disease-related death at a rate given by δ 1 to join the class DðtÞ of the deceased or are detected and quarantined to join the class QðtÞ. In this paper, we assume that individuals in class E who are suspected to have been exposed to COVID-19 are detected at a rate given by σ 1 joining the class Q whereas individuals in class I are detected at a rate given by σ 2 to join the class Q. Upon entering the class QðtÞ individuals will either recover at a rate given by ρ 2 or can experience disease-related death at a rate given by δ 2 to join the class DðtÞ of the deceased. We assume that people who recover do not go back to the susceptible class. The total human population is thus given by The flow diagram for the COVID-19 model is given in Figure 1. The description of model variables, parameters and assumptions combined with the model flow diagram (Figure 1) leads to the following set of nonlinear ordinary differential equations: where all model parameters are assumed to be positive.

Positivity of solutions
We consider the positivity of system (2.2). We prove that all the state variables remain non-negative and the solutions of system (2.2) with positive initial conditions will remain non-negative for all t > 0. We state the following.
Thus t > 0 and it follows from the first equation of system (2.2) that giving From the second equation of system (2.2), we have In a similar way, it can be shown that IðtÞ > 0, QðtÞ > 0,RðtÞ > 0, DðtÞ > 0 for all t > 0 and this completes the proof.

Invariant region
Adding the first five equations of (2.2) gives dN dt ¼ 0: Thus we have the feasible region for system (2.2) defined by It is easy to verify that the region Ω is positively invariant with respect to system (2.2).

The disease-free equilibrium point and the basic reproductive number
The model has a disease-free equilibrium resembling a scenario without disease in the community. The basic reproduction number is defined as the average number of new infections produced by a single infectious individual in a completely susceptible population over the duration of the infectious period. Usually denoted R 0 , the basic reproductive number tells whether the disease under study will persist or die out. In general, for values of R 0 < 1, the disease will die out and if R 0 > 1 the disease will persist in the community. Following the next-generation matrix approach by Van den Driessche and Watmough (Van den Driessche & Watmough, 2002) we have (3:1) R 1 indicates that a fraction γ ðγþσ 1 Þ of exposed individuals progress to the infective class and will spend an average time of 1 ðσ 2 þδ 1 þρ 1 Þ , the contact rate is β; R 2 indicates that a fraction σ 1 ðσ 2 þδ 1 þρ 1 Þ , from the exposed class will progress to the quarantine class and spend an average time of 1 ðσ 2 þρ 2 Þ , the contact rate is βη and finally R 3 indicates that a fraction σ 2 σ 2 þδ 1 þρ 1 Þ of infected individuals will progress to the quarantine class and spend an average time of 1 ðσ 2 þρ 2 Þ , and the contact rate is of infection is βη.

