Daily records of the COVID-19 cases were obtained from a public repository (https://www.worldometers.info/coronavirus/country cameroon/). The data included information dating from March 6, 2020, when the first COVID-19 cases were confirmed in Cameroon. The data revealed that the country has successively experienced 3 epidemic peaks (Figure 1): on April 3 (203 positive cases), on May 22 (555 cases), and on June 15 (1183 cases). The high peak observed on June 15, 2020 resulted from a large screening campaign conducted by the government. On June 24, the total number of confirmed cases was 12592, among whom 10100, 313, and 2179 were individuals who recovered, those who died, and active cases, respectively. From June 25 to July 6, the total active cases remained constant as the country did not provide record of new COVID-19 cases.

By July 30, the total number of cases reported were 17255, among whom 391, 15320, and 1544 died, recovered, or remained active cases, respectively, bringing the cure rate to 88.8% and a lethality rate to 2.3%. However, the number of cases started increasing from July 7 to 30, with regular increase of confirmed cases, suggesting the continuity of the epidemic. Data showed a constant increase of confirmed cases with a short period of under-reporting of cases. Under-reporting was noted during the month of July. A slight increase in the number of deaths was observed in July 19 (Figure 1). The epidemic in Cameroon revealed 3 peaks and more peaks are expected due to difficulties in maintaining confinement measures and re-opening of learning institutions.

Active cases can be either symptomatic with mild, severe, or critical disease condition, or asymptomatic. Cabore et al. estimated the risk of exposure (RoE) in Cameroon at 0.461 18, considering it as the risk of getting infected at time t, time t being considered as a reported date. This estimation was derived from a function of gathering events, sanitation and hygiene practices, weather, and distribution, i.e. movement of an infected individual.

Two approaches were adopted: a SIRD model, and a discrete-time Markov chain with different transition probabilities.

The first modelling experiment used the data from May 18 to May 31 (two weeks before the end of lockdown) to project the epidemic trend in June, July and August. The second modelling experiment used the data from June 1 to June 24 to project the trend of the pandemic in July and August 2020. The risk of an increased number of deaths was assessed using data from March 6 to June 24, 2020. Some modelling experiments like the transition probabilities were calculated on aggregated data over time (from March 6 to July 30, 2020) since individual data were not available and were used to estimate the total number of infected persons during the month of July 2020.

The SIRD model was used, first, to examine the evolution of the epidemic during the last two weeks in May (throughout the period of May 18 to June 24), and, secondly, during the whole month of June after the first re-opening of learning institutions to end the 2019/2020 academic year, with the purpose of assessing the impact of learners on the epidemic trends, and to predict the trend of the epidemic in Cameroon 1018.

Details regarding the model are found in the appendices, equation A.1, as well the calculation of R0 24. The model assumed a social distancing effect α, which was assessed during the studied period 18. Social distancing effect of 1 implies a complete confinement, but an inverse effect can also be expected since gathering events are still allowed and learning institutions were re-opened during the period of data analysis.

Given a day and an infected individual, there are different intra-individual status and circumstances of disease transmission. An infected individual can become a closed case (i.e., death), or recovered/discharged, or remain an active case (who is not yet completely recovered and not dead). An active case can have either mild symptoms or serious/critical condition that could either lead to death or recovery. We assumed that a recovered individual could get infected again if re-exposed to the virus.

We were interested in estimating the transition probability among different possible states and the probability of death given the other conditions. The state of death was regarded as an absorbent state since an individual cannot leave that state once he or she is classified as dead. Active case and recovered states were assumed to be transient as individuals can leave these states and return to them after some time. These multiple states occur as a random process usually known as a stochastic process.

We applied a Markov chain with multiple states (3 and 10) to estimate the probability of the death state to occur, and the probability of death number to increase. It is known that Markov chain property states that, the probability of an event to occur in the future, given the past and the present conditions, depends only on the present condition 25. Long-term simulation of a Markov chain allows to reach the stationary distribution, where no more variation is found in the time series It usually relies on discarding the first half of simulation (called burn-in process), and probability estimates are based on the last iterations.

In this paper, 1 denotes the active case, 2 the recovery state, and 3 the death state. It was assumed that a recovered individual is still at risk of being re-infected. Therefore, possible transitions are represented by the following sequences: 1 1, 1 2, 1 3, 2 1. Discrete Markov chain models assume that the current state of observation fully depends on the state of the observations at the previous time step per discrete time interval. Mathematically, this means that the probability to observe state i at time t transitioning to state j at time t + 1, or the t-step forward transition probability along the course of observations, is denoted by: pi, j (t) = Pr [I(t + 1) = j|I(t) = i], where Pr denotes the probability, and I(t) is the identification of observed state at time t. The single time-step transition probability is defined as pi, j (1), and is also referred to as the daily transition probability. The sum p11+p12+p13 is the total percentage of state of infection on day t of an individual who remained in one of the three states at time t+1. This sum is always equal to 100%, or 1.

