Mathematical Modelling of the Impact of Misdiagnosis on Dengue-Malaria Co-Infection Dynamics

Malaria and dengue fever are among the leading causes of Undifferentiated Febrile Illness (AUFI) which account for majority visits to the hospital in Africa. The diseases are spread through biting of mosquitoes making coifection a very common phenomena in tropical and subtropical regions due to high prevalence of mosquito populations. A common challenge of managing both diseases in Africa is misdiagnosis of dengue fever for malaria due to similarities in symptoms and limited resources for accurate diagnosis. This research focused on developing a mathematical epidemiological model to investigate the impact of misdiagnosis of dengue fever as malaria on the general spread dynamics of malaria and dengue fever. The analytical properties of the spread dynamics were studied. The model was determined to be mathematically well stated and biologically well posed by ensuring the positivity and boundedness of the solutions. The co-infection-free Equilibrium was determined to exist using the jacobian matrix method. The co-infection reproductive number was calculated using the next generation matrix approach. The existence of the endemic equilibrium was determined. The normalized forward approach was used to determine the sensitivity indices and identified the parameters that most likely to be targeted for control of the malaria-dengue fever coinfection spread dynamics. Numerical simulations were conducted to obtain the numerical values of the sensitivity analysis and the numerical solutions to the spread dynamics.

• Basic Reproductive Number
• Coinfection Dynamics
• Malaria-Dengue Fever
• Equilibrium Points
• Mathematical Modelling
• Numerical Simulations

1. Introduction

The monoinfection of either malaria or dengue fever by itself is fatal, however, the coinfection of both diseases has been proven to be more fatal [1]. Since they are both spread by mosquitoes, their prevalence in Africa has become more prevalent [2]. The two diseases even share common signs such as fever, headache, myalgia, nausea, and diarrhoea. As a consequence, the coinfection of the diseases is either ignored or leads to misdiagnosis of coinfection as malaria since malaria is more pronounced when compared to dengue fever [3]. In resource limited countries such as Kenya, the probability of presumptive treatment of dengue fever as malaria is very high [4].

More so, coinfection is often is treated as a mono infection due to the clinical similarities in the manifestation of the disease. Misdiagnosis is more prone in coinfection spread dynamics than in regions that have a prevalence of one disease. Concurrent infection often occurs and obscures the symptoms of either of the initial conditions and the treatment regime of coifection is not the same as the treatment of the mono infection of either disease. As a result, the delayed implementation of the correct treatment regime leads to complications that can be fatal [5]. Increased movement of populations arising from efficient transport systems is posed to further aggravate the situation by introducing the disease to new environments where the diseases were not prevalent leading to further coinfection.

Malaria and dengue are the leading causes of Acute Undifferentiated Febrile Illness (AUFI), which is the most frequent reason for seeking healthcare services in Africa [6]. The World Health Organization (WHO) reported 263 million malaria cases across the globe with Africa accounting for 94% of the cases. The same reports indicated 597,000 deaths across the globe with Africa accounting for 95% of the deaths [7]. Dengue fever has also been in the rise in sub-Saharan Africa with around 200,000 suspected cases and 90,000 confirmed cases that resulted in 900 deaths [8][9]. Most of the cases in Africa go unreported leading to inaccurate statistics.

One of the key challenges of the management of both diseases is misdiagnosis which is more prevalent in Africa due to the prevalence of both types of mosquitoes that spread the disease [5]. As a result, this clinical gap presents an opportunity for research in mathematical epidemiological modelling of malaria-dengue fever coinfection. Therefore, this study focused on developing a mathematical model that investigates the coinfection dynamics of the two diseases in one geographical area.

2. Model Description and Formulation

In this model, the total human population at any given time t, denoted by ( N h ) is divided in ten human beings sub-classes. These are susceptible humans ( S h ) , individuals exposed to malaria ( E m ) , individuals infected by malaria ( I m ) , and individuals who recover from malaria ( R m ) . On the other side, we have individuals exposed to dengue ( E d ) , individuals misdiagnosed ( M d ) , individuals infected by dengue ( I d ) , and individuals who recover from dengue ( R d ) . At the center of the model are individuals infected with malaria and dengue co-infection ( I c ) and individuals who recover from malaria and dengue fever ( R c ) . Susceptible humans are those that are at risk of developing an infection. Infectious individuals are those that have symptoms of either of the disease or both disease. Exposed individuals are those that have been bitten by an infected mosquito but have not become infectious. Recovered are those that show temporal immunity. Co-infection are those individuals that are infected with both malaria and dengue fever. Total human population is represented by

N h = S h + E d + M d + I d + E m + I m + I c + R d + R m + R c

A schematic presentation of the malaria dengue fever spread dynamics was presented in Figure 1 below.

