In this paper, a deterministic mathematical model for Chikungunya virus (Chikv) transmission and control is developed and analyzed to underscore the effect of vaccinating a proportion of the susceptible human, and vertical transmission in mosquito population. The disease free, and endemic equilibrium states were obtained and the conditions for the local and global stability or otherwise were given. Sensitivity analysis of the effective reproductive number, R c (the number of secondary infections resulting from the introduction of a single infected individual into a population where a proportion is fairly protected) shows that the recruitment rate of susceptible mosquito (ΛM) and the proportion of infectious new births from infected mosquito (β) are the most sensitive parameters. Bifurcation analysis of the model using center manifold theory reveals that the model undergoes backward bifurcation (coexistence of disease free and endemic equilibrium when Rc ＜ 1 ). Numerical simulation of the model shows that vaccination of susceptible human population with imperfect vaccine will have a positive impact and that vertical transmission in mosquito population has a negligible effect. To the best of our knowledge, our model is the first to incorporate vaccinated human compartment and vertical transmission in (Chikv) model.
Chikungunya is a mosquito-borne viral disease that was first observed in Tanzania in 1952 [
In recent years, the virus has risen from relatively obscurity to become a global public health menace affecting millions of persons throughout the tropical and subtropical regions of the world and as such has also become a frequent cause of travel associated febrile illness [
Diagnosis is by confirming the presence of anti-Chikungunya antibody in the patient. At the moment, there is no vaccine or treatment for the disease. Protection is by covering of exposed skin with long pants and long sleeved shirts, insect repellents and insecticide treated mosquito nets. Since the beginning of the 19th century, mathematical model has become a veritable tool in the study of vector-borne diseases [
Pongsumpun and Sangsawang [
In this work, we proposed a deterministic mathematical model for the spread, and control of Chikv. Our model attempt to bridge identified gaps in the works cited above. Specifically, our model incorporated an imperfect vaccinated human compartment and vertical transmission in the mosquito population.
The chic model is represented by nine non-linear ordinary differential equation consisting of human-sub population and mosquito sub-population. The human sub-population is divided into; susceptible human S H , vaccinated human V H , exposed human E H , infected symptomatic human I 1 , infected asymptomatic human I 2 , recovered Human R, such that the total human population, N H = S H + V H + E H + I 1 + I 2 + R . While the mosquito sub-population is divided into; susceptible mosquito S M , exposed mosquito E M , and infected mosquito I 3 , such that the total mosquito population, N M = S M + E M + I 3 .
The parameters of the model and their values are given in
The susceptible human sub-population is generated at a constant rate Λ H , which includes birth and immigration. The vaccinated population is generated as members of the susceptible population receive vaccination at the rate ν , a proportion of the vaccinated with time lose their immunity at the rate ψ as their vaccine wanes and move back to the susceptible population. Member of the
susceptible and vaccinated populations acquire infection at the rate α 1 b m I M N H and α 1 b m I M ( 1 − ε ) N H respectively and move to the exposed population, where
α 1 is the probability of infection, b m biting rate of mosquito and ε (where 0 < ε < 1 ) is the efficacy of the imperfect vaccine. Members of the exposed population move to either symptomatic infectious population at the rate σ 1 or to asymptomatic infectious population at the rate ( 1 − σ 1 ) . The recovered population is generated as both symptomatic and asymptomatic infected populations recover with lifelong immunity at the rate γ . All human population are decreased by natural death at the rate μ 1 , except the two infected populations that are decreased by disease induced death at the rate δ .
The susceptible mosquito population is generated by Λ M , this population is decreased by birth from infected mosquito (vertical transmission) at the rate β Λ M ; and as its members take a blood meal from either symptomatic or asymptomatic infected human (horizontal transmission) at the rate α 2 . The exposed mosquito population progresses to infected mosquito population at the rate σ 2 . It is assumed that births from infected mosquito do not pass through the exposed class. All sub-populations of mosquito die naturally at the rate μ 2 .
