Madagascar remains vulnerable to coronavirus attack. Despite the small number of cases detected, the impact of the new coronavirus on the Malagasy health system is considerable. The country is still at risk of an unmanageable and dangerous health crisis.

As soon as the virus appeared in Madagascar, the CCO took the initiative to introduce restrictions in order to limit the spread of the virus. Studies have shown that this initiative is early and has not provided more information on how the virus spreads.

According to official data, during the first wave, between March and August 2020, the outbreaks are in two regions: Analamanga region (Antananarivo, capital) and Antsinana region (Tamatave) (Figure 1).

Much of the data used in this article comes from these two strategic regions of Madagascar: Antananarivo the capital and Toamasina the economic capital. Figure 2 and Figure 3 represent the evolution of new cases observed in these two regions.

These graphs show that the epidemic has not had the same dynamic in these two major regions of Madagascar. It first hit the east side of the big island before entering the highlands.

For the entire territory of Madagascar, the SRAS-COV2 coronavirus epidemic has experienced several phases. The numbers of new cases and deaths were considered constant during the first trimester. These numbers followed an upward phase from June 2020. The impact of the epidemic was major in July 2020. The peak of this first wave is assumed to have reached around the beginning of August 2020, see Figure 4.

Figure 5 describes the cumulative data observed for all 22 regions of Madagascar.

The effectiveness of SIR (Susceptible-infected-removed) models has already been shown in epidemiological modelling problems. Several modifications have been proposed based on hypotheses on how the disease is transmitted [7] [8].

To limit the spread of the COVID-19 virus, Madagascar’s strategy is to isolate all suspected cases and hospitalize confirmed case. Given the inadequacy or lack of mass screening (PCR test), the effects of this strategy remain questionable. Many people with the virus are asymptomatic and show no signs of infection.

In line with this strategy, we propose in this article a model with seven compartments:

• S (Susceptibles): individuals who are not infected but at some risk of becoming infected;

• E (Exposed): individuals who have been in contact with the virus but not yet infectious until after a latency period;

• I (Infectious): individuals who can infect and are not yet hospitalized or isolated,

• H (Hospitalized): confirmed and hospitalized (isolated) cases;

• A (Asymptomatic): infected individuals but do not have symptoms (not isolated);

• C (Cured): healed confirmed cases;

• D (Death): deaths due to COVID-19.

Diagram in Figure 6 describes the evolution of our system according to the flows between the different compartments. The particularity of this model among others are first of all the separation of healthy (susceptible) individuals and individuals who are not yet sick but already in contact with the virus (Exposed). We have also put in another compartment infected individuals asymptomatic but who can transmit the virus.

Using the assumptions of Manou-Abi and Balicchi in [15] all the models’ parameters is summarized in Table 1.

The basic reproductions number R₀ corresponding to the model will be estimated using the officiad data recenced by the CCO-COVID-19. It represents the average number of secondary cases produced by a fundamental case over its infectious period. It is the initial growth rate of the epidemic.

As a reference, we will consider the deterministic case adapted to the context of.

Madagascar. This type of model is widely used in mathematical modeling of the spread of the COVID-19 [5] [16].

At a moment t, let S ( t ) , E ( t ) , I ( t ) , A ( t ) , H ( t ) , C ( t ) et D ( t ) the number of individuals in the compartments “Susceptible”, “Exposed”, “Infections”, “Asymptomatic”, “Hospitalized”, “Cured” et “Death”.

Assume that the total number of population is constant, i.e.,

N = N ( t ) , (1)

with N ( t ) = S ( t ) + E ( t ) + I ( t ) + A ( t ) + H ( t ) + C ( t ) + D ( t ) , for any momentt.

In a time interval [ t , t + d t ] , the evolution of the system is described by Equation (2):

{ d S ( t ) d t = − α S ( t ) I ( t ) N , d E ( t ) d t = α S ( t ) I ( t ) N − β E ( t ) , d I ( t ) d t = β E ( t ) − p γ I ( t ) − ( 1 − p ) λ I ( t ) , d A ( t ) d t = p γ I ( t ) , d H ( t ) d t = ( 1 − p ) λ I ( t ) − ( 1 − q ) δ H ( t ) − q θ H ( t ) , d C ( t ) d t = q θ H ( t ) , d D ( t ) d t = ( 1 − q ) δ H ( t ) . (2)

with:

- ( 1 − p ) λ I ( t ) d t the number of confirmed cases (isolated, hospitalized),

- p γ I ( t ) the number of asymptomatic cases,

- α ( S ( t ) I ( t ) ) / N d t the number of new infections in the interval [ t , t + d t ] .

