key: cord-1006972-zccp5q72
authors: Dwomoh, Duah; Iddi, Samuel; Adu, Bright; Aheto, Justice Moses; Sedzro, Kojo Mensah; Fobil, Julius; Bosomprah, Samuel
title: Mathematical modeling of COVID-19 infection dynamics in Ghana: Impact evaluation of integrated government and individual level interventions
date: 2021-01-22
journal: Infect Dis Model
DOI: 10.1016/j.idm.2021.01.008
sha: 98fd629153d6ab6ff6ca5a1ab162a29de0de3339
doc_id: 1006972
cord_uid: zccp5q72

The raging COVID-19 pandemic is arguably the most important threat to global health presently. There is currently no vaccine or therapeutics and several interventions, mostly preventive measures have been proposed to reduce the spread of infection but the efficacy of these interventions, and their likely impact on the number of COVID-19 infections is unknown. In this study, we proposed the SEIQHRS model (susceptible-exposed-infectious-quarantine-hospitalized-recovered-susceptible) model that predicts the trajectory of the epidemic to help plan an effective control strategy for COVID-19 in Ghana. We provided a short-term forecast of the early phase of the epidemic trajectory in Ghana using the generalized growth model. We estimated the effective basic Reproductive number Re in real-time using three different estimation procedures and simulated worse case epidemic scenarios and the impact of integrated individual and government interventions on the epidemic in the long term using compartmental models. The maximum likelihood estimates of Re and the corresponding 95% confidence interval was 2.04 [95% CI: 1.82–2.27; 12th March-7th April 2020]. The Re estimate using the exponential growth method was 2.11 [95% CI: 2.00–2.24] within the same period. The Re estimate using time-dependent (TD) method showed a gradual decline of the basic Reproductive number since March 12, 2020 when the first 2 index cases were recorded but the rate of transmission remains high (TD: Re = 2.52; 95% CI: [1.87–3.49]). The current estimate of Re based on the TD method is 1.74 [95% CI: 1.41–2.10; (13th May 2020)] but with comprehensive integrated government and individual level interventions, the Re could reduce to 0.5 which is an indication of the epidemic dying out in the general population. Our results showed that enhanced government and individual-level interventions and the intensity of media coverage could have a substantial effect on suppressing transmission of new COVID-19 cases and reduced death rates in Ghana until such a time that a potent vaccine or drug is discovered.

COVID-19 pandemic remains a major public health concern globally as the number of new cases and case fatalities keep increasing every day. As of 16 th May 2020 at 16.13GMT, there were 4,628,879 cases and 308,655 deaths with 1,759,729 recoveries confirmed worldwide (Worldometer, 2020) . The global spread has been rapid, with 213 Countries and Territories around the world now having reported at least one case (Worldometer, 2020) . In Ghana, the first two index cases were recorded on the 12 th March 2020 and by 13 th May of 2020, the total number of cases was 4,120 including 320 recoveries and 18 deaths 1 .

National governments have instituted measures in response to the pandemic. Such measures included lockdowns (partial or full) and enhanced safety protocols to control transmission. As part of an integrated intervention, the Government of Ghana adapted a partial lockdown in the two most affected cities, Greater Accra and Greater Kumasi for two weeks, which was extended for a further one week. This was concurrence with individual control measures (social distancing, enhanced contact tracing, handwashing and the use of hand sanitizers, and intensive media education), and government interventions (closure of schools, provision of personal protective equipment (PPE) to health workers, quarantine, self-isolation, hospitalization, and treatment). All the aforementioned interventions have been implemented in Ghana with the aim to prevent the spread of COVID-19 but no studies to date have evaluated its long term impact on the expected number of cases and deaths to inform policy decision.

The global nature of this pandemic with countries at difference levels of the epidemic curves, suggests that the crisis could be long. Clearly, enonomies cannot immediately return to how things were prior to the pandemic. But it is equally clear that the lockdown cannot persist indefinitely, because of its damaging impact on the economy and social life. We need to adapt and minimise the harm it can do, and continue to suppress the spread of the virus while ensuring that children are educated, that businesses can reopen, and that society can function. This will require continuous monitoring of transmission so that remedial measures can be re-instituted should there be a resurgence.