| Parameters | Meaning | Value | Reference |
|---|---|---|---|
| Λ H | Recruitment rate of susceptible human | 0.073 | [ |
| α 1 | Contact rate of susceptible human when bitten by Aides Mosquitoes | 0.24 | [ |
| μ 1 | Natural death of human | 0.000039 | [ |
| σ 1 | Progression rate of exposed human to Symptomatic and Asymptomatic | 0.33 | [ |
| δ 1 | Death rate of human due to virus infection | 0.02 | Assumed |
| γ | Recovery rate of infectious human | 0.68 | [ |
| Λ M | Birth rate of Susceptible Aides Mosquitoes | 83.75 | [ |
| β | Proportion of infectious new birth from infected Aides Mosquitoes | 0.00005 | Assumed |
| α 2 | The rate at which susceptible Aides become infectious | 0.24 | [ |
| σ 2 | Progression rate of exposed Aedes | 0.285 | Assumed |
| μ 2 | Natural death rate of Aides | 0.0714 | [ |
| ν | The rate at which susceptible human receive vaccine | Variable | |
| ψ | The rate at which vaccine wane | Variable | |
| ε | Vaccine efficacy where 0 < ε < 1 | Variable | |
| b m | Biting rate of mosquito | 0.25 | [ |
From the model formulation, and schematic diagram
d S H d t = Λ H + ψ V H − α 1 b M S H I M N H − ( ν + μ 1 ) S H , (1)
d V H d t = ν S H − α 1 b M ( 1 − ε ) V H I M N H − ( ψ + μ 1 ) V H , (2)
d E H d t = α 1 b M I M N H ( S H + ( 1 − ε ) V H ) − ( σ 1 + μ 1 ) E H , (3)
d I 1 d t = σ 1 E H − ( γ + μ 1 + δ ) I 1 , (4)
d I 2 d t = ( 1 − σ 1 ) E H − ( γ + μ 1 + δ ) I 2 (5)
d R d t = γ I 1 + γ I 2 − μ 1 R , (6)
d S M d t = Λ M − α 2 b M S M ( I 1 + I 2 ) N H − β Λ M I M − μ 2 S M , (7)
d E M d t = α 2 S M b M ( I 1 + I 2 ) N H + β Λ M I M − ( σ 2 + μ 2 ) E M , (8)
d I M d t = σ 2 E M − μ 2 I M . (9)
Adding (1) to (6) gives
d N H d t = Λ H − δ ( I 1 + I 2 ) − μ 1 N H . (10)
Also adding (7) to (9), gives
d N M d t = Λ M − μ 2 N M . (11)
where
N H ( t ) = S H ( t ) + V H ( t ) + E H ( t ) + I 1 ( t ) + I 2 ( t ) + R ( t ) , (12)
N M ( t ) = S M ( t ) + E M ( t ) + I M ( t ) . (13)
(12) and (13) are the total human population and Aides mosquito population respectively.
For the Chikungunya model (1) to (9) to be epidemiological meaningful, it is necessary to prove that all its state variables are non-negative for all time. This means that the solution of the model Equations (1) to (9) with non-negative initial data will remain non-negative for all time t > 0 .
Lemma 1.
The closed set
D = { ( S H , V H , E H , I 1 , I 2 , R , S M , E M , I M ) ∈ ℜ + 9 : S H , V H , E H , I 1 , I 2 , R ≤ Λ H μ 1 ； S M , E M , I M ≤ Λ M μ 2 ; } . (14)
is positively-invariant and attracting with respect to the basic model Equations (1) to (9).
Proof
From Equations (10) and (11);
d N H d t ≤ Λ H − μ 1 N H , d N M d t ≤ Λ A − μ 2 N M .
It follows that d N H d t < 0 and d N M d t < 0 if N H ( t ) > Λ H μ 1 and N A ( t ) > Λ M μ 2 respectively. Thus a standard comparison theorem as in Lakshmikantham and Martynyuk, [
N H ( t ) ≤ N H ( 0 ) e − μ 1 ( t ) + Λ M μ 1 ( 1 − e − μ 1 ( t ) ) and N M ( t ) ≤ N M ( 0 ) e − μ 2 ( t ) + Λ M μ 2 ( 1 − e − μ 2 ( t ) ) . In particular N H ( t ) ≤ Λ H μ 1 and N M ( t ) ≤ Λ M μ 2 if N H ( 0 ) ≤ Λ H μ 1 and N A ( 0 ) ≤ Λ M μ 2 respectively. Thus D is positively-invariant. Further, if N H ( 0 ) > Λ H μ 1 , and N M ( 0 ) > Λ M μ 2 , then either the solution enters D in finite time or N H ( t ) approaches Λ H μ 1 , and N M ( t ) approaches Λ M μ 2 , and the infected variables E H , I 1 , I 2 , E A , I 3 approaches 0.
Hence D is attracting, that is all solutions in ℜ + 9 eventually enters D. Thus in D, the basic model Equations (1) to (9) is well posed epidemiologically and mathematically according to [
Lemma 2. Let the initial data F ( 0 ) ≥ 0 ,
where
F ( t ) = ( S H , V H , E H , I 1 , I 2 , R , S M , E M , I M ) .
Then the solution F ( t ) of the Chikungunya virus model (1) to (9) are non-negative for all t ≥ 0 . Furthermore form (10) and (11),
lim t → ∞ sup N H ( t ) = Λ H μ 1 + δ and lim t → ∞ sup N M ( t ) = Λ M μ 2 .
