The emergence of novel coronavirus (Covid-19) in 2019 sprung a sudden outbreak across the globe, presenting clinical and public health management challenges which led to global cancellation of conferences, travel restrictions, social distancing and closure of institutions. Thus, in considering the grave implications of the continuous spread of coronavirus disease, a SEIHRD epidemic model was formulated to gain insight into disease transmission dynamics with impacts of proposing control measures. The model captures the impact of undetected infectious individuals and detected hospitalized individuals with saturated treatment on the spread, death and recovery of Covid-19 patients in Nigeria. The model epidemic threshold and equilibria are determined, and their stabilities are analyzed. The model is validated by fitting it to data from January 28 to December 5, 2020. Results obtained suggest that decreasing the transmission rate for infective alone is not sufficient to eradicate the disease because of the presence of backward bifurcation, and we recommend that Nigerians must also adhere strictly to Covid-19 protocols in mitigating the spread and demise of coronavirus.
The emergence of strain severe acute respiratory syndrome Coronavirus-2 (SARS-COV-2) commonly known as Covid-19 on December 31, 2019, sprung a sudden outbreak that span globally across (Europe, North America, Asia, South America, Africa and Oceania) affecting 215 countries/territories which rapidly becomes a devastating pandemic. As of June 2020, over 6,861,727 people were affected by coronavirus and over 3,361,466 deaths globally due to the virus [
The outbreak of Covid-19 was reported in Wuhan, Hubei province, China on December 31, 2019, followed by Europe, America, Asia, Africa and across the globe. Coronavirus spreads rapidly across 47 African countries with Nigeria being the third largest affected African country having recorded over 84,811 confirmed cases and 1264 deaths. Nigeria first documented Covid-19 case was recorded on February 29th 2020, after a 44-year-old Italian citizen on January 27th 2020 arrived at the Murtala Muhammed International Airport, from Milan, Italy traveling to his company site at Ogun State. Starting with the official index from February, Nigerian began to experience increase in number of new cases, which led Nigeria government to respond to the outbreak by adhering strictly to Nigeria Centre for Disease Control (NCDC) and World Health Organization (WHO) preventive guidelines via enforcing total lock-down on human and economy activities in states with high index cases, while imposing travel restrictions, social distancing, closure of institutions and others unprecedented preventive measures in order to mitigate the spread of disease.
Coronaviruses are large family of zoonotic virus that causes disease in both humans and animals, In humans, there are three existing families known as coronaviruses which includes severe acute Respiratory Syndrome SARS-CoV discovered in (2002), Middle Eastern Respiratory Syndrome MERS-CoV (2012) and Covid-19 caused by SARS-CoV-2 (2019). In animals like cows and pigs, they cause diarrhea, while in mice they cause hepatitis and encephalomyelitis. The history of these viruses have their members generally associated with vector-transmitted diseases in humans and causes illness ranging from the common cold to severe respiratory disease which includes symptoms such as dry cough, fever, shortness of breath, and breathing difficulties. However, in more severe cases infection can cause pneumonia, kidney failure, SARS and death [
Managing this serious epidemic requires the appropriate deployment of limited human resources across all cadres of health care and public health as recommended in [
However, each country or city has its own shortfall on precautionary measures and maximal capacity for the treatment of coronavirus disease. This study is aimed to rigorously evaluate the impact of undetected infectious patients and limited supplies of medical facilities for treatment of hospitalized or isolated Covid-19 patients in Nigeria. Hence, this paper is organized as follows. The proposed model is formulated in Section (2) and Section (3) for model properties and analysis. Section (4) describes the steady-state analysis, Section (5) describes the model fitting and sensitivity analysis, Section (6) describes the numerical analysis and the conclusion is presented in Section (7).