Figure 1

Figure 1. Malaria Dengue fever Coinfection model flow chart.

Susceptible are recruited through Λ, susceptibles individuals are exposed to dengue fever and malaria fever at rate of β h * , which is the force of infection. Individuals exposed to malaria fever move to infectious population at the rate of λ . Individuals exposed to dengue transfer to misdiagnosed population at rate of η . The misdiagnosed individuals transfer to infectious dengue individual at rate of ε The exposed individuals to dengue move to infectious dengue population at rate of τ . Infectious dengue individuals move to misdiagnosis at rate of κ . Individuals infected by dengue and malaria fever move to co-infection human population at rate of ε c and ρ c respectively. Individuals exposed to dengue and malaria move to co-infection population at rate of η c and δ c respectively. Individual infected by dengue fever and malaria fever move to dengue recovered population and malaria recovered at rate of γ and α respectively.

Co-infected individuals move to dengue recovered and malaria recovered at rate of γ c and α c respectively. The recovered dengue and malaria individuals move to co-infected recovered at the rate of ω and ϕ respectively. individuals from co-infected recovered move to susceptible human population at rate of Ψ. Natural death rate in all compartment is μ . The diseases induced death rates for misdiagnosed individuals, infected dengue, co-infection and infected malaria are δ d , σ d , σ c , σ m respectively.

The vector population is divided into four compartments namely the susceptible vectors ( S v ) . The susceptible vectors entail the mosquitoes which are susceptible to both dengue virus and plasmodium bacteria. Mosquitoes that bite infected individuals proceed to the latent stage where they fit the description of an exposed compartment labelled ( E v ) . At this stage the dengue virus or the plasmodium bacteria has not replicated enough to make the mosquito to be infectious. When disease agent in the mosquito replicates to enough agents the mosquito can move to either the Infectious dengue mosquito compartment ( I v d ) with mosquitoes infected with dengue fever, or moves to the infectious malaria compartment ( I v m ) compartment for mosquitoes infected with plasmodium bacteria. The total vector population is given by N v = S v + E v + I v d + I m d .

The susceptible mosquitoes are recruited through natural hatching of eggs at the rate Π. The susceptible mosquitoes become infectious with either plasmodium or dengue fever through the force of infection β v * and thus move to the latent stage. The latent mosquitoes become infectious of dengue fever at a rate ξ 1 and become infectious with malaria at a rate ξ 2 . All vectors in all compartments are susceptible to natural attrition at a rate of μ v . A summary of the parameters is defined in Table 1 below.

Table 1. Numerical values for malaria-dengue Co-infection model parameters.

Table 1

Param	Description	Value (day −1 )	Reference
Λ	Human recruitment rate	0.00011	[ 10 ]
β *	Force of infection rate	0.000035	[ 11 ]
η	rate of exposed dengue individuals transfer to misdiagnosed	0.00500	[ 12 ]
ε	rate of misdiagnosed individuals move to infectious human population	0.00200	[ 13 ]

Continued

Table 2

λ	rate at which individuals exposed to malaria move to infectious malaria population	0.00100	[ 14 ]
τ	rate at which individual exposed to dengue move infectious dengue population	0.00142	[ 15 ]
κ	rate at which infectious dengue individuals move to misdiagnosis	0.00100	[ 12 ]
ε c	rate of infectious dengue individuals moving to infectious co-infection	0.00020	[Estimated]
ρ c	rate of infectious malaria individuals moving to infectious co-infection	0.00300	[Estimated]
η c	rate of exposed dengue individuals move to infectious co-infection	0.00100	[Estimated]
δ c	rate of exposed malaria individuals move to infectious co-infection	0.00100	[ 16 ]
γ	rate of recovered dengue individual from infectious dengue population	0.00142	[ 17 ]
α	rate of recovered malaria individuals from infectious malaria population	0.07100	[ 18 ]
γ c	rate of dengue recovered individuals from co-infection	0.00125	[ 19 ]
α c	rate of malaria recovered individuals from co-infection	0.00500	[ 20 ]
ω	rate of individual moving to co-infection recovered from dengue recovered	0.00100	[ 20 ]
ϕ	rate of individual moving to co-infection recovered from malaria recovered	0.00100	[ 19 ]
μ	rate of natural death rate	0.00004	[ 9 ]
δ d	induced death rate for misdiagnosed	0.00200	[Estimated]
σ d	disease induced death rate for dengue	0.00040	[ 21 ]
σ c	diseases induced death rate for co-infection	0.00500	[ 22 ]
σ m	disease induced death rate for malaria	0.00100	[ 21 ]