Proof
t 1 = sup { t > 0 : F ( t ) > 0 ∈ [ 0 , t ] } . Thus t 1 > 0 . It follows from (1) that
d d t { S H ( t ) exp [ α 1 b m ∫ 0 t 1 I M N H ( ξ ) d ξ + ( ν + μ 1 ) t ] } = ( Λ H + ψ V H ) exp [ α 1 b M ∫ 0 t 1 I M N H ( ξ ) d ξ + ( ν + μ 1 ) t ] , (15)
So that,
d d t S H ( t 1 ) exp [ α 1 b m ∫ 0 t 1 I M N H ( ξ ) d ξ + ( ν + μ 1 ) t 1 ] − S H ( 0 ) = ∫ 0 t 1 ( Λ H + ψ V H ) exp [ α 1 b m ∫ 0 P I M N H ( ξ ) d ξ + ( ν + μ 1 ) p ] d p (16)
Hence,
S H ( t 1 ) = S H ( 0 ) exp [ − α 1 b m ∫ 0 t 1 I M N H ( ξ ) d ξ + ( ν + μ 1 ) t 1 ] + exp [ − α 1 b m ∫ 0 t 1 I M N H ( ξ ) d ξ + ( ν + μ 1 ) t 1 ] ∫ 0 t 1 ( Λ H + ψ V H ) exp [ α 1 b M ∫ 0 P I M N H ( ξ ) d ξ + ( ν + μ 1 ) p ] d p > 0. (17)
Similarly, it can be shown that F > 0 , for all t > 0 .
For the second part of the proof, note that,
0 < V H ( t ) ≤ N H ( t ) , 0 < E H ( t ) ≤ N H ( t ) , 0 < I 1 ( t ) ≤ N H ( t ) , 0 < I 2 ( t ) ≤ N H ( t ) , 0 < R ( t ) ≤ N H ( t ) , 0 < S M ( t ) ≤ N M ( t ) , 0 < E M ( t ) ≤ N M ( t ) , 0 < I M ( t ) ≤ N M (t)
From Equations (10) and (11),
Λ H μ 1 + δ ≤ lim t → ∞ inf N H ( t ) ≤ lim t → ∞ sup N H ( t ) = Λ H μ 1 + δ , (18)
and
Λ M μ 2 ≤ lim t → ∞ inf N M ( t ) ≤ lim t → ∞ sup N M ( t ) = Λ M μ 2 . (19)
as required.
The basic model (1) to (9) has a DFE, E 0 obtained by setting the right-hand sides of the model equations to zero, which gives:
E 0 = ( S H * , V H * , E H * , * I 1 * , I 2 * , S M * , E M * , I M * ) = ( Λ H ( ψ + μ 1 ) ( ψ + μ 1 + ν ) μ 1 , ν Λ H ( ψ + μ 1 + ν ) μ 1 , 0 , 0 , 0 , 0 , Λ M μ 2 , 0 , 0 ) (20)
The linear stability of E 0 can be established using the next generation Matrix operator method on the system (I) to (9). Using the notation in [
F = [ 0 0 0 0 α 1 b m [ S H * + ( 1 − ε ) V H * ] N H * 0 0 0 0 0 0 0 0 0 0 0 α 2 b m S M * N H * α 2 b m S M * N H * 0 0 0 0 0 0 0 ] , (21)
and,
V = [ − K 3 0 0 0 0 − σ 1 K 4 0 0 0 ( 1 − σ 1 ) 0 K 4 0 0 0 0 0 K 5 0 0 0 0 − σ 2 K 6 ] . (22)
where,
K 1 = ν 1 + μ 1 , K 2 = ψ + μ 1 , K 3 = σ 1 + μ , K 4 = γ + μ 1 + δ , K 5 = σ 2 + μ 2 , K 6 = μ 1 − β Λ M (23)
R c = 1 2 ( M 1 + M 2 + M 3 M 4 ) , (24)
M 1 = Λ M β K 3 K 4 K 5 N H * , (25)
M 2 = Λ M β K 3 K 4 K 5 N H * , (26)
M 3 = 4 K 3 K 4 K 5 K 6 α 2 σ 2 b m S M * ( α 1 b m S H * + ( 1 − ε ) ) N H * , (27)
M 4 = β K 3 K 4 K 5 K 6 N H * . (28)
Hence using theorem 2 of [
Theorem 1 The disease free equilibrium, E 0 of the model (2.1) to (2.9) is locally asymptotically stable (LAS) if R c < 1 , and unstable if R c > 1 .
Consider the feasible region:
D 1 = { X ∈ D 1 : S H ≤ S H * , V H ≤ V H * , R ≤ R * , S M ≤ S M * } , (29)
X = { S H , V H , E H , I 1 , I 2 , R , S M , E M , I M } . (30)
Lemma 3. The region D 1 is positively invariant for the Chikungunya model Equations (1) to (9).
