COVID-19 outbreak and control in Kenya- Insights from a mathematical model

The coronavirus disease 2019 (COVID-19) pandemic reached Kenya in March 2020 with the initial cases reported in the capital city Nairobi and in the coastal area Mombasa. As reported by the World Health Organization, the outbreak of COVID-19 has spread across the world, killed many, collapsed economies and changed the way people live since it was ﬁrst reported in Wuhan, China, in the end of 2019. As of May 25,2020 It had led to over 100,000 conﬁrmed cases in Africa with over 3000 deaths. The trend poses a huge threat to global public health. Understanding the early transmission dynamics of the infection and evaluat-ing the eﬀectiveness of control measures is crucial for assessing the potential for sustained transmission to occur in new areas. We employed a SEIHCRD mathematical transmission model with reported Kenyan data on cases of COVID-19 to estimate how transmission varies over time. The model is concise in structure, and successfully captures the course of the COVID-19 outbreak, and thus sheds light on understanding the trends of the outbreak. The next generation matrix approach was adopted to calculate the basic reproduction number ( R 0 ) from the model to assess the factors driving the infection . The results from the model analysis shows that non-pharmaceutical interventions over a relatively long period is needed to eﬀectively get rid of the COVID-19 epidemic otherwise the rate of infection will continue to increase despite the increased rate of recovery.


Introduction
The current outbreak of COVID-19 pandemic has caused many fatalities, affected global economy and changed the way people live since it was first reported. COVID-19 had infected at least 4,801,202 people by May 20,2020 with the total number of deaths standing at 318,935 and that of recoveries at 964,161 and had affected over 213 countries worldwide according world health organization (17). In Africa, the virus has spread to dozens of countries within weeks. In Kenya the number of people tested of the virus by May 20, 2020 were 49,405 with 1029 cases, 366 recoveries and 50 deaths (17). In February 2020, WHO decleared the disease COVID-19, a global sures including school closure, wearing of masks, social distancing, city lockdown, mass testing, hospitalization and quarantine of patients. The parameter values may be improved when more information is available. Nevertheless, our model is a preliminary conceptual model, intending to lay a foundation for further modelling studies, but we can easily tune our model so that the outcomes of our model are in line with previous studies (2-4; 7).
The rest of the paper is organized as follows: in Section 2, we describe the formulated model. In Section 3, we carry-out model analysis including equilibrium analysis. The model is fitted to COVID-19 data in Section 4. The effects of social distancing and mass testing are also investigated in Section 4. In Section 5, we provide dicussions and recommendations.

Model Description and Formulation
To describe the dynamics of COVID-19 in Kenya, we develop a seven disease state compartmental SEIHCRD-model describing the movement of individuals from one state to another starting from the susceptible class S, that is, individuals with no history of infection by the disease. Individuals get infected with the virus and move into compartment E, referred to as exposed, who are asymptomatic. Then the exposed can develop symptoms and move to compartment I, referred as infected individuals. The Infected individuals will take themselves to be hospitalized or quarantine themselves at home. Those who get hospitalized move to class H. The hospitalized can get worse and move to ICU class denoted as C or recover and move to compartment R. The last class is the Death,represented by compartment D containing those who succumb to COVID-19. Figure 1 shows the structure of model The total population at any time t, is denoted by N (t) and is given by 3   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 The rate of generation of new COVID-19 cases is modelled by λS, where λ is the force of infection given by

Description of the variables and parameters used in the model
The variables and parameters description for the model are summarized in Tables 1 and 2 :  proportion of susceptible entering the country from other countries κ rate of recovery after being exposed β 0 Effective contact rate between susceptible and infected individuals . β 1 Effective contact rate between susceptible and hospitalized individuals β 2 Effective contact rate between susceptible and those in ICU γ transition rate from exposed to infectious α hospitalization rate ω recovery rate after treatment δ transfer rate to ICU σ recovery rate from ICU η effects of social distancing ζ effects of mass testing µ death rate due to COVID-19. Given the flow diagram in Figure 1, the parameter description in Table 2, we have the following system of non-linear ordinary differential equations: subject to the following initial conditions Proof. Considering the first equation in system (3), we have Upon integrating, Applying the above procedure to the rest of the equations in model system (3), we obtain: 5   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 63 64 Clearly, all the state variables, S, E, I, H, C, R, D of model system (3) are non-negative for all time t > 0.