There is evidence to suggest that, among symptomatic cases, the probability of having mild, moderate, severe, or critical disease is approximately 40%, 40%, 15% and 5%, respectively (Epidemiology group, Park M). Hospitalisation rates vary depending on hospitalisation policy and capacity in a given region or country, but it is estimated that 30% of symptomatic patients need hospitalisation, with the highest case fatality rate (CFR) in critical cases (up to 89% without intervention) and 49% for severely ill patients 2627.

Cabore et al. described a ten-stage disease modelling that encompasses all states for a COVID-19 case with the description of no point-in-time and time dependence, as follows: a country’s population is deemed susceptible (S1 , the initial Markov state), while the exposed population (those at risk of getting infected at any given time) are the next Markov state, S2. Individuals who get infected are the next state (S3). If they are not infected (including recovered individuals), they will return to the previous Markov state (S2) to face a continued risk of exposure. An infected individual can be represented by five mutually exclusive Markov states: asymptomatic (S4), mild symptoms (S5), moderate symptoms (S6), severe symptoms (S7) or critical symptoms (S8). ‘Absorbent states’ are death (S9) or recovery (S10).

Since an infected person can be classified into one of the enumerated disease conditions with probability 0.8, 0.08, 0.08, 0.03, 0.01, for asymptomatic, mild, moderate, severe, and critical states, respectively 18, the probability of being an active case and remaining active varies from 0.01 to 0.8. For the transition matrix, p11 is set to 0.2 (the mean of those values). The probability of recovery also varies depending on the state of the disease (Figure 4). It decreases when the disease state becomes worse. It is equal to 1 for mild, asymptomatic, or moderate cases, and 0.5 for severe condition, and 0.12 for critical state, respectively. An active case can recover with a probability of p12=0.72, i.e. the mean of the transition probabilities of moving from infection status to recovered stage. Death usually occurs in severe and critical states with a higher probability among severe cases, with p13 = 0.69 (i.e., the mean of 0.5 and 0.88). Since the sum of the rows of the transition matrix should be equal to 1, the probability of being infected and died was derived from the probability of being infected and remained infected (0.2), and the probability of moving from infection to recovering (0.72) i.e. 1-0.2-0.72=0.076.

The estimate of RoE in Cameroon accounted for several factors, such as the sanitation and hygiene practice, gathering events, re-opening of learning institution, and other factors of vulnerabilities (Cabore et al., 2020). Since a recovered individual can get re-infected with probability that ranged from 7 to 23%22, and could reach 14% (a country report accessible at https://www.todayonline.com/world/14-patients-who-recovered-covid-19-test-positive-again-guangdong-report), it was assumed that a recovered individual will be re-infected with probability 0.15 (that is the mean value of the range 7 to 23%). Given this consideration, an individual will remain recovered with a probability p22 of 0.85 (i.e., 1 – 0.15). In the other hand, for the purpose of comparison, it was also assumed that a recovered person will remain recovered.

The Markov chain and the probabilities of all transition states are illustrated in Figure 2 (A and B). We used each transition matrix to simulate the probability of being in each state over 91 days (which cover the month of July and August 2020). In addition, p13 was also calculated from time-series data of infected/recovered/death persons.

On the left:10-state disease with one absorbent state (death), with the assumption that a recovered person can get infected again. On the right: three-state transitioning probabilities built from the 10-states. This figure was built using the R package markovchain. Transition probabilities on the left were extracted from a published study 1822. In the right, the probability of being infected and died was derived from the probability of being infected and remained infected (0.2), and the probability of being infected and recovered (0.72) i.e. 1-0.2-0.72.

To assess whether the epidemic curve will continue to increase, we studied the number of active cases/recovered/death from March 6 to June 24. The trend was viewed as a time series, and each section of the data was classified into 3 categories: “increase”, “drop”, or “constant” over time, starting from March 6 to June 24. The decomposition yielded a sequence of three states from which a first order Markov chain was fitted to obtain a transition probability between increase, drop and constant states 2829. We analysed the probability of having an increase in the number of active cases/recovered/death by August 2020 by running 92 iterations of the Markov chain starting from June 24.