Proof
From Equations (1) to (9) and (20),
we have that, the only non-zero compartments at disease free equilibrium are;
d S H d t = Λ H + ψ V H − α 1 b M S H I M N H − ( ν + μ 1 ) S , d V H d t = ν S H − α 1 b M ( 1 − ε ) V H I M N H − ( ψ + μ 1 ) V H , d S M d t = Λ M − α 2 b M S M ( I 1 + I 2 ) N H − β Λ M I M − μ 2 S M (31)
Such that,
d S H d t = Λ H + ψ V H − α 1 b m S H I M N H − ( ν + μ 1 ) S H , ≤ Λ H + ψ V H − ( ν + μ 1 ) S H ≤ ( ν + μ 1 ) [ Λ H ( ψ + μ 1 ) ( ψ + μ 1 + ν ) μ 1 + ψ ν Λ H ( ψ + μ 1 + ν ) μ 1 − S H ] = ( ν + μ 1 ) ( S H * + ψ V H * − S H ) , (32)
Hence,
S H ( t ) ≤ S H * + ψ V H * − [ S H * − ψ V H * − S H ( 0 ) ] e − ( ν + μ 1 ) t . (33)
Thus if N H * = Λ H μ 1 and S H ( 0 ) ≤ S H * + ψ V H * for all t ≥ 0 , then S H ( t ) ≤ S H * + ψ V H * for all t ≥ 0 .
Similarly, it follows from Equation (7) of our model and (20) where S M * = Λ M μ 2 .
We have that,
d S M d t = Λ M − α 2 b M S M ( I 1 + I 2 ) N H − β Λ M I M − μ 2 S M ≤ Λ M − μ 2 S M ≤ μ 2 [ Λ M μ 2 − S M ] = μ 2 ( S M * − S M ) . (34)
Hence,
S M ( t ) ≤ S M * − [ S M * − S M ( 0 ) ] e − μ 2 t . (35)
Thus if N M * = Λ M μ 2 and S M ( 0 ) ≤ S M * for all t ≥ 0 , then S M ( t ) ≤ S M * for all t ≥ 0 .
In summary, we have shown that D 1 is positively invariant and attracting with respect to the solutions of our model Equations (1) to (9).
Theorem 2
The DFE of the basic model (1) to (9) is Global Asymptotical Stability (GAS) in D 1 , whenever R C ≤ 1 .
Proof
To prove the GAS of the DFE we adopt the approach in [
Let X = ( S H , V H , R , S M ) and Z = ( E H , I 1 , I 2 , E M , I M ) and group our model Equations (1) to (8) into:
d X d t = F ( X , 0 ) , d Z d t = G ( X , Z ) . (36)
where F ( X , 0 ) is the right hand side of S ˙ H , V ˙ H , R ˙ , S ˙ M with E H = I 1 = I 2 = E M = I M = 0 and G ( X , Z ) , the right hand side of E ˙ H , I ˙ 1 , I ˙ 2 , E ˙ M , I ˙ M . Next we consider the reduced system:
d X d t = F ( X , 0 ) given as,
d S H d t = Λ H − μ 1 S H , d V H d t = ν S H − ( ψ + μ 1 ) V H , d R d t = − μ 1 R , d S M d t = Λ M − μ 2 S M . (37)
Let X * = ( S H * , V H * , R * , S M * ) = ( Λ H ( ψ + μ 1 ) ( ψ + μ 1 + ν ) μ 1 , ν Λ H ( ψ + μ 1 + ν ) μ 1 , 0 , Λ M μ 2 ) (38)
be an equilibrium of (37) we show that X * is a global stable equilibrium in D 1 .
To do this, we solve the Equations (37), which gives
S H ( t ) = ( Λ H ( ψ + μ 1 ) ( ψ + μ 1 + ν ) μ 1 + ψ V H * ) − ( Λ H ( ψ + μ 1 ) ( ψ + μ 1 + ν ) μ 1 + ψ V H * ) e − ( ( ψ + μ 1 + ν ) μ 1 ) t + S H ( 0 ) e ( ( ψ + μ 1 + ν ) μ 1 ) t , S H ( t ) → Λ H ( ψ + μ 1 ) ( ψ + μ 1 + ν ) μ 1 + ψ V H * , (39)
as t → ∞ .
V H ( t ) = ν Λ H ( ψ + μ 1 + ν ) μ 1 − ν Λ H ( ψ + μ 1 + ν ) μ 1 e − ( ( ψ + μ 1 + ν ) μ 1 ) t + V H ( 0 ) e ( ( ψ + μ 1 + ν ) μ 1 ) t , V H ( t ) → ν Λ H ( ψ + μ 1 + ν ) μ 1 , (40)
as t → ∞ .