To effectively evaluate Covid-19 disease dynamics with saturated treatment, a deterministic mathematical model (SEIHRD) comprising six distinct compartments of susceptible S ( t ) , exposed E ( t ) , infected I ( t ) , hospitalized H ( t ) , recovered R ( t ) and deceased D ( t ) is formulated with inclusion of saturated treatment function as applicable to similar pandemic diseases see [
δ 2 ), the effect of saturated treatment is captured by g ( H ( t ) ) = α H ( t ) 1 + κ H ( t )
where α is the cure rate κ is the saturation factor that measures the effect of hospitalized being delayed for treatment. The death compartment D ( t ) comprises of persons who died from the disease in both compartment I ( t ) and H ( t ) over time, with μ define as the natural death rate in all model compartment, σ is the burial rate of decease patient in H ( t ) . While η 2 is the rate at which dead Covid-19 person infects susceptible individuals, and this connotes the denial of releasing the deceased to family members for burial.
Thus, the force of infection is given as:
λ ( t ) = γ ( I ( t ) + η 1 H ( t ) + η 2 D ( t ) ) S ( t ) + E ( t ) + I ( t ) + H ( t ) + D ( t )
N = S ( t ) + E ( t ) + I ( t ) + H ( t ) + R ( t ) + D ( t ) , ∀ t ≥ 0 (1)
The SEIHRD epidemic model with saturated treatment function consists of the following equation
S ( t ) = A − ( λ ( t ) − μ ) S ( t ) E ( t ) = λ ( t ) S ( t ) − ( a + μ ) E ( t ) I ( t ) = a ( 1 − θ ) E ( t ) − ( ω 1 + d 1 + μ ) I ( t ) H ( t ) = a θ E ( t ) − g ( H ( t ) ) − ( ω 2 + d 2 + μ ) H ( t ) R ( t ) = g ( H ( t ) ) + ω 1 I ( t ) + ω 2 H ( t ) D ( t ) = d 1 I ( t ) + d 2 H ( t ) − σ D ( t ) (2)
where g ( H ( t ) ) = α H ( t ) 1 + κ H ( t ) , ∀ α > 0 , κ ≥ 0
The schematic diagram of model system (2) is represented in
| Variable | Interpretation |
|---|---|
| S | Population of susceptible individuals |
| E | Population of individuals exposed to Covid-19 |
| I | Population of infectious individuals |
| H | Population of infected and Hospitalized individuals |
| R | Population of Recovered Individuals |
| D | Number of death due to Covid-19 |
| Parameter | Interpretation |
| A | Recruitment rates into the population |
| η 1 , ( η 2 ) | Effective contact rates in (I & H) |
| γ | Force of infection with Susceptible |
| a | Transmission rate form E to I |
| θ | Fraction of those Hospitalized |
| d 1 , ( d 2 ) | Disease mortality rate in (I & H) |
| w 1 , ( w 2 ) | Rate of recovery from (I & H) |
| σ | Burial rate of Covid-19 patient |
| κ | Saturation factor that measure |
| Hospitalized been delayed for treatment | |
| α | Cure rate for Hospitalized individuals |
| 1 / μ | Human average life span |
However, since the recovered compartment in System (2) does not contribute to Covid-19 disease transmission, R ( t ) may be decoupled from other variables in the closed system and the new system becomes:
S ( t ) = A − λ ( t ) − μ S ( t ) E ( t ) = λ ( t ) − ( a + μ ) E ( t ) I ( t ) = a ( 1 − θ ) E ( t ) − ( ω 1 + d 1 + μ ) I ( t ) H ( t ) = a θ − g ( H ( t ) ) − ( ω 2 + d 2 + μ ) H ( t ) D ( t ) = d 1 I ( t ) + d 2 H ( t ) − σ D ( t ) (3)
Q ( t ) = S ( t ) + E ( t ) + I ( t ) + H ( t )
In this section, it’s pertinent to show that all state variables of model (3) are biological meaningful and non-negative for all time (t). In other words, the solution of model (3) with positive initial solution will remain positive for all t ≥ 0 and this can be established using the following lemma.
Theorem 1. Let the initial solution S ( 0 ) > 0 , E ( 0 ) > 0 , I ( 0 ) > 0 , H ( 0 ) > 0 , D ( 0 ) > 0 , then the solutions S , E , I , H , D of model (3) are positive for all t ≥ 0 . Furthermore,
lim t → ∞ sup Q ( t ) ≤ A μ and lim t → ∞ D ( t ) ≤ A ( d 1 + d 2 ) μ σ
Lemma 2. The region
Ω = { ( S ( t ) , E ( t ) , I ( t ) , H ( t ) , D ( t ) ) ∈ ℝ 5 : Q ( t ) ≤ A μ , D ( t ) ≤ A ( d 1 + d 2 ) μ σ }
is attracting and positively invariant for model 3.