The force of infection for the coinfection model was taken to be β v * = β v d * + β v m * where

β v d * = b h β v h d I h d N h

where

b h is the number of mosquito bites that one human has per unit time,

β v h d is the dengue transmission rate from mosquitoes to human beings,

I h d infected human beings with dengue fever.

and

β v m * = b v T β h v m I h m N h

where

b v is the number of human bites that one mosquito has per unit time,

β h v m is the malaria transmission rate from mosquitoes to human beings,

T is the temperature coefficient of biting rate effectiveness given by 0 ≤ T ≤ 1 .

While the force of infection of the human population β h * = β h m * + β h d * , defined as

β h m * = b h T β h v m I v m N v

where

b h is the number of mosquito bites that one human has per unit time,

β h v m is the malaria transmission rate from mosquitoes to human beings,

T is the temperature coefficient of biting rate effectiveness given by 0 ≤ T ≤ 1 .

and

β h d * = b h β h v d I v d N v + b h θ M h d N h

where

b h is the number of mosquito bites that one human has per unit time,

β h v d is the dengue transmission rate from human beings to mosquitoes,

θ infectiousness modifiers for misdiagnosis ( 0 < θ ≤ 1 ),

I v d dengue infected mosquitoes population,

M h d Misdiagnosed human population.

2.1. Assumptions

The Model Utilized the Following Assumptions:

1) The population mixes homogeneously.

2) The parameters are uniform across all strata of population, therefore the biting rate of mosquitoes is uniform despite evidence of variation in mosquito preferences in biting.

3) There are no vertical transmissions of plasmodium.

4) There is no misdiagnosis of malaria due to the large number of malaria cases making health practitioners more proficient on malaria diagnosis.

5) There is no simultaneous recovery from both disease.

6) The contribution of the transmission of either disease by individuals coinfected is insignificant hence ignored.

2.2. Model Equations

The model description in Figure 1 gives rise to a system of non-linear differential equations below that describe the malaria-dengue fever disease host-vector transmission dynamics.

(1) { d S d t = Λ − β * S I d − β * S I m − μ S + Ψ R c d E d d t = β * S I d − ( μ + η c + η + τ ) E d d M d d t = η E d + κ I d − ε M d − ( μ + δ d ) M d d I d d t = τ E d + ε M d − κ I d − ( μ + σ d ) I d − ε c I d − γ I d d R d d t = γ I d + γ c I c − μ R d − ω R d d E m d t = β * S I m − μ E m − λ E m − δ c E m d I m d t = λ E m − ρ c I m − ( μ + σ m ) I m − α I m d R m d t = α I m + α c I c − Φ R m − μ R m d I c d t = δ c E m + ρ c I m + ε c I d + η c E d − ( μ + δ c ) I c − α c I c − γ c I c d R c d t = Φ R m + ω R d − μ R c − Ψ R c d S v d t = Π − μ v S v − β * S v d E v d t = β v * S v − ξ 1 E v − ξ 2 E v − μ v E v d I v m d t = ξ 2 E v − μ v I v m d I v d d t = ξ 1 E v − μ v I v d

with the initial conditions

(2) S ( 0 ) ≥ 0 , E d ( 0 ) ≥ 0 , M d ( 0 ) ≥ 0 , I d ( 0 ) ≥ 0 , R d ( 0 ) ≥ 0 , E m ( 0 ) ≥ 0 , I m ( 0 ) ≥ 0 , R m ( 0 ) ≥ 0 , I c ( 0 ) ≥ 0 , R c ( 0 ) ≥ 0 , S v ( 0 ) ≥ 0 , E v ( 0 ) ≥ 0 , I v m ( 0 ) ≥ 0 , I v d ( 0 ) ≥ 0

3. Basic Properties of the Model

In this section, we study the posedness of the epidemiological model as stated by investigating the positivity and invariant region span by the model.