Measurement of transmission intensity is important to describe the intensity of the disease outbreak, to assess the impact of public health measures on transmission, to understand epidemiological patterns, and for planning intervention studies. The single most important measure of transmission intensity is the basic 1 https://ghanahealthservice.org/covid19/archive.php.

Reproductive number, . The describes the number of cases, on average, an infected person will cause during their infectious period. To contain the virus, the must be kept below 1, and this means minimising the risk of spreading the virus at every turn. When is greater than 1, the virus will spread. It is equally important to bring down the number of new cases per day. Lowering the number of new cases per day as much as possible, together with an below 1, will mean fewer cases and reduced harm.

The for COVID-19 was recently estimated in the South Korea (Choi, Ki, & Health, 2020) , China and other countries, but limited evidence supports the applicability of outside the region for which the value was estimated (Ridenhour, Kowalik, & Shay, 2018) . Since is a function of effective contact rate, any factor having the potential to influence the contact rate, including individual behavioral changes, other human social behavior, and organization, including population density (e.g., rural vs. urban), social organization (e.g., integrated vs. segregated), as well as the innate biological characteristics of particular pathogens, and seasonality (e.g., wet vs. rainy season for vector-borne infections), will ultimately affect (Delamater, Street, Leslie, Yang, & Jacobsen, 2019) .

In this study, we described the epidermic dynamics and estimated the for COVID-19 in an entirely susceptible Ghanaian population using a mathematical model. We provided estimates of the initial basic

Reproductive number (at the beginning of the outbreak) and the time-dependent Reproductive number at any time during the outbreak using four different estimation procedures: next-generation matrix (Odo Diekmann, Heesterbeek, & Roberts, 2010) , exponential growth method (Wallinga & Lipsitch, 2007) , maximum likelihood (White et al., 2009) , and time-dependent Reproductive number method (Wallinga and Teunis (2004) ). We also assessed the effects of individual and government level integrated interventions on cumulative incidence and new cases per day in Ghana.

J o u r n a l P r e -p r o o f 5 2. Methods

The outbreak data for Ghana were cumulative numbers of confirmed cases and deaths obtained from the Ghana Health Service (GHS). The data from the GHS is a daily time series data of reported COVID-19

cases. These dataset do not include individual patient-level information, but aggregate number of total suspected and confirmed cases tested at the two main laboratory facilities at the University of Ghana 

Although researchers do not fully understand the transmission dynamics of COVID-19, we formulated the SEIQHRS model ( Figure 1 ) based on the available information and the basic understanding of the biological mechanism through which infection spreads in the population. We partitioned the Ghanaian population into five distinct classes ( =susceptible, =Exposed, =Infectious, =Selfisolation/Quarantined, =Hospitalized, =Recovery). We defined susceptible populations as all uninfected individuals at time but there is the likelihood there may be infected at time + where is the incremental continuous-time point. We included newborn babies from all compartments into the susceptible class since there is not enough evidence on vertical transmission of the virus from the pregnant woman to the unborn baby. After successful transmission of COVID, the susceptible individuals S enter the exposed class where they are infected but not yet infectious to others. They stay in the exposed class before they become infectious and move to the infectious class or identified through contact tracing and quarantined at class . The hospitalized class is the proportion of the population who have been infected but have been identified through contact tracing or self-report to health facilities and hospitalized.

We specified three modes of recovery based on the current understanding of the transmission dynamics.

First, a fraction of the population who are exposed may recover from the infection without necessarily J o u r n a l P r e -p r o o f 6 infecting others and therefore do not transit through the infectious category but enter directly into the recovery class because of boosted immunity. Second, those who become infectious may also recover and move to the recovery class (R) or die from COVID-19. Finally, those in the quarantine-based hospitals and those in self-isolation may also recover and join the recovery class. Individuals who live in Ghana and those arriving from other countries (Ghanaian citizens and non-citizens) who have not yet contracted COVID-19 were classified as susceptible. Thus, the total population size, , is = + + + + + .