R ( t ) = R ( 0 ) e − μ 1 t , R ( t ) → 0 , (41)
as t → ∞ .
S M ( t ) = Λ M μ 2 − Λ M μ 2 e − μ 2 t + S M ( 0 ) e − μ 2 t , S M ( t ) → Λ M μ 2 , (42)
as t → ∞ .
This asymptotic dynamics is independent of initial conditions in D. Hence the solution of xxx converges globally in D 1 .
Next we are required to show that G ( X , Z ) satisfies the following two conditions in [
G ( X , 0 ) = 0 , G ( X , Z ) = D Z G ^ ( X * , 0 ) Z − G ^ ( X , Z ) , G ^ ( X , Z ) ≥ 0 , (43)
where,
( X * , 0 ) = ( Λ H ( ψ + μ 1 ) ( ψ + μ 1 + ν ) μ 1 , ν Λ H ( ψ + μ 1 + ν ) μ 1 , 0 , Λ M μ 2 ) . (44)
and D Z G ( X * , 0 ) is the Jacobian of G ( X , Z ) taken with respect to ( E H , I 1 , I 2 , E M , I M ) and evaluated at ( X * , 0 ) , which is an M-Matrix (the off diagonal elements are non-negative).
Thus,
D Z G ( X * , 0 ) = ( − k 3 0 0 0 Q 1 σ 1 − k 4 0 0 0 1 − σ 1 0 − k 4 0 0 0 α 2 b m S M * N H * α 2 b m S M * N H * − k 5 0 0 0 0 σ 2 − k 6 ) , (45)
G ^ ( X , Z ) = ( 0 0 0 0 Q 2 I M 0 0 0 0 0 0 0 0 0 0 0 α 2 b M N H * S M * Q 3 α 2 b M N H * S M * Q 3 0 0 0 0 0 0 1 β Λ M ) , (46)
where,
Q 1 = α 1 b m S H * + ( 1 − ε ) + V H * N H * , Q 2 = N H * S H * + ( 1 − ε ) + V H * ( 1 − N H * S H * + ( 1 − ε ) + V H * S H + ( 1 − ε ) V H N H ) , Q 3 = ( 1 − N H * S M * S M N H I 1 ) , Q 4 = ( 1 − N H * S M * S M N H I 2 ) . (47)
Further S H ≤ S H * , V H ≤ V H * and S M ≤ S M * in D 1 . Thus, it follows that ( 1 − S H S H * ) > 0 , ( 1 − V H V H * ) > 0 and ( 1 − S M S M * ) > 0 . Hence G ^ ( X , Z ) ≥ 0 .
Therefore, by the theorem 2 in [
Here we present the sensitivity index of the parameters of the effective reproductive number ( R C ) . Sensitivity tells us how important each parameter is to disease transmission. Such information, is crucial not only to experimental design, but also to data assimilation and reduction of complex nonlinear model [
| Parameter | Sensitivity index |
|---|---|
| α 1 | 0.06 |
| α 2 | 0.37 |
| μ 1 | 0.63 |
| μ 2 | 0.87 |
| σ 1 | 0.02 |
| σ 2 | 0.87 |
| β | 1.25 |
| ψ | −0.12 |
| ε | −0.86 |
| δ | 0.13 |
| γ | 0.5 |
| Λ H | 0.62 |
| Λ M | 1.6 |
| ν | 0.46 |
From
Endemic Equilibrium
Let E 1 = ( S H * * , V H * * , E H * * , R * * , I 1 * * , I 2 * * , R * * , S M * * , E M * * , I M * * ) , (48)
represents any arbitrary endemic equilibrium of the model (1) to (9). Further, let