Proof:
Given that Q ( t ) = S ( t ) + E ( t ) + I ( t ) + H ( t ) yields
d Q ( t ) d t ≤ A + μ Q ( t ) (4)
Applying the standard comparison theorem, the following solution was obtained:
Q ( t ) ≤ A μ + ( Q ( 0 ) − A μ ) exp ( − μ t ) , ∀ t ≥ 0 (5)
If Q ( 0 ) ≤ A μ then the upper bound of Q ( t ) is A μ when t → ∞ . If Q ( 0 ) ≥ A μ , then Q ( t ) decreases to A μ , when t → ∞ and enters Ω or approaches asymptotically.
Similarly, since I ( t ) ≤ Q ( t ) and H ( t ) ≤ Q ( t ) from the last equation of model (3), we have
D ( t ) ≤ A ( d 1 + d 2 ) μ + σ D ( t ) (6)
whose solution is:
D ( t ) = 1 σ ( A ( d 1 + d 2 ) μ − σ ( A ( d 1 + d 2 ) μ σ − D ( 0 ) ) exp − σ t ) .
Thus, as t → ∞ , D ( t ) ≤ A ( d 1 + d 2 ) μ σ , Hence the feasible solution of model (3)
enters the region Ω . In this region, the model can be considered to be epidemiological and mathematically well posed.
At equilibrium, The disease free equilibrium (E0) of model (3) is obtained by setting ( E ( t ) = I ( t ) = H ( t ) = D ( t ) = 0 ) and the L H S = 0 which results to:
E 0 = ( S * , E * , I * , H * , D * ) = ( A μ , 0 , 0 , 0 , 0 ) (7)
The basic reproduction number R0 also called the control reproduction number of the model (3), measures the average number of new Covid-19 cases generated by a typical infectious individual introduced into a population where a certain fraction is protected. The linear system of Equation (7) is established using the next generation matrix method on model (3) to compute the reproduction number see ( [
F = [ 0 γ γ η 1 γ η 2 0 0 0 0 0 0 0 0 0 0 0 0 ] , V = [ a + μ 0 0 0 a ( θ − 1 ) B 1 0 0 − a θ 0 B 2 + α 0 0 − d 1 − d 2 σ ]
where, B 1 = ω 1 + d 1 + μ and B 2 = ω 2 + d 2 + μ .
The reproduction number is defined by the quantity R 0 = ρ ( F V − 1 ) , where ρ is the spectral radius,
R 0 = γ a [ ( 1 − θ ) ( B 2 + α ) ( σ + d 1 η 2 ) + θ B 1 ( η 1 σ + d 2 η 2 ) ] B 1 σ ( B 2 + α ) ( a + μ ) , R I = a γ ( 1 − θ ) B 1 ( a + α ) , R H = a γ η 1 θ ( B 2 + α ) ( a + μ ) , R D = a γ η 2 σ B 1 ( a + μ ) ( d 1 ( 1 − θ ) + θ B 1 d 2 B 2 + α ) (8)
This implies that by interpretation R0 is the sum of three constituent reproduction number RI, RH and RD respectively which give the contribution of undetected-infected cases, detected-hospitalized cases and deceased cases to transmission of Covid-19.
Lemma 3. The Disease Free Equilibrium E0 of model (3) is locally asymptotically stable if the basic reproduction number R 0 < 1 and unstable if R 0 > 1 .
Proof.