Invariant Region
Restating equation (1), we have Substituting the derivatives in system (3) and simplifying, we obtain Integrating equation 4, we get where Clearly, there exists a bounded positive invariant region for model system (3). Let us denote this region as Ω ∈ R 7 + , where, Therefore, any solution of our system (3) that commences in Ω, at any time t ≥ 0 will always remain confined in that region. The region Ω is therefore positively invariant and attracting with respect to COVID-19 model system (3). The deterministic model in (3) is hence mathematically and biologically well-posed.

Equilibria analysis of the model
The basic reproduction number R 0 , is defined as the number of secondary infections produced by one infective that is introduced into an entirely susceptible population at the disease free equilibrium (5). The next generation matrix approach is frequently used to compute R 0 , see (15). System (3) has a disease-free equilibrium (DFE) given by The matrix F V −1 is called the next generation matrix. The (i, k) entry of F V −1 indicates the expected number of new infections in compartment i produced by the infected individual originally introduced into compartment k. 6   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 64 The (i, k) entry of F V −1 indicates the expected number of new infections in compartment i produced by the infected individuals originally introduced into compartment k. The model reproduction number, R 0 ,which is defined as the spectral radius of F V −1 , and denoted by ρ(F V −1 ) is evaluated to: Hence, Here R 0 is the sum of three terms each representing the average new infections contributed by each of the three infectious classes. R I represents the new cases generated by infected individuals in compartment I, R H represents new cases generated by patients hospitalized, and R C represents new cases from patients hospitalized and in ICU.
From R 0 in equation (7), each term is multiplied by (1 − η) which represents the Government control measures, hence if the measures are followed, then the emergency of new corona cases is reduced. Therefore the government campaign of "social distancing"and "hygiene"is very important in preventing the development of new cases.
From Theorem 2 in (15), we have the following result: Theorem 2. The DFE, E 0 of the system of equations (3) is locally asymptotically stable when R 0 < 1 and unstable otherwise.
Question: What is likely disease burden (total infections, total hospital admissions, total ICU admissions, total deaths) if epidemic is not contained by a country?
• The model will give, at any time, the estimated number of new infections, the total infections, the total population hospitalized, the patients in ICU and the number recovered together with mortality estimates.
• The model to predict expected cases of COVID-19 in future (say two weeks time)

Numerical Simulations
In this section, we present a series of numerical results of system (3) using COVID-19 reported cases data in Kenya to predict and estimate the incidence of the virus in the country.  The parameter values which are calculated from the given data in Table 3 are displayed in Table  4 and the initial conditions for the populations are as given in Table 5,   Using parameter values in Table 4, we identified how different input parameters affect the reproduction number R 0 of our model as shown in Figure 2.  Table 4 From Figure 2, it is evident that an increase in government action and recovery rate are most likely to reduce the number of new cases and severity of COVID-19 infections. A lot more emphasis on government action is therefore necessary. Such actions include compulsory face masks, regularly and proper washing of hands with soap and water or with alcohol-based hand sanitizers, cessation of movement and patrolled night curfews. To improve the rates of recovery, good care for COVID-19 patients is warranted. Infected individuals should be taken to isolation centres for proper treatment and care.

Application of the model to COVID-19 data in Kenya
Results of parameter sensitivity, further indicate the need to reduce contacts between the susceptibles and the infected COVID-19 persons. This is achievable through compulsory social distancing and imposed night curfews in the country.

Impact of mass testing on the populations
(a) Effects of mass testing on asymptomatic population (b) Effects of mass testing on symptomatic population Figure 6: The impact of mass testing on the populations denoted by γ and α in our model Figure 6 presents the projections and the impact of mass testing on the non-hospitalized COVID-19 cases. From Figure 6, it is shown that with mass testing, the asymptomatic and symptomatic patients who are not yet in hospital will be identified and either hospitalized or quarantined hence preventing further transmission. This implies that correct information based on an adequate diagnosis system would be desired for the Kenyan government to act appropriately.    3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 63 64 A combination of adherence to existing government control measures and improved medical environment is likely to yield most recoveries from COVID-19 infections as shown in Figure 5.

Discussion and recommendation
In this study, we applied the SEIRHCRD compartmental model to the daily reported cases of COVID-19 to determine the transmission dynamics of COVID-19 in Kenyan population over time. The model Simulation shows that ignoring safety guidelines such as social distancing, wearing of masks, frequent washing of hands with water and soap or using alcohol-based hand sanitizers and cutting down on travel has devastating effect on the disease dynamics. The model results also give insights to health policy-makers and Government on the effective approaches and implementable actions that can enhance the prevention, preparedness and readiness for future emergencies of COVID-19 and similar diseases.