A case study for COVID-19 spread in Ghana
In this section, we perform numerical simulations of system (1). We model the COVID-19 outbreak in Ghana, which started on 12 March 2020 as officially confirmed by the government of Ghana. We include the governmental action of lockdown as a stepwise function that takes zero before lockdown and nonzero for different lockdown stages. We adopt the transmission rate given in Lin et al. (Lin et al.) that includes both the impact of governmental actions (such as lockdown, wearing of face masks, encouraging hygienic practices, etc.) and the public perception (represented by PðtÞ) of risk regarding the number of critical cases and deaths. Thus, the transmission rate is given as follows: (4:1) The term ð1 À P=NÞ κ captures the effects of public perception of the risk to contract the disease based on severe cases reported. Here, κ is a parameter controlling the strength of the response. β 0 is the baseline transmission rate and α is the efficacy of "governmental actions". A value of α close to one implies high efficacy and the reverse is true for values of α close to zero. Thus, we now have the following set of nonlinear ordinary differential equations: (4:2) where ω À 1 is the mean duration of public reaction. The flow diagram for the extended COVID-19 model is now given in Figure 2.
After the official confirmation of COVID-19 outbreak in Ghana on 12 March 2020, the initial lockdown was instituted starting on Monday 30 March 2020 and ended on 19 April 2020. From then onwards the lockdown was relaxed but social distancing and wearing of face masks continue to be enforced by the government. We include the governmental action of lockdown as a stepwise function which takes zero before lockdown and non-zero for different lockdown stages. We also capture immigration of people before the lockdown where we assume that these people might have been susceptible or exposed to COVID-19 upon their entering into the country. Due to the travel restrictions imposed when the lockdown started, we assume that there was no more immigration of people after 30 March 2020. Thus, we set Λ�0 before the lockdown; α ¼ 0 before the lockdown; The COVID-19 data for Ghana used in this study was obtained from the reports presented by the government of Ghana Figure 3 and reported by Worldometer, 2020). We perform curve fitting using the data given in Table 1.
Estimates of unknown parameter values and intervals used are shown in Table 2. Curve fitting process allows us to quantitatively estimate the trend of the outcomes of COVID-19 in Ghana. We make use of the leastsquares curve fit routine (lsqcurvefit) in Matlab with optimization to estimate our unknown model parameters. Cumulative cases for COVID-19 in Ghana are estimated using the function where t kÀ 1 1 and t k denote the start and end of the time interval, respectively. We fit the model to the cumulative cases of COVID-19 in Ghana before, during and after the lockdown. In order to obtain the optimal fit under this combined scenario, we set the conditions as given in (4.3). We use the optimal value for α during the lockdown to measure the potential impact of lockdown in reducing the peak of cumulative cases. Figure 4 illustrates the trends in the cumulative COVID-19-detected cases in Ghana. We observe from Figure 4 that system (4.2) fits well with the data from Table 1. Estimated parameter values are shown in Table 2.
The baseline value of the basic reproduction number R 0 ¼ 2:0497.
To explore the impact of immigration, governmental intervention and public perception of risk regarding the number of COVID-19 critical cases and deaths in Ghana, we will simulate model (4.2) using parameter Pð0Þ ¼ 20 with Λ and α set as given in (4.3).  Figures 5-8 illustrate the simulation of cumulative cases in the absence and presence of 21-day lockdown. Figure 6 clearly shows the scenario that had there not been any intervention instituted then the number of cumulative COVID-19 cases might have reached close to 22000 cases by 21 May 2020. We observe from Figure 8 that in the absence of any form of intervention the curve will flatten approximately at the end of April 2021 whereas in the presence of intervention the curve will flatten at around the end of July 2021. It should be noted that the form of intervention considered is in the form of a 21-day nationwide lockdown. Figure 8 indicates that the lockdown lengthens the period of reaching peak infections thereby giving time for policymaking and management of the pandemic. Figures 9 and 10 illustrate the simulation of daily detected cases in the absence and presence of 21-day lockdown assuming a constant detection rate for the entire time period under consideration. Figure 9 indicates the daily detected cases for a shorter time period whilst Figure 10 is projected for a longer period of time. Figure 10 shows that in the absence of intervention, the peak daily detected cases may reach approximately 12500 around October 2020 whereas in the presence of intervention the peak daily detected cases may reach approximately 11500 around February 2021. Figures 11 and 12 illustrate the effect of varying the parameters σ 1 and σ 2 , respectively, on the number of daily detected cases. Figure 11 illustrates how an increase of σ 1 by levels of 10% and 20% can influence the number of daily detected cases in the presence of a 21-day lockdown. We observe that if the detection rate had been at levels 10% or 20% higher than the current rate then the daily reported cases would have dramatically reduced to less than 50 cases by 21 May 2020. As can be seen, detection of exposed individuals has more impact as compared to detection of infectious individuals. This emphasizes the need for more contact tracing of those suspected to have been exposed to COVID-19 disease so as to reduce the spread of the disease. This is a reflection that individuals in class E are the main drivers of the pandemic. Thus, more effort should be directed towards increasing efficient test kits and contact tracing of exposed individuals so as to reduce the number of detected cases.  Figure 13 shows that a 10% increase in the level of σ 1 flattens the curve on approximately 21 May 2020 with close to about 11000 cases whereas a 20% increase in the level of σ 1 flattens the curve on approximately 5 May 2020 with close to about 10500 cases. As observed in Figure 8, the current level of σ 1 will flatten the curve at approximately the end of July 2021. Figures 14-16 show that the parameter σ 2 has less impact on the number of daily detected cases and cumulative detective cases as compared to σ 1 . As can be observed in Figures 14-16, σ 2 is observed to impact in the later stages of the pandemic unlike σ 1 which has greater impact in the earlier stages of the pandemic.

Conclusion
In this paper, a deterministic model to investigate the impact of detection of exposed and infected individuals on the transmission dynamics of COVID-19 in Ghana was formulated.  Figure 11. Effect of increasing the current detection rate σ 1 by a level of up to 20% on the number of daily detected cases in the presence of 21-day lockdown. study of the disease spread in Ghana. The least-squares curve fit routine is used to fit the model to the available cumulative COVID-19 cases in Ghana and trends of outcomes quantitatively estimated. Estimates of important parameters were obtained within plausible ranges. Different case scenarios on the impact of the 21-day lockdown were analysed by considering a step-wise function, which takes zero before lockdown and non-zero for different lockdown stages. Using estimated and known parameter values of the model, we performed numerical simulations for cumulative and daily detected cases of COVID-19 in Ghana in the absence and presence of lockdown. Results from this study confirmed with the existing data from Ghana would be useful in informing government policy direction and management regarding the control of COVID-19 in Ghana and other countries. The model can be extended to incorporate asymptomatic infectious, mildly infected and severely infected individuals so as to specifically capture the contributions of these individuals to the disease dynamics.