The experience we gather from the contact tracing being carried out by the Government of Ghana and commentary from some infectious disease experts globally showed that some individuals have come into contact with SARS-CoV-2 positive patients but were tested negative for SARS-CoV-2. We therefore assume that although people might have the infection, they may not instantaneously be infectious to others. As such we have specified and distinguished the exposed or the latent individuals from those who are infectious to others based on the traditional definition of an exposed person in infectious disease modeling. We agreed that there may be some exposed individuals capable of transmitting the infection but their overall impact of the dynamics of infection is assumed to be negligible in this study. Furthermore, studies have shown that individuals who recover from viral diseases usually would have either long-life immunity, short term immunity, or in some cases there may be no immunity at all (Keeling & Rohani, 2011 ). Since enough is not known about COVID-19, this study assumed that individuals who recover from COVID-19 may have short term immunity but once the immunity wanes; these individuals will become susceptible to the same or a different viral strain of COVID-19 (Lan et al., 2020) . For the purpose of assessing government intervention, we define the quarantine-based hospitalized class to consists of all exposed and infectious individuals who were identified through contact tracing, isolation, and hospitalization. We further assumed that individuals leave the hospitalize class only after they have fully recovered from COVID-19 or when they suffer from COVID-19 disease-induced death or die naturally. 

To provide a detailed understanding of the transmission of COVID-19 and the control strategies required to mitigate its spread and relative impact on the Ghanaian population and examine the epidemiology of the infection, we detailed the parameterization of the SEIQHRS model from which useful public health guidance can be generated (Table 1 ). Our SEIQHRS model has eleven parameters: the birth rate ( ), natural death rate ( , rate of transmission from being exposed to becoming infectious within the infectious community ( ). The rate of transmission from the quarantine class is and that of transmission from the hospitalized class is denoted by ! . " = # $ %×'×( is the time-dependent integrated individual-level intervention and the intensity of media coverage which is parameterized by two non-negative rates ) and * = +, , ,!,-., +, , ,!,-. * is the proportion of COVID-19 positive cases and deaths in the population. ) is the rate of intensity of adherence to prevention guidelines by the individual and media coverage, We assume that the transmission rate also decreases with time as suggested by Chowell, Hengartner, Castillo-Chavez, Fenimore, and Hyman (2004) , and this assumption also seems plausible because of the integrated interventions put out by the Government of Ghana. This assumption is similar to what was reported during the SARS epidemic of 2002-2003 where many people began to wear face masks to try to prevent disease spread through contact with airborne particles (Riley et al., 2003) .

The parameter, " ∈ [0, 1] basically models the impact of behavioral change on reducing the effective reproduction number, cumulative number of cases and deaths similar to previous studies that model the impact of behavioral change as a result of intensive media education and increased rate social distancing based on the change in the daily cumulated number of cases and deaths (Shen et al., 2015; Tuncer et al., 2018) . " = 0 indicates perfect adherence to COVID-19 prevention protocols (positive behavioural change) and 5 represents the rate of recovery among the infectious class, 6 is COVID-19 induced death rate, and 7 is the rate of COVID-19 waning immunity (rate of moving from recovered class to the susceptible class). The rate at which an exposed individual recovers without necessarily becoming infectious to others is denoted by 8. The rate at which infectious and exposed individuals are removed from the exposed and infectious classes respectively and quarantine is 9 and : respectively. The community infectious and quarantine classes are hospitalized at a rate of ; and < respectively. The quarantined and hospitalized individuals recover at a rate of = and respectively.

J o u r n a l P r e -p r o o f 8 Furthermore, in determining the effective contact rate and the duration of infectiousness, this study assumed homogenous population mixing and contact patterns where all population members are equally likely to come into contact with one another because of the typical Ghanaian culture, social attributes, and settings. Admittedly, heterogeneous mixing, where there is possible variation in contact patterns among age subgroups, social class, religious affiliations, or geographic regions would have been appropriate but data available to estimate the parameter of interest for these different subgroups of the population would have been near impossible. Furthermore, we assumed that the rate at which susceptible individuals are quarantined is the same as the rate at which susceptible individuals will return to the susceptible class after laboratory results have proved negative. We assumed unstructured populations, that is, we did not account for age or sex structure and therefore assumed a fixed value for the transmission rate using available data from the cumulative number of cases. However, our model can be adopted to include difference sociodemograhic/economic classes, which would require splitting each variable of the model into specific classes if data are available.