To proof lemma (3), the Jacobian of model 3 is obtained as:
J 0 = [ − μ 0 − γ − γ η 1 − γ η 2 0 − a − μ γ γ η 1 γ η 2 0 a ( 1 − θ ) − μ − d 1 − w 1 0 0 0 a θ 0 − μ − d 2 − w 2 − α 0 0 0 d 1 d 2 − σ ] (9)
Jacobian matrix J0, have one negative eigenvalues ( − μ ) and the remaining can be obtained through the characteristics equation below:
Γ ( λ ) = λ 4 + a 1 λ 3 + a 2 λ 2 + a 3 λ + a 4
a 1 = σ + μ + a + B 1 + B 2 a 2 = [ ( 1 − R H ) ( B 2 + α ) + ( 1 − R I ) B 1 ] ( a + μ ) + ( α + μ + a + B 1 + B 2 ) σ + B 1 ( B 2 + α ) a 3 = [ ( 1 − R I ) B 1 σ + ( 1 − R H ) ( B 1 + σ ) ( B 2 + α ) ] ( a + μ ) − R I B 1 ( B 2 + α + d 1 η 2 ) ( a + μ ) − d 2 γ θ a 4 = ( 1 − R 0 ) ( B 2 + α ) ( a + μ ) B 1 σ
From Equation (8) it can be seen that R I < R 0 , R H < R 0 and R D < R 0 and this implies that the coefficient a 2 , a 3 , a 4 are positive whenever R 0 < 1 , and thus all the coefficients are positive. Further, the criteria of Rough-Hurtwiz for the fourth order polynomial is a i > 0 for i = 1 , 2 , 3 , 4 and a 1 a 2 a 3 − a 1 2 a 4 − a 3 2 > 0 can be easily satisfied by using the above coefficients. So, the model (3) at the disease free equilibrium is locally asymptotically stable if R 0 < 1 .
To explore the possibility of existence of the positive equilibria (say, equilibria where at least one of the infected variable is non-negative)
E 1 = ( S ∗ , E ∗ , I ∗ , H ∗ , D ∗ ) (10)
representing any arbitrary equilibrium of model (3).
By some simple calculation we obtained the following:
E ∗ = ϰ H ∗ a ( H ∗ + 1 ) θ , I ∗ = ϰ ( 1 − θ ) H ∗ B 1 ( κ H ∗ + 1 ) θ D ∗ = [ ϰ ( 1 − θ ) d 1 + B 1 d 2 θ ( κ H ∗ + 1 ) ] H ∗ B 1 σ ( κ H ∗ + 1 ) θ ,
S ∗ = [ ( 1 − θ ) ( d 1 + σ ) ϰ + B 1 ( κ H ∗ + 1 ) ( d 2 + σ ) θ a + σ B 1 ϰ ] ( a + μ ) ϰ H ∗ ζ 1
with
ϰ = B 2 κ H ∗ + B 2 + α ζ 1 = [ a σ ( γ − B 1 − γ θ ) ϰ + γ B 1 ( κ H ∗ + 1 ) ( η 1 σ + d 2 ) θ a + γ B 2 η 2 ( κ H ∗ + 1 ) ( 1 − θ ) a + α d 1 ( 1 − θ ) a + σ μ ϰ B 1 ] a ( κ H ∗ + 1 ) θ
and H ∗ is the positive solution for the following polynomial:
a 0 ( H ∗ ) 3 + a 1 ( H ∗ ) 2 + a 2 H ∗ + a 3 = 0 (11)
with the coefficient of H ∗ given as
a 0 = γ B 2 κ 2 ( a + μ ) [ B 2 ( 1 − θ ) ( γ σ + μ σ + d 1 ( η 2 γ + μ ) ) + θ B 1 ( ( σ + d 2 ) μ + γ ( d 2 η 2 + η 1 σ ) ) − B 1 B 2 σ ] a 1 = 2 B 2 κ ( B 2 + α ) [ ( 1 − θ ) ( σ + d 1 ) ( a + μ ) μ + γ ( d 1 η 2 + σ ) ( 1 − θ ) μ − ( a + μ ) ( 1 − R 0 ) B 1 σ ] + 2 κ B 1 θ μ B 2 [ ( a + μ ) ( d 2 + σ ) + γ ( η 2 d 2 + η 1 σ ) ] + κ B 1 θ α ( a + μ ) [ ( d 2 η 1 + η 1 σ ) γ + μ ( d 2 + σ ) ] − A K 2 B 2 θ B 1 [ B 2 ( d 1 η 2 + σ ) ( a + μ ) R I + a γ θ ( d 2 η 2 + η 1 σ ) − B 2 σ ( a + μ ) ]