λ H * * = α 1 b m I M * * N H * * , λ M * * = α 2 b M ( I 1 * * + I 2 * * ) N H * * . (49)
be the forces of infection of humans and vectors at steady state, respectively. Solving (1) to (9) in terms of λ H * * and λ M * * , we have;
S H * * = Λ H ( λ H * * + k 2 ) ( λ H * * + k 2 ) ( λ H * * + k 1 ) + ψ ν , V H * * = ν Λ H ( λ H * * + k 2 ) ( λ H * * + k 1 ) + ψ ν , E H * * = Λ H λ H * * ( λ H * * + k 2 ) + ( 1 − ε ) ( ( λ H * * + k 2 ) ( λ H * * + k 1 ) − ψ ν ) k 3 , I 1 * * = σ 1 Λ H λ H * * ( ( λ H * * + k 2 ) + ( 1 − ε ) ) ν ( ( λ H * * + k 2 ) ( λ H * * + k 1 ) − ψ ν ) k 3 k 4 , I 2 * * = ( 1 − σ 1 ) Λ H λ H * * ( ( λ H * * + k 2 ) + ( 1 − ε ) ) ν ( ( λ H * * + k 2 ) ( λ H * * + k 1 ) − ψ ν ) k 3 k 4 , R * * = γ Λ H λ H * * ( ( λ H * * + k 2 ) + ( 1 − ε ) ) ν ( ( λ H * * + k 2 ) ( λ H * * + k 1 ) − ψ ν ) k 3 k 4 μ 1 , S M * * = k 5 k 6 Λ M λ M * * ( k 5 k 6 + β Λ M σ 2 ) + μ 2 k 5 k 6 , E M * * = k 6 Λ M λ M * * λ M * * ( k 5 k 6 + β Λ M σ 2 ) + μ 2 k 5 k 6 , I M * * = σ 2 Λ M λ M * * λ M * * ( k 5 k 6 + β Λ M σ 2 ) + μ 2 k 5 k 6 . (50)
Substituting (20) into (19) we have;
λ M * * = α 2 b m Λ M λ H * * ( ( λ H * * + k 2 ) + ( 1 − ε ) ) ν ( ( λ H * * + k 2 ) ( λ H * * + k 1 ) − ψ ν ) k 3 k 4 , λ H * * = A ( λ H * * ) 4 + B ( λ H * * ) 3 + C ( λ H * * ) 2 + D ( λ H * * ) − E . (51)
where,
A = ( α 2 b m Λ H k 5 k 6 + β Λ M σ 2 ) ( μ 1 + k 4 μ 1 + γ ) , (52)
B = ( T 2 k 5 k 6 + β Λ M σ 2 ) ( k 3 k 4 ( μ 1 k 2 + ψ ) ) − ( α 1 α 2 ( b m ) 2 Λ H σ 2 ) k 3 k 4 μ 1 , (53)
C = [ α 2 b m Λ H σ 2 ( k 2 + ( 1 − ε ) ν ) k 5 k 6 + β Λ M σ 2 ( k 3 k 4 μ 1 k 2 + k 1 μ 1 + k 4 μ 1 + γ ( 1 − ε ) ν ) ] − α 1 α 2 ( b m ) 2 Λ H σ 2 k 3 k 4 μ 1 ( k 1 + k 2 ) + α 1 α 2 ( b m ) 2 Λ H σ 2 ( k 2 + ( 1 − ε ) ν ) k 3 k 4 μ 1 , (54)
D = ( α 2 b m Λ H σ 2 ( k 2 + ( 1 − ε ) ν ) ( α 2 b m Λ H k 5 k 6 + β Λ M σ 2 ) ( k 3 k 4 ( μ 1 k 2 + ν ) ) ) − ( α 1 α 2 ( b m ) 2 Λ H σ 2 ( k 2 + ( 1 − ε ) ν ) ( k 1 + k 2 ) k 3 k 4 μ 1 ) + ( k 1 k 2 k 3 k 4 − ψ ν k 3 k 4 μ 1 ) ( α 1 α 2 ( b m ) 2 Λ H σ 2 ) , (55)
E = α 1 α 2 ( b m ) 2 Λ H σ 2 ( k 2 + ( 1 − ε ) ν ) k 1 k 2 k 3 k 4 − ψ ν k 3 k 4 μ 1 (56)
Theorem 3.6. The Chikungunya basic model (1) to (9) undergoes backward bifurcation whenever the coefficient a in equation is positive.