The following system of differential equations (1.1-1.6) describes the transmission dynamics of COVID-19 in Ghana. This system of ODEs was solved numerically via the standard Runge-Kutta method using the function ode from the package deSolve (Soetaert, Petzoldt, & Setzer, 2010) . The study only considered constant-valued parameters with the exception of the proportion of the population adhering to the integrated government interventions.

Limited data and lack of clear understanding of the etiology of the disease spread place severe restrictions on the attempt to model the dynamics of COVID-19. Nonetheless, this study relied on previous studies and the ongoing discussions from experts at the Ghana Health Service and around the world to initialize the model parameters. The transmission rate in the community was estimated from the observed number of cases using the non-linear least square estimation via Levenberg-Marquardt algorithm that minimizes the sum of squared error between the observed cumulative number of cases and predicted cumulative cases. Other parameters were carefully chosen to fit our definition of worst-case scenario. The systems of differential equations were analysed using the "desolve" package in R (Soetaert et al., 2010) and the estimation of # using the method of maximimum likelihood, time-dependent procedure and the exponential growth methods were done using the " " package in (Obadia, Haneef, Boëlle, & making, 2012) . The estimate using the exponential growth and the maximum likelihood methods is dependent on the choice of generation time and therefore sensitivity analysis was conducted to determine the how changes with generation time for exponential growth and the maximum likelihood methods.

J o u r n a l P r e -p r o o f The rate at which the individual in the susceptible, exposed, infectious, and recover classes suffer natural death. In this study, also represents the population crude birth rate. The rate at which an exposed person becomes infectious. 1 N g represents the average duration of the exposed or latent period (that is the average amount of time spent in the class).

The rate of recovery of an infectious person who lives in the community. The rate at which an exposed person is quarantined

The rate at which an exposed person recovers without necessarily becoming infectious or being quarantined-that is the rate of self-recovery from COVID-19 without recover without medical intervention-Self immunity boosting

√ : Parameter used in the projection; ×∶ ∶ ∶ ∶ Parameter set to zero; †: Not used 2.5 Estimation of the basic Reproductive number

We derived the effective basic Reproductive from our dynamic system in model 1.1-1.6. We used the next-generation matrix u proposed by Diekmann, Heesterbeek, and Metz (1990) ; (Diekmann et al., 2010) to estimate the basic reproductive number which is the dominant eigenvalue of u. To compute , we created a sub-model that only considers the 'disease' compartments, a subset of the equations in the SEIQHRS model. The disease compartments are those that include individuals that are in any stage of infection which, for the SEIQHRS model, includes both the exposed, infectious, and quarantine individuals. We only used the latent class as our transmission state of our dynamic models as that was the only state that described the production of new COVID-19 infections and used both the latent, infectious, quarantine and hospitalized classes to represent the transition states; that is changes in the state among 

, w = " w c w ‚ w ƒ w i … = † ‡ ‡ − N + 8 + + : N − 9 + 5 + + 6 + ;

:

− jr + o™ + or + oer + rq + rš š + n + p o + oe + š + j + q ™ + s + š + r −o š + n + p o + oe + š + j + q −• š + n + p › − jrp + ™nq + o™p + orp + oerp + ™qp + rqp + š™q + šrp l + š + j š + n + p ™ + s + š + r o + oe + š + j + q − nq + op + qp + šq l + š + j š + n + p o + oe + š + j + q −p l + š + j š + n + p −

The dominant eigenvalue of the Next Generation Matrix u = −vw $c is equal to which we obtained by solving = " Κ . " is the spectral radius (i.e the eigenvalue with the largest magnitude) of Κ. We solved |u − | = 0 to derive the basic reproductive number. The effective reproductive number is given by = ™ ™ + s + š + r o + oe + š + j + q + jr + o™ + or + oer + rq + rš š + n + p 9 + 5 + + 6 + ; ™ + s + š + r + jrp + ™nq + o™p + orp + oerp + ™qp + rqp + š™q + šrp l + š + j š + n + p ™ + s + š + r o + oe + š + j + q

= ™ ™ + s + š + r o + oe + š + j + q +

x r š + n + p ™ + s + š + r + o™ š + n + p 9 + 5 + + 6 + ; ™ + s + š + r • + x rp l + š + j š + n + p ™ + s + š + r + ™q l + š + j ™ + s + š + r o + oe + š + j + q + o™p l + š + j š + n + p ™ + s + š + r o + oe + š + j + q • ¡ ¢ = ¡ ¢ + ¡ ¢ + ¡ ¢ is the contribution of infectious individuals in the general population, quarantine class, and hospital infectives to the overall . 