a 2 = ( a + μ ) ( B 2 + α ) 2 [ ( d 1 + σ ) ( 1 − θ ) + γ ( d 1 η 2 + σ ) ( 1 − θ ) − B 1 σ ( 1 − R 0 ) ] − B 1 θ [ ( a + μ ) ( σ + d 2 ) + γ ( d 2 η 2 + σ η 1 ) ( 1 + A κ a θ ) ] + A κ θ B 1 [ B 2 σ − ( a + μ ) σ ( B 2 + α ) ( R 0 − 1 ) − B 2 ( d 1 η 2 + σ ) ( a + μ ) R I ] a 3 = γ A θ ( a + μ ) ( B 2 + α ) ( 1 − R 0 ) B 1 σ (12)
From Equation (12), it follows that a0 and a1 are always positive whether 1 < R 0 > 1 since all parameters are non negative. Thus, the number of possible positive real roots of Equation (11) depends on the signs of a2 and a3. This is analyzed using Descartes’ law of signs on polynomial (11) as presented in
However, theorem (4) summarizes the conditions for existence of endemic equilibrium points for Covid-19 model (3):
Theorem 4. The Covid-19 model system (3) has:
(i) no endemic equilibrium otherwise, if all the coefficient of Equation (11) are all positive;
(ii) a unique endemic equilibrium, if R 0 > 1 ;
(iii) a two (2) endemic equilibrium points, if R 0 < 1 ;
Thus, It is clear from Theorem 2 Case (ii) that the model has a unique endemic equilibrium whenever R 0 > 1 . Furthermore, Case (iii) establishes the existence
| a 0 > 0 , a 1 > 0 | a 2 < 0 | a 3 > 0 ( R 0 < 1 ) | 0 |
|---|---|---|---|
| a 3 < 0 ( R 0 > 1 ) | 1 | ||
| a 2 > 0 | a 3 > 0 ( R 0 < 1 ) | 2 | |
| a 3 < 0 ( R 0 > 1 ) | 1 |
of multiple equilibrium points which indicates the possibility of backward bifurcation (where the locally-asymptotically stable DFE co-exists with a locally-asymptotically stable endemic equilibrium (see, for instance, [
Δ = ( − 27 a 0 2 ) a 3 2 + ( 18 a 0 a 1 a 2 − 4 a 1 3 ) a 3 + ( a 1 2 a 2 2 − 4 a 0 a 2 3 ) = 0 (13)
solving for the critical value of R0 denoted by Rc
R c = 1 − a 3 γ A θ ( a + μ ) ( B 2 + α ) B 1 σ (14)
and the positive roots of a3 in polynomial (13) which can be viewed as a quadratic equation which yields
a 3 = − b − b 2 − 4 a c 2 a (15)
where a = − 27 a 0 2 < 0 , b = 18 a 0 a 1 a 2 − 4 a and c = a 1 2 a 2 2 − 4 a 0 a 2 3 when R 0 = R c , the largest positive solution of a3 is substituted into Equation (14) thus,
R c = 1 − a 3 γ A θ ( a + μ ) ( B 2 + α ) B 1 σ = 1 − − 18 a 0 a 1 a 2 + 4 a 1 3 − ( 18 a 0 a 1 a 2 − 4 a 1 3 ) 2 + 108 a 0 2 ( a 1 2 a 2 2 − 4 a 0 a 2 3 ) − 54 a 0 2 ( γ A θ ( a + μ ) ( B 2 + α ) B 1 σ ) = 1 − 18 a 0 a 1 a 2 − 4 a 1 3 + ( 18 a 0 a 1 a 2 − 4 a 1 3 ) 2 + 108 a 0 2 ( a 1 2 a 2 2 − 4 a 0 a 2 3 ) 54 a 0 2 ( γ A θ ( a + μ ) ( B 2 + α ) B 1 σ ) (16)
therefore Equation (16) gives the analytical expression of a threshold value for the reproduction number, and backward bifurcation would occur for values of Ro, such that R c < R 0 < 1 . The associated bifurcation diagram is depicted in
Lemma 5. The Covid-19 model system (3) exhibits backward bifurcation whenever case (iii) of theorem 4 holds and that R c < R 0 < 1 .