A 3-state Markov model was applied on the data during the period starting from March 6 to July 30, 2020to study the evolution of the disease, to assess the impact of control measures recommended by the Cameroonian government, and to estimate the total number of cases of COVID-19 in Cameroon. Yang Liu (2020) provided a comprehensive set of formula to derive the transition probabilities for COVID-19 using a 3-state Markov chain such that, as time t increases, the probability to attain the absorbent death state will converge to the true case fatality rate (CFR), as CFR naïve estimates being restricted to resolved cases 30. For each day, daily transition probabilities for the 3-states denoted pi, j were calculated. These empirical probabilities were then used to forecast over several iterations (91 iterations starting from June 24, including July, August and September 2020), the total number of COVID-19 cases 31.

Given the assumptions and statistical data, we used several innovative methods embedded in the R Epidemics Consortium (RECON) suite tools in R (https://www.repidemicsconsortium.org/), and the markovchain package, to analyse the data. In addition, Markov chain simulations were based on 50000 iteration among which the first 25000 were discarded to allow for convergence. The software R (version 3.6.3) was used to implement the models 32.

Two weeks before learning institutions re-opened, i.e. from May 18 to May 31, 2020 (a time period characterized by the relaxation of measures initially imposed by the government to the population at the onset of the pandemic in Cameroon), the SIRD model estimated a basic reproduction number R0 of 3.08, and a social distancing effect of 0.50.

Before the peak of the epidemic that occurred on June 15, R0 was estimated as 2.75, with α = 0.479, implying a small decrease compared to two weeks later. This decrease could be due to the total number of reported infected cases that remained constant for more than four days. The estimated number of infected individuals ranged from 17632 to 26424 (median=21587) persons.

From June 16 to June 24, the model estimated R0 as 2.84 (α=0.45), which yielded the number of infected individuals ranging from 28100 to 36628, suggesting the continuity of the disease severity. Projection of COVID-19 cases revealed an increase rate throughout the month of August (Figure 3).

There was an absence of reporting between June 25 and July 6, 2020. The projected cases (June 1 to July 30) was made by adding the total observed cases between June 25 and July 30, to June1-June 24 data. The model projected an increase in the number of death by mid-August 2020. The dotted dark-green curve and the orange dashed line provide the interval of the epidemic evolution in Cameroon by Mid-August 2020.

A long-term projection based on the data from May 1 to June 24, 2020 shows an exponential epidemic trend by mid-August 2020 (Figure 3). In addition, we assessed the time when the last epidemic peak will be reached by carrying out a long-term simulation with the SIRD model. It appeared that the total number of infected cases displayed a downward trend by January 2021, especially in the context of the implementation of strict measures such as social distancing (Figure 4).

Figure 4A shows the epidemic trend from May 1 to June 24, 2020 using reported cases. Projection in Figure 4B is based on the whole data from May 1 to June 24, and Figure 4C is based on the projection that used data from June 1 to June 24.

The consequence of this increase in the number of infected individuals could inevitably lead to an increase in the number of deaths. The projection during the month of May and June showed that the mortality rate associated with Covid-19 was increasing rapidly (Figure 3). This observation suggests that modelling the risk of death can contribute to provide to decision makers, some insights of the disease progression.

After discretizing each reported time series data (from March 6 to June 24) into several sub-time series data showing either a decrease, an increase, or constant values, we looked at the probability of having an increased number of active cases, recovered cases, and deaths over the months of July, August and September 2020. Indeed, for each COVID-19 data (active case, recovered or death), a 3 - dimensional discrete Markov Chain was built using the sequence generated by the following states: “constant”, “drop”, and “increase. For each sequence, we were interested in the increasing probability. The simulation carried out over 92 days starting from June 24 revealed an increase in the number of deaths throughout August and September (Figure 7). In addition, the probability of the number of deaths to increase converged to 0.42 by September 2020.

The computed transition probabilities from March 6 to July 30, 2020 yielded a mean of p11=0.33, p12=0.65, and p13=0.02. These transition probabilities are graphically shown in appendix B (Figure B1). We used these transition probabilities to forecast the total COVID-19 cases between July 1 and July 30. According to the projection, the epidemic worsens and increases the number of infected individuals significantly during the period between July 1 and July 30, suggesting constant activity of the COVID-19 virus in Cameroon (Figure 8).

Legend: the transition probabilities were computed from the time series of reported COVID-19 cases. The projected number of total cases was significantly higher than the observed total cases. It was expected more than 24000 cases by July 30, 2020.