Proof. To prove this theorem, we use the Centre Manifold theory as in Castillo-Chavez and songs [
Let S H = x 1 , V H = x 2 , E H = x 3 , I 1 = x 4 , I 2 = x 5 , R = x 6 , S M = x 7 , E M = x 8 , I M = x 9 so that N H = x 1 + x 2 + x 3 + x 4 + x 5 + x 6 and N M = x 7 + x 8 + x 9 . Further by using vector notation
X = ( x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 + x 9 ) T Equations (1) to (9) can be written
as d X d t = ( f 1 + f 2 + f 3 + f 4 + f 5 + f 6 + f 7 + f 8 + f 9 ) T as follow:
d x 1 d t = Λ H + ψ V H − α 1 b M x 1 x 9 x 1 + x 2 + x 3 + x 4 + x 5 + x 6 − k x 1 , d x 2 d t = ν x 2 − α 1 b M ( 1 − ε ) x 2 x 9 x 1 + x 2 + x 3 + x 4 + x 5 + x 6 − k 2 x 2 , d x 3 d t = α 1 b M x 9 x 1 + x 2 + x 3 + x 4 + x 5 + x 6 ( x 1 + ( 1 − ε ) x 2 ) − k 3 x 3 , d x 4 d t = σ 1 x 3 − k 4 x 4 , d x 5 d t = ( 1 − σ 1 ) x 3 − ( γ + μ 1 + δ ) x 5 , d x 6 d t = γ x 4 + γ x 5 − μ 1 x 6 , d x 7 d t = Λ M − α 2 b M x 7 ( x 4 + x 5 ) x 1 + x 2 + x 3 + x 4 + x 5 + x 6 − β Λ M x 9 − μ 2 x 7 , d x 8 d t = α 2 S M b M ( x 4 + x 5 ) x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + β Λ M x 9 − k 5 x 8 , d x 9 d t = σ 2 x 8 − k 6 x 9 . } (57)
Because it’s not always convenient to use R C = 1 as bifurcation parameter, we choose P = P * where P * = α 2 b m as the bifurcation parameter such that,
P * = 1 2 ( M 4 M 1 + M 2 + M 5 ) , (58)
where
M 5 = 4 k 3 k 4 k 5 k 6 α 2 x 1 * ( α 7 b m x 1 * + ( 1 − ε ) x 2 * ) . (59)
The Jacobian of (57) evaluated at E 0 with α 2 b m = P * , denoted by J * is given
J * = [ − k 1 0 0 0 0 0 0 0 − Q 5 ν − k 2 0 0 0 0 0 0 − Q 6 0 0 − k 3 0 0 0 0 0 0 0 0 σ 1 − k 4 0 0 0 0 0 0 0 1 − σ 1 0 − k 4 0 0 0 0 0 0 0 γ γ − μ 1 0 0 0 0 0 0 Q 7 Q 7 0 μ 2 0 − β Λ M 0 0 0 Q 7 Q 7 0 0 − k 5 0 0 0 0 0 0 0 0 σ 2 − k 6 ] (60)
where,
Q 5 = α 1 b m x 1 * x 1 + x 2 + x 3 + x 4 + x 5 + x 6 , Q 6 = α 1 b m ( 1 − ε ) x 2 * x 1 + x 2 + x 3 + x 4 + x 5 + x 6 , Q 7 = α 2 b m x 7 * x 1 + x 2 + x 3 + x 4 + x 5 + x 6 . (61)
It follows that (60) has a right eigenvector denoted by v = v 1 , v 2 , v 3 , v 4 , v 5 , v 6 , v 7 , v 8 , v 91 , where
v 1 = − Q 5 ( k 1 k 2 + ν ψ ) + ψ ( Q 6 k 1 + ν Q 5 ) v 9 ( k 1 k 2 − ν ψ ) k 1 , v 2 = − ( Q 6 k 1 + ν Q 5 ) v 9 k 1 k 2 − ν ψ , v 3 = Q 1 v 9 k 3 , v 4 = σ 1 Q 1 v 9 k 3 k 4 , v 5 = ( 1 − σ 1 ) Q 1 v 9 k 3 k 4 ,
v 6 = γ Q 1 k 4 ( Q 1 + ( 1 − σ 1 ) ) v 9 k 3 k 4 2 , v 7 = ( β Λ M k 4 2 k 3 + Q 7 Q 1 k 1 ) v 9 k 3 k 4 2 μ 2 , v 8 = Q 7 Q 1 k 4 ( σ 1 + ( 1 − σ 1 ) ) v 9 k 3 k 4 2 k 5 , v 9 = v 9 . (62)
And a left eigenvector given by w = w 1 , w 2 , w 3 , w 4 , w 5 , w 6 , w 7 , w 8 , w 91 , where
w 1 = ν w 2 k 1 , w 2 = w 2 , w 3 = Q 7 w 7 + γ w 6 k 3 k 4 , w 4 = w 5 = γ w 6 − Q 7 w 7 k 3 k 4 ,
w 6 = w 6 , w 7 = w 7 , w 8 = w 9 = 0. (63)
Computation of a
∂ 2 f 1 ∂ x 1 ∂ x 9 = α 1 b m x 1 * ( x 1 * + x 2 * ) 2 − α 1 b m x 1 * + x 2 * ,
∂ 2 f 1 ∂ x 2 ∂ x 9 = ∂ 2 f 1 ∂ x 3 ∂ x 9 = ∂ 2 f 1 ∂ x 4 ∂ x 9 = ∂ 2 f 1 ∂ x 6 ∂ x 9 = α 1 b m x 1 * ( x 1 * + x 2 * ) 2 , ∂ 2 f 1 ∂ x 1 ∂ x 2 = k 1 − ψ x 1 * + x 2 * + 2 ( ψ x 2 * − k 3 x 1 * ) ( x 1 * + x 2 * ) 3 , ∂ 2 f 7 ∂ x 4 ∂ P * = ∂ 2 f 7 ∂ x 5 ∂ P * = ∂ 2 f 7 ∂ x 7 ∂ P * = x 7 x 1 * + x 2 * , ∂ 2 f 7 ∂ x 4 ∂ x 7 = ∂ 2 f 7 ∂ x 5 ∂ x 7 = μ 2 ( x 1 * + x 2 * ) 2 − P * x 1 * + x 2 * . (64)
a = ∑ k , i , j = 1 n v k w i w j ∂ 2 f k ∂ x i ∂ x j ( 0 , 0 ) = v 1 w 9 α 1 b m x 1 * ( x 1 * + x 2 * ) 2 ( w 1 + w 2 + w 3 + w 4 + w 5 + w 6 ) − v 1 w 2 ( w 9 α 1 b m x 1 * + x 2 * + w 2 ( k 3 + ψ x 1 * + x 2 * + 2 ψ x 2 * + k 3 x 1 * ( x 1 * + x 2 * ) 3 ) ) − v 7 w 7 ( w 4 + w 5 ) ( μ 2 ( x 1 * + x 2 * ) 2 − P * x 1 * + x 2 * ) . (65)
b = ∑ k , i , j = 1 n v k w i ∂ 2 f k ∂ x i ∂ P * = v 7 ( w 4 + w 5 + w 7 ) ( x 7 * x 1 * + x 2 * ) . (66)