Proof.
By applying the following simplification and change of variables on model (3). Let S = x 1 , E = x 2 , H = x 3 , I = x 4 , D = x 5 and S ˙ = F 1 , E ˙ = F 2 , H ˙ = F 3 , D ˙ = F 5 , D ˙ = F 5 so that N = x 1 + x 2 + x 3 + x 4 + x 5 . Further, by using vector notation X = ( x 1 , x 2 , x 3 , x 4 , x 5 ) T model (3) is transformed as follows:
F 1 = A − γ ( x 3 + η 1 x 4 + η 2 x 5 ) x 1 + x 2 + x 3 + x 4 + x 5 − μ x 1 F 2 = γ ( x 3 + η 1 x 4 + η 2 x 5 ) x 1 + x 2 + x 3 + x 4 + x 5 − ( a + μ ) x 2 F 3 = a ( 1 − θ ) x 2 − ( ω 1 + d 1 + μ ) x 3 F 4 = a θ − α x 4 1 + κ x 4 − ( ω 2 + d 2 + μ ) x 4 F 5 = d 1 x 3 + d 2 x 4 − σ x 5 (17)
Therefore by considering the case when R o = 1 and chosen γ as a bifurcation parameter, while solving for γ ∗ from R o = 1 gives the critical value as:
γ ∗ = B 1 σ ( B 2 + α ) ( a + μ ) a ( 1 − θ ) ( B 2 + α ) ( σ + d 1 η 2 ) + a θ B 1 ( η 1 σ + d 2 η 2 ) (18)
and evaluating the Jacobian matrix (9)) at E0 when γ = γ ∗ has a simple zero eigenvalue ( λ 5 = 0 ), whose corresponding right eigenvector w = ( w 1 , w 2 , w 3 , w 4 , w 5 ) T and left eigenvector v = ( v 1 , v 2 , v 3 , v 4 , v 5 ) T are given by
w 1 = − a + μ μ , w 2 = 1 , w 3 = a ( 1 − θ ) B 1 , w 4 = a θ B 2 + α , w 5 = a θ [ ( d 2 η 2 + η 1 σ ) − ( d 1 η 1 − d 2 ) σ ] σ ( d 1 η 2 + σ ) ( B 2 + α ) (19)
furthermore,
v 1 = 0 , v 2 = a ( 1 − θ ) ( B 2 + α ) ( d 1 η 2 + σ ) + a B 1 θ ( d 2 η 2 + η 1 σ ) B 1 ( B 2 + α ) ( a + μ ) η 2 , v 3 = σ + η 2 d 2 η 2 B 1 , v 4 = η 1 σ + η 2 d 2 η 2 ( B 2 + α ) , v 5 = 1
Since v1 is zero, there is no need to find the derivative of F1 and the associated non-zero partial derivatives of the right hand side of (17) is given by:
∂ 2 f 2 ∂ x 2 ∂ x 3 = − γ μ A , ∂ 2 f 2 ∂ x 2 ∂ x 4 = − γ η 1 μ A , ∂ 2 f 2 ∂ x 3 2 = − 2 γ μ A , ∂ 2 f 2 ∂ x 3 ∂ x 4 = − γ μ ( 1 + η 1 ) A ∂ 2 f 2 ∂ x 3 ∂ x 5 = − γ μ ( 1 + η 2 ) A , ∂ 2 f 2 ∂ x 4 2 = − 2 γ η 1 μ A , ∂ 2 f 2 ∂ x 4 ∂ x 5 = − γ μ ( η 1 + η 2 ) A , ∂ 2 f 2 ∂ x 5 2 = − 2 γ η 2 μ A , ∂ 2 f 2 ∂ x 3 ∂ γ ∗ = 1 , ∂ 2 f 2 ∂ x 4 ∂ γ ∗ = η 1 , ∂ 2 f 2 ∂ x 5 ∂ γ ∗ = η 2
However, using the express in [
a = ∑ k , i , j = 1 n v k w i w j ∂ 2 f k ∂ x i ∂ x j ( 0 , 0 ) b = ∑ k , i = 1 n v k w i ∂ 2 f k ∂ x i ∂ γ ∗ ( 0 , 0 ) .