We also estimated the effective basic reproductive number , and the generation time using the maximum likelihood-based method proposed by Forsberg White and Pagano (2008) . The generation time is the time lag between infection in a primary case and a secondary case. The estimates were based on readily available daily reported case data. The estimation of R , and the generation time for COVID-19 based on the maximum likelihood is important since the estimate can be obtained in real-time to quantify the impact of government intervention and inform an appropriate public health response to the COVID-19. . We imputed unobserved cases at the beginning of the epidemic to correct for missing data.

This study employed the exponential growth method proposed by Wallinga and Lipsitch (2007) 

The study also estimated time-varying basic reproductive numbers for COVID-19 using the technique proposed by Wallinga and Teunis (2004) . 

We employed a two-parameter generalized-growth model (GGM) (Viboud, Simonsen, & Chowell, 2016) to is È , that is, the cumulative number of cases per unit time (days). We estimated both ) and ‰ using the non-linear least square estimation procedure in Stata 16 (StataCorp, College Station, TX, USA). This was achieved by fitting the model to cumulative time series data between 12 March 2020 to 7 th April 2020. The estimated parameters of the model were then used to forecast the early growth of the epidemic in Ghana.

This study developed and utilized a mathematical model that incorporates intervention strategies for COVID-19. Although there is no vaccine yet for COVID-19, the Government of Ghana and other countries have initiated several interventions to reduce new infections. Our study simulated two modes of interventions. The first is the individual level integrated intervention which includes social distancing, use of face mask, use of hand sanitizers, self-confinement, and use of running water with soap to wash hands.

The integrated government interventions include the provision of personal protective equipment, contact tracing, self-isolation based on contact tracing of an index positive case, quarantining, and the intensity of media coverage in selected areas. Key among these interventions in Ghana was partial lockdown placed J o u r n a l P r e -p r o o f within the Greater Accra and the Greater Kumasi in the Ashanti Region, but we do not study its impact since it was lifted in less than three weeks and also we do not foresee the government re-introducing the lockdown to other parts of the country as it is perceived to be highly unsustainable in the country 4 . The individual and government integrated interventions work instantaneously although the process is complex, capital and labor-intensive, and requires a great deal of commitment. Almost all the interventions focus on the susceptible population and the population of infectives as the interaction between the two subclasses would likely lead to onward transmission of the virus to the larger population. Studies have shown that isolation is effective against any infectious disease even when the etiology of the disease is largely unknown (Keeling & Rohani, 2011) . We model five scenarios of the intervention to examine their likely impact on the dynamics of the epidemic. Specifically, by studying the different scenarios we examine the likely impact of government and individual-level interventions on the worse-case scenario of COVID-19 in Ghana.

Scenario 1: First, we modeled the worse-case scenario where we assume that there is no government or individual level interventions and that the individual can only recover from COVID-19 through pure natural process (immune system boosting). We defined this scenario as the normal-life situation.

Scenario 2: Second, we modeled the impact of enhanced contact tracing on worst-case scenario 1 and further assume that isolation and treatment centers can transmit the infection to the general public and medical staff at a rate that is one-five of the community transmission rate.

Scenario 3: Third, we assessed the impact of enhanced contact tracing on worst-case scenario 1 and further assume that the rate at which isolation and treatment centers will transmit the infection to the general public and medical staff will be reduced by at most 10% (5% reduction from the isolation centers and 5% reduction from hospitals). This is achieved by providing personal protective equipment to frontline line health workers and enforcing strict adherence to isolation protocols to prevent transmission of infection at the centers.

Scenario 4: We modeled scenario 3 in addition to effective case management. That is, we introduced a gradual increase in the rate of recovery among the quarantine and hospitalized classes.