By grouping asymptomatic cases and healed people into R(t) (“removed”):

R ( t ) = C ( t ) + A ( t ) , (3)

the simulation of the model with an extract from the population, in an environment where the virus circulates, gives the graphs of Figure 7.

To estimate the infection rate β and recovery rate γ in the SIR sub-process, we use the method introduced by Kermarck and McKendick in [17] using the observed data.

(The parameters β and γ of the SIR sub-process).The number of basic reproductions R₀ is obtained by

R 0 = β / γ , (4)

after training the ODE system of Equation (3) with the data of the COVID-19 in Madagascar.

Note:If R₀ < 1, the infected individual contaminates less than one other individual on average. This is why the epidemic is disappearing from the population.

If R₀ > 1, then the virus continues to spread in the population.

For Madagascar, by running the model with the observed data and its population of about 26,026,000, we obtain:

The estimated number of basic reproductions with official data is quite small. This may be due to the specificity of the Malagasy on the vulnerability to the COVID-19. Figure 8 shows the simulation of the model with the epidemic parameters of Madagascar. Without intervention, the dynamics stabilize over the long term, see Figure 9.

The spread and speeds of transmission of the new coronavirus vary widely from country to country. It depends on many factors such as demographic, socioeconomic, geographical, biologic and especially epidemic [22]. At the same time, individuals strive to find good ways to deal with the disease and adapt their behaviors to the different requirements of the riposte strategy.

We assume that factors external to the virus affect its rate of spread and turn it into a random variable. Thus, the process formed by transmission speeds ( α ( t ) ) , t ≥ 0 can be considered a stochastic process in continuous time with:

α ( t ) d t = α d t + σ d B ( t ) (5)

where B is a standard Brownian movement [23].

α ( t ) d t represents the number of infection transmitted by an infectious individual during a period dt. It is a normally distributed random variable [15]:

α ( t ) d t ~ N ( α d t , σ 2 d t ) .

The works of Norris [18] and Durret [23] are good references to the theories on continuous time Markov chain. We use results that are demonstrated in these works.

For the SEIAHCD process of Figure 6, for each individual, consider the following events:

i) Exposition: a susceptible individual became exposed through contact with an infected individual, under a probability of p₁

ii) Infection: an exposed individual became infected under a probability of p₂

iii) Recovery: an infected individual has passed the incubation period and become an asymptomatic case, with a probability of p₃

iv) Hospitalization: an infected individual is hospitalized with a probability of p₄

v) Healing: an individual hospitalized is cured of COVID-19

vi) Death: an individual hospitalized died of COVID-19

Events i), ii), iii), iv), v) et vi) correspond to jumps in processes S ( t ) , E ( t ) , I ( t ) , A ( t ) , H ( t ) , C ( t ) et D ( t ) . Consider the following subset of events:

- Events after exposition:

G 1 = { exposition , infection , removed } , where

removed = { revovery , hospitalization } .

- Events after infections:

G 2 = { recovery , hospitalization , retired } , where

retired = { death , healing } , with

- Events after hospitalization:

G 3 = { death , healing } .

In our modeling approach, we assume that events in each subset are independent uniformly distributed. Probabilities of jumps corresponding to the different events can be written:

- For G₁:

( exposition ) ~ p 1 p 1 + p 2 + p 3 + p 4 , ( infection ) ~ p 2 p 1 + p 2 + p 3 + p 4 ,

( removed ) ~ p 3 + p 4 p 1 + p 2 + p 3 + p 4 .

- For G₂

( recovery ) ~ p 3 p 3 + p 4 + p 5 + p 6 , ( hospitalization ) ~ p 4 p 3 + p 4 + p 5 + p 6 ,

( retired ) ~ p 5 + p 6 p 3 + p 4 + p 5 + p 6 .

- For G₃:

( healing ) ~ p 5 p 5 + p 6 , ( death ) ~ p 4 p 3 + p 4 + p 5 + p 6 .

with the below assumption and with an additional hypotheses that the transition time are exponentially distributed, we can consider that processes S ( t ) , E ( t ) , I ( t ) , A ( t ) , H ( t ) , C ( t ) et D ( t ) can be modeled by continuous time Markov chain. Algorithm 1 can simulate this model with the parameters estimated using official data. The result is in Figure 10.

The introduction of random effects into Equations (2) leads to a stochastic differential equation system. The deduced model is the stochastic equivalent of the modified SEIR model [24].

In a general way, a classic differential equation in explicit form

d x t d t = b t ( x t )

transforms into a stochastic differential equation by adding random noise.