Vaccine was believed to confer life-long immunity until 1990s. This was the norm as it was approximately correct for most available vaccine for infectious children diseases. But most vaccines used for combating adult infectious diseases today are defective and thus immunity conferred on the recipients wane with time. It is expected that the future Chikv vaccine will also be defective and hence the need to assess its effectiveness in R C a community. In this paper, the vaccine impact analysis is done by differentiating effective reproductive number with respect to the proportion p of susceptible individuals vaccinated at
equilibrium, according to [
To further verify the analytical results in the model, the ode 45 code embedded in matlab was used to simulate some parameters of the model.
with various values of the vertical transmission rate ( β ) .
A deterministic mathematical model for Chikungunya virus dynamics was developed using the standard incidence approach. The model assumed that the offspring of infected mosquito is infected at birth (vertical transmission) and also through blood meal from symptomatically and as-symptomatically infected human (horizontal transmission). For the subhuman population, only horizontal transmission was considered and the virus infection in human is assumed fatal, though with a very low rate. The disease free and endemic equilibrium was obtained and analyzed for both local and global asymptotically stability. The analysis shows that the model undergoes backward bifurcation when the effective basic reproductive number R C ≤ 1 . Numerical simulation of the model shows that the effect of vertical transmission of the mosquito sub-population in the dynamics of the virus is negligible, even when the rate is high as shown in Figures 4(A)-(D). Further, the contour plot of the effective basic reproductive number R C with respect to the vaccine efficacy ε and the proportion of susceptible vaccinated (
The authors declare no conflicts of interest regarding the publication of this paper.
Onuorah, M.O., Obi, E.I. and Babangida, B.G. (2019) Modelling the Effects of Vertical Transmission in Mosquito and the Use of Imperfect Vaccine on Chikungunya Virus Transmission Dynamics. Applied Mathematics, 10, 245-267. https://doi.org/10.4236/am.2019.104019
Castilo-Chaevz and Song [
Consider the following general system of ordinary differential equations with a parameter ϕ .
d x d t = f ( x , ϕ ) : R n × R → R n and f ∈ C 2 ( R n × R )
where 0 is an equilibrium point of the system (that is, f ( 0 , ϕ ) = 0 for all ϕ ) and
(A1) A = D x f ( 0 , 0 ) = ( ∂ f i d x j ( 0 , 0 ) ) is the linearization matrix of the system 2.10 around the equilibrium 0 with ϕ evaluated at 0;
(A2) Zero is a simple eigenvalues of A and other eigenvalues of A have negative real parts;
(A3) Matrix A has a right eigenvector w and left eigenvector v (each corresponding to zero eigenvalues).
Let f k be the kth component of f and
To do this we need the values of a and b given below:
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 ∂ x ϕ ( 0 , 0 )
then, the local dynamics of the system around equilibrium point 0 is totally determined by the signs of a and b, particularly,
1) a > 0 , b > 0 , when ϕ < 0 with | ϕ | ≪ 1 , 0 is locally asymptotically stable and there exists a positive unstable equilibrium; when 0 < ϕ ≪ 1 , 0 is unstable and there exists a negative, locally asymptotically stable equilibrium;
2) a < 0 , b < 0 , when ϕ < 0 with | ϕ | ≪ 1 , 0 is unstable; when 0 < ϕ ≪ 1 , 0 is locally asymptotically stable equilibrium and there exists a positive unstable equilibrium;
3) a < 0 , b > 0 , when ϕ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly a negative unstable equilibrium becomes positive and locally asymptotically stable.