Scenario 5: We modeled scenario 4 in addition to improved individual level interventions-social distancing, use of hand sanitizers, use of face mask, washing hands with running water with soap, and avoidance of social gathering and intensive media coverage.

As of May 13 th , 2020, Ghana had a total of 5,530 confirmed cases of COVID-19 with 674 recoveries and a total of 24 COVID-19 related deaths resulting in a case fatality rate of 0.43%. The Greater Accra region is currently the epicenter with 4,204 confirmed cases. The highest daily number of new cases recorded is 367. The Regional distribution of cases can be found at the official Ghana Health Service website (https://ghanahealthservice.org/covid19/).

The generalized growth model (GGM) (Chowell, Sattenspiel, Bansal, & Viboud, 2016) which is characterized by the initial growth rate ), the deceleration parameter ‰ were jointly estimated using Table 2 below. Note: For the TD method, daily estimates were averaged over the first 38 days and confidence interval was obtained using 1000 bootstrap simulations. The Bias corrected Maximum Likelihood (ML) method corrects the bias in the Reproductive number estimate occurring in method ML when the epidemic curve is not observed from the first case on (Obadia et al., 2012) . The confidence limit of the next generation matrix was obtained using Markov Chain Monte Carlo.

This study varied the generation time between 4-14 days with a standard deviation of 1 day. The graph of the basic Reproductive number versus mean generation time in days showed that the Ro estimates increases with increasing mean generation time for both ML and EG as expected (Nishiura, Castillo-Chavez, Safan, & Chowell, 2009) (Figure 4) . To obtain a correct estimate of the average generation time and to make data-driven policy decisions, there is the urgent need for more data on generation time (Peak et al., 2020) . 

Five different scenarios were studied to understand the trajectory of the epidemic. We forecast a range of possible future scenarios by assessing the impact of various individual and integrated government interventions. This section presents the results of the different intervention scenarios and the expected impact on COVID-19 deaths and total cumulative cases.

Scenario 1: It represents the worse case scenario. We defined the worst-case scenario as the socialcultural and political life of the individual and the country at large before the first two index cases of COVID-19 were reported in Ghana. That is before the 12 th of March 2020, there were no government and individual-level interventions to address COVID-19. Had the situation remained the same, we explore what would have been the total number of expected cases and deaths assuming that the epidemic lasted a

year-long (12 th March 2021). The worse case scenario assumes that because of the immune system, some Figure 5 ). Figure 6 ). Figure 9 ). 3. Impact of enhanced contact tracing on worst-case scenario 1 and further assume that the rate at which isolation and treatment centers will transmit the infection to the general public and medical staff will be reduced by 10% (5% reduction from the isolation centers and 5% reduction from hospitals). This is achieved by providing PPEs to frontline line health workers and enforcing strict adherence to isolation protocols to prevent transmission of infection at the centers. The results in Table 3 were obtained from using parameter estimates from Table 1 .

J o u r n a l P r e -p r o o f

The main focus of the study is to understand the dynamics of COVID-19 infection in Ghana using mathematical models that incorporate waning immunity to COVID-19, naural birth and death rates, integrated individual and government interventions and the fact that an exposed person may recover without necessarily been infectious to other due to boosted immunity. The proposed model allow for possible tramsmission of infection from quarantine, isolation and treatment centers similar to what was reported by (Giordano et al., 2020) . Our model further assume that individuals who recovered from COVID-19 may become susceptible again which is in sharp contrast to (Giordano et al., 2020) . We Previous studies have used mathematical models to study the dynamics of COVID-19 (Giordano et al., 2020; Lin et al., 2020; Munayco et al., 2020; Roosa et al., 2020) The effectiveness of these integrated interventions as found in this study have been reported elsewhere (Prem et al., 2020) . We have shown in this study that separate interventions could reduce the burden of COVID-19 but it was evident that we need both the government and the individual level interventions to reduce the impact of COVID-19 to the barest minimum.

Although 

Authors declare no competing interests.

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We acknowledge all the staff of Noguchi Memorial Institute of Medical Research and the Kumasi Center for Collaborative Research who are actively doing contact tracing and testing all suspected COVID-19 cases in Ghana. We acknowledge Mr. Armachie Joseph, a PhD student with the Department of Biostatistics for assisting the team in the matrix calculations.