This noise can be taken into account by a process of Winner W_t [25]:

d x t d t = μ ( x t ) + σ ( x t ) d W t , (6)

where μ ( X , θ ) the average et σ 2 ( X , θ ) the covariance matrix.

Algorithm 1. Simulation of the continuous time Markov chain SEIAHCD model.

1) First, consider the sub-model SEI (Susceptible, Exposed, Infected). Let ( x 1 x 2 x 3 ) , associated random variables. For a small interval of time ∆t; transition probabilities are written:

P ( S + k , E + j , I + m ) = { p 1 = α S I N Δ t ,                                                                     ( k , j , m ) = ( − 1 , 1 , 0 ) p 2 = β E Δ t ,                                                                         ( k , j , m ) = ( 0 , − 1 , 1 ) p 3 = ( p γ I + ( 1 − p ) λ I ) Δ t ,         ( k , j , m ) = ( 0 , 0 , − 1 ) 1 − ( ∑ i = 1 3 p i ) Δ t ,                                                                 ( k , j , m ) = ( 0 , 0 , 0 ) 0 ,                                                                                                                   otherwise (7)

E ( Δ x ) = ∑ i = 1 3 p i ( Δ x ) i = p 1 ( Δ x ) 1 + p 2 ( Δ x ) 2 + p 3 ( Δ x ) 3 = p 1 ( − 1 1 0 ) + p 2 ( 0 − 1 1 ) + p 3 ( 0 0 − 1 )

E ( Δ x ) = ( − α S I N α S I N − β E β E − p γ I − ( 1 − p ) λ I ) Δ t = μ ( x 1 , x 2 , x 3 ) Δ t

E ( ( Δ x ) ( Δ x ) ) = ∑ i = 1 3 p i ( Δ x ) i ( Δ x ) i T = p 1 ( − 1 1 0 ) ( − 1 1 0 ) + p 2 ( 0 − 1 1 ) ( 0 − 1 1 ) + p 3 ( 0 0 − 1 ) ( 0 0 − 1 ) = ( p 1 − p 1 0 − p 1 p 1 + p 2 − p 2 0 − p 2 p 2 + p 3 ) = ( α S I N − α S I N 0 − α S I N α S I N + β E − β E 0 − β E β E + p γ I + ( 1 − p ) λ I ) Δ t = σ 2 ( x 1 , x 2 , x 3 ) Δ t

We obtain:

{ d S ( t ) d t = − α S ( t ) I ( t ) N + σ 1 , 1 d W d t + σ 1 , 2 d W d t d E ( t ) d t = α S ( t ) I ( t ) N − β E ( t ) + σ 2 , 1 d W d t + σ 2 , 2 d W d t + σ 2 , 3 d W d t d I ( t ) d t = β E ( t ) − p γ I(t) − (1 − p) λ I(t) + σ 3 , 2 d W d t + σ 3 , 3 d W d t (8)

where a Wienner i process.

2) For the sub-model “after infections”:

Note ( x 4 x 5 ) the random variables associated with sub-model AH.

For a small interval ∆t. Transition probabilities are:

P ( A + k , H + j ) = { p 4 = p γ I Δ t ,                                                                                 ( k , j ) = ( 1 , 0 ) p 5 = ( 1 − p ) λ I Δ t ,                                                         ( k , j ) = ( 0 , 1 ) p 6 = ( ( 1 − q ) δ H + q θ H ) Δ t ,               ( k , j ) = ( 0 , − 1 ) 1 − ( ∑ i = 4 6 p i ) Δ t ,                                                                       ( k , j , m ) = ( 0 , 0 ) 0                                                                                                                       otherwise

E ( Δ x ) = ∑ i = 4 6 p i ( Δ x ) i = p 4 ( Δ x ) 4 + p 5 ( Δ x ) 5 + p 6 ( Δ x ) 6 = p 4 ( 1 0 ) + p 5 ( 0 1 ) + p 6 ( 0 − 1 )

E ( Δ x ) = ( p γ I ( 1 − p ) λ I − q θ H − ( 1 − q ) δ H ) Δ t = μ ( x 1 , x 2 , x 3 ) Δ t

E ( ( Δ x ) ( Δ x ) ) = ∑ i = 4 6 p i ( Δ x ) i ( Δ x ) i T = p 4 ( 0 1 ) ( 0 1 ) + p 5 ( 1 0 ) ( 1 0 ) + p 6 ( − 1 0 ) ( − 1 0 ) = ( p 5 + p 6 0 0 p 4 ) = ( ( 1 − p ) λ I + q θ H + ( 1 − q ) δ H 0 0 p γ I ) Δ t = σ 2 ( x 1 , x 2 , x 3 ) Δ t .