Below are the computation for a and b:
a = ∑ k , i , j = 1 5 v k w i w j ∂ 2 f k ∂ x i ∂ x j ( 0 , 0 ) a = − a ( 1 − θ ) μ η 2 A [ σ β 1 + 2 a σ ( 1 − θ ) β 1 2 + a θ σ ( 1 + η 1 ) β 1 ( β 2 + α ) + a θ ( 1 + η 2 ) T β 1 ] − a θ σ μ η 2 A [ η 1 β 2 + α + η 2 T σ + 2 a θ η 1 ( β 2 + α ) 2 + 2 a θ η 2 T 2 σ 2 + a θ ( η 1 + η 2 ) T σ ( β 2 + α ) ] + 2 ( d 2 η 2 + η 1 σ ) a 2 θ 2 α κ η 2 ( β 2 + α ) 3 (20)
b = ∑ k , i = 1 5 v k w i ∂ 2 f k ∂ x i ∂ γ ∗ ( 0 , 0 ) b = a ( 1 − θ ) ( B 2 + α ) ( d 1 η 2 + σ ) + a B 1 θ ( d 2 η 2 + η 1 σ ) B 1 ( B 2 + α ) ( a + μ ) η 2 ⋅ [ a ( 1 − θ ) β 1 + a θ η 1 β 2 + α + a θ η 2 T σ ] (21)
where T = ( d 1 η 2 + η 1 σ ) + ( d 2 − d 1 η 1 ) σ ( β 2 + α ) ( d 1 η 2 + σ ) .
From Theorem 4, It is established that the direction of bifurcation is determined by the sign of a and b. Furthermore, since the coefficient of b is always positive ( b > 0 ), it implies that whenever a > 0 the model exhibit a backward bifurcation and if a < 0 the model exhibits a forward bifurcation. The proof is complete □ .
Furthermore,
2 ( d 2 η 2 + η 1 σ ) a 2 θ 2 α κ η 2 ( β 2 + α ) 3 > a ( 1 − θ ) μ η 2 A [ σ β 1 + 2 a σ ( 1 − θ ) β 1 2 + a θ σ ( 1 + η 1 ) β 1 ( β 2 + α ) + a θ ( 1 + η 2 ) T β 1 ] + a θ σ μ η 2 A [ η 1 β 2 + α + η 2 T σ + 2 a θ η 1 ( β 2 + α ) 2 + 2 a θ η 2 T 2 σ 2 + a θ ( η 1 + η 2 ) T σ ( β 2 + α ) ] (22)
from Equation (22), we can point out that several parameters when stronger than some level will invoke backward bifurcation. Hence the effect of the infected being delayed for treatment, say κ , is one of the factors which lead to the backward bifurcation amist ( a , θ and α ).
Thus the existence of backward bifurcation is crucial and important in planning on how to control Covid-19 disease, the occurrence of a backward bifurcation at R 0 = 1 makes control more difficult which implies that the disease free equilibrium is not globally stable. One control measure often used is the reduction of susceptibility to infection produced by vaccination along with educational programs such as encouragement of better hygiene and preventive measures [
Parameter values use in plotting
a sudden rise as we have observed in the second wave of Covid-19 cases as already warned by Nigeria minister of health [
In this section, we perform parameter estimation and sensitivity analysis to provide some level of reliability to simulation of the proposed model.