We obtain:

{ d A ( t ) d t = p γ I ( t ) + σ 1 , 1 d W 1 d t d H ( t ) d t = ( 1 − p ) λ I ( t ) − ( 1 − q ) δ H ( t ) − q θ H ( t ) + σ 2 , 2 d W d t (9)

where W the independent processes of Winner.

3) For the last sub-model (compartments after hospitalization):

Note ( x 6 x 7 ) random variables corresponding to the CD submodel. For a small inteval ∆t, the transition probabilities are:

P ( c + k , d + j ) = { p 6 = q θ H Δ t ,                                         ( k , j ) = ( 1 , 0 ) p 7 = ( 1 − q ) δ I Δ t ,                     ( k , j ) = ( 0 , 1 ) 1 − ( ∑ i = 6 7 p i ) Δ t ,                                   ( k , j , m ) = ( 0 , 0 ) 0 ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​                                                                                           otherwise

E ( Δ x ) = ∑ i = 6 7 p i ( Δ x ) i = p 6 ( 1 0 ) + p 7 ( 0 1 )

E ( Δ x ) = ( q θ H 0 ) Δ t + ( 0 ( 1 − q ) δ H ) Δ t = μ ( x 1 , x 2 , x 3 ) Δ t

E ( ( Δ x ) ( Δ x ) ) = ∑ i = 6 7 p i ( Δ x ) i ( Δ x ) i T = p 6 ( 1 0 ) ( 1 0 ) + p 7 ( 0 1 ) ( 0 1 ) = ( p 6 0 0 p 7 ) = ( q θ H 0 0 ( 1 − q ) δ H ) Δ t = σ 2 ( x 1 , x 2 , x 3 ) Δ t

The system becomes:

{ d C ( t ) d t = q θ H + σ 1 , 1 d W d t d D ( t ) d t = ( 1 − q ) δ H + σ 2 , 2 d W d t (10)

where W a Wienner process.

Figure 11 represents a simulation of the model through a numerical resolution of stochastic differential Equations (8)-(10).

1) Estimate by the method of maximum likelihood:

In [26], the authors propose maximum likelihood method to estimate the number of basic reproductions of an epidemic. This technique is based on the assumption that the number of secondary cases caused by an index case is a Poisson distribution, with the expected value R₀.

Let ( I 0 , I 1 , ⋯ , I T ) the observation on consecutive time and ω generation time distribution. The number of basic reproductions R₀ is estimated by a maximization of the log-likelihood:

L L ( R ) = ∑ t = 1 T log ( exp ( − μ t ) μ t I t I t ) (11)

where µ t = R ∑ t − i t I t − i ω i .

The likelihood function must be calculated over a period of exponential growth, and the R-square deviance can be used to select the best period. For Madagascar, using the observed data, the maximum likelihood estimate (MLE) of the basic reproduction number is:

R 0 = 1.1027

with 95% of confidence interval.

2) Bayesian estimate:

Introduced by Bettencourt and Ribeiro [27], this technique allows sequential estimate of the number of reproduction R. It is based on the assumption that the number of infections I ( t ) is a discrete random variable and I ( t + 1 ) is a Poisson random variable: P ( λ ) , where

λ = I ( t ) exp ( γ ( R − 1 ) )     et     1 γ

the average duration of the infectious period.

With a non informative prior distribution of the basic number of reproduction R₀, we have implemented an algorithm that can calculate the posterior distribution:

P ( R 0 | I 0 , I 1 , ⋯ , I t + 1 ) = P ( I t + 1 | R 0 , I 0 , I 1 , ⋯ , I t ) P ( R 0 | I 0 , I 1 , ⋯ , I t ) P ( I 0 , ⋯ , I t + 1 ) (12)

Knowledge about R₀ is updated as new data is observed. On each new day, the posterior distribution after the previous day is considered to be the prior distribition. At any given moment, the Bayesian estimate can be calculated with the highest probability density interval.

As before, the method requires that the epidemic be in a period of exponential growth, i.e., it does not take into account the depletion of sensitivity. It implicitly uses an exponential distribution for generation time; and assumes a random mix in the population [28].

For our case, on the graph of Figure 12, the grayed area corresponds to the

confidence interval and the black curve to the average number of reproductions in time function R ( t ) . According to these results, the basic reproduction number in the Malagasy territory is lower than the references in neighboring countries [29] [30].