In particular, we first fit the model system (3) with observed Nigeria Covid-19 data from NCDC available at [
Sensitivity analysis of the basic reproduction number of Covid-19 model to each of the parameter value is carried out, with the aim to examine behavioral of the model in response to the changes in parameter. However, contribution of parameters to the basic reproduction number reveals their effectiveness or significance to spread of Covid-19 disease in the populace with respect to time. In the present case, the focus is given to determine how changes in the model parameters impact the effective (R0). Thus, this is done through the normalized
forward-sensitive index, and this is defined as:
Γ R 0 ℵ = ∂ R 0 ∂ Π × Π R 0 (23)
Using (23), the sensitivity index value calculated to any parameter say ( Π ) are shown in the last column of
The sensitivity analysis of the basic reproduction number with respect to the model parameters is carried out in section (5.2) and Covid-19 model has been validated by fitting observed data in section (5.1), Thus we perform the numerical simulation of model system (3) using Runge-Kutta fourth order scheme, to elucidate the qualitative analysis of the model system based on variables and parameter values presented in
Using the following initial condition for the state variable S ( 0 ) = 170000000 , E ( 0 ) = 30840297 , I ( 0 ) = 43048 , H ( 0 ) = 21524 , D ( 0 ) = 1485 for the simulation of our proposed model, and also noting the demographic population of Nigeria to be 201,000,000 people.
In
At different increasing level of transmission rate from expose to infected individual (a), the cumulative death case depicts in
| Parameter | Value | Reference | Sensitivity Index |
|---|---|---|---|
| η 1 | 2 | Estimated | 0.00059 |
| η 2 | 0.05 | Estimated | 0.0078 |
| γ | 0.487 | [ | 1 |
| a | 0.196 day−1 | [ | 0.0841 |
| θ | 0.025 day−1 | [ | −0.0250 |
| d 1 | 0.142 day−1 | [ | −0.6912 |
| d 2 | 0.071 day−1 | [ | −0.00003 |
| w 2 | 0.0103 day−1 | [ | −0.000073 |
| w 1 | 0.043 day−1 | [ | −0.2117 |
| σ | 0.6 | Estimated | −0.00784 |
| μ | 0.018 | [ | −0.1727 |
| α | 1 | Estimated | −0.000546 |
| κ | 4 | Estimated | |
| A | 8.4 | Estimate |
more individual becoming symptomatic and this connotes high mortality rate, as a result of certain proportion of individuals ( 1 − θ ) who self isolate and limited medical resources available for hospitalized patients.
In
Also
might grew weary and decide to source for other means of survival.
In this work, we presented a compartmental, deterministic model to assess the impact of undetected infected individuals and infected hospitalized individuals with saturated treatment on the spread of Covid-19 disease transmission dynamics in Nigeria. The model’s basic properties such as positivity and boundedness of solutions, the basic reproduction number as well as the steady states were determined and analysed. The model reveals continuum of disease-free equilibria which was shown to be locally-asymptotically stable whenever the associated basic reproduction number (denoted by R0) is less than unity, or globally-asymptotically stable for the value of the basic reproduction number less than the critical value ( R c = 0.96 ). The model also reveals that even if the transmission rate for exposed Covid-19 patients is lower, controlling the diseases in Nigeria will prove difficult because of the presence of backward bifurcation. The model fitted well to the NCDC (Nigeria center for disease control) cumulative data from Feb. 28 to Dec. 5, 2020. However, the sensitivity analysis of the model parameters reveals that the most sensitive parameters are the force of infection with susceptibility and the rate of transmission from exposing compartment to the infected. Thus, any control measure that can significantly reduce these parameters will contribute effectively to eradicating the disease from the population. Also, the model simulation reveals that an increase in transmission rate for infected Covid-19 patients yields an increase in cumulative death and rapid spread of the disease in the population. Thus decreasing the saturated treatment alone as shown by decreasing η 1 in
Sincere thanks to the members of OALib Journal for their professional performance, and special thanks to managing editor Yaqin YAN for a rare attitude of high quality.
The authors declare no conflicts of interest regarding the publication of this paper.
Oke, I.I., Oyebo, Y.T., Fakoya, O.F., Benson, V.S. and Tunde, Y.T. (2021) A Mathematical Model for Covid-19 Disease Transmission Dynamics with Impact of Saturated Treatment: Modeling, Analysis and Simulation. Open Access Library Journal, 8: e7332. https://doi.org/10.4236/oalib.1107332