3.1. The Invariant Region

Theorem 1Let’s consider the space

(3) Ξ M D = { ( S ( t ) , E d ( t ) , M d ( t ) , I d ( t ) , R d ( t ) , E m ( t ) , I m ( t ) , R m ( t ) , I c ( t ) , R c ( t ) , ∈ ℝ + 10 ) :                       ( S ( t ) > 0 , E d ( t ) > 0 , M d ( t ) ≥ 0 , I d ( t ) ≥ 0 , R d ( t ) ≥ 0 , E m ( t ) > 0 , I m ( t ) ≥ 0 ,                       R m ( t ) ≥ 0 , I c ( t ) ≥ 0 , R c ( t ) ≥ 0 , S v ( t ) ≥ 0 , E v ( t ) ≥ 0 , I v m ( t ) ≥ 0 , I v d ( t ) ≥ 0 , ∈ ℝ + 14 ) }

then the solutions of ( S ( t ) , E d ( t ) , M d ( t ) , I d ( t ) , R d ( t ) , E m ( t ) , I m ( t ) , R m ( t ) , I c ( t ) , R c ( t ) , S v ( t ) , E v ( t ) , I v m ( t ) , I v d ( t ) ) should be non-negative for all time t ≥ 0 . Considering the fact that the initial conditions ( S ( t ) > 0 , E d ( t ) > 0 , M d ( t ) ≥ 0 , I d ( t ) ≥ 0 , R d ( t ) ≥ 0 , E m ( t ) > 0 , I m ( t ) ≥ 0 , R m ( t ) ≥ 0 , I c ( t ) ≥ 0 , R c ( t ) ≥ 0 , S v ( t ) ≥ 0 , E v ( t ) ≥ 0 , I v m ( t ) ≥ 0 , I v d ( t ) ≥ 0 ) are non-negative, then the generation solutions ( S ( t ) , E d ( t ) , M d ( t ) , I d ( t ) , R d ( t ) , E m ( t ) , I m ( t ) , R m ( t ) , I c ( t ) , R c ( t ) ) should be non-negative at any time t ≥ 0 .

Proof. We considered the total human population N ( t ) at any given time t defined as

(4) N ( t ) = S ( t ) + E d ( t ) + M d ( t ) + I d ( t ) + R d ( t )   + E m ( t ) + I m ( t ) + R m ( t ) + I c ( t ) + R c ( t )

differentiating both sides of Equation (4) we obtained

(5) d N ( t ) d t = d S ( t ) d t + d E d ( t ) d t + d M d ( t ) d t + d I d ( t ) d t + d R d ( t ) d t   + d E m ( t ) d t + d I m ( t ) d t + d R m ( t ) d t + d I c ( t ) d t + d R c ( t ) d t

In the event that there is no disease in the community then Equation (5) reduces to

(6) d N d t = Λ − μ S − μ E d − μ M d − μ I d − μ R d − μ E m − μ I m − μ R m − μ I c − μ R c

which was further simplified to

(7) N ( t ) ≤ Λ μ − ( Λ − μ N ( 0 ) μ ) e − μ t

As t → ∞ then N ( t ) → Λ μ , which is the upper bound while zero is the lower

bound of the population. As a consequence, the positive bounded region is expressed in the form in Equation (8) below

(8) 0 ≤ N ( t ) ≤ Λ μ

Therefore, the positive invariant region Π h was found to be expressed as shown in the Equation (9) below

(9) Ξ h = { ( S ( t ) , E d ( t ) , M d ( t ) , I d ( t ) , R d ( t ) , E m ( t ) , I m ( t ) , R m ( t ) , I c ( t ) , R c ( t ) , ∈ ℝ + 10 ) :                       S ( t ) + E d ( t ) + M d ( t ) + I d ( t ) + R d ( t ) + E m ( t ) + I m ( t ) + R m ( t ) + I c ( t ) + R c ( t ) ≤ Λ μ }

Considering the total vector population N v ( t ) at any given time t defined as

(10) N v ( t ) = S v ( t ) + E v ( t ) + I v d ( t ) + I v m ( t )

differentiating both side Equation (10) with respect to time we obtained the following equation

(11) d N v ( t ) d t = d S v ( t ) d t + d E v ( t ) d t + d I v d ( t ) d t + d I v m ( t ) d t

which was simplified as

(12) N v ( t ) ≤ Π μ − ( Π − μ N v ( 0 ) μ ) e − μ t

At t → t then N v → Π μ with zero as the lower bound and Π μ as the upper

bound of the population. As a result the bounded region for the vector population is given by

(13) 0 ≤ N v ( t ) ≤ Π μ v

As a result the positive invariant region for the vector population was given by

(14) Ξ v = { ( S v ( t ) , E v ( t ) , I v d ( t ) , I v m ( t ) , ∈ ℝ + 4 ) : S v ( t ) + E v ( t ) + I v m ( t ) + I v d ( t ) ≤ Π v μ v }

which is positively invariant region for the vector population ☐

In conclusion, the system of differential Equation (1) has solutions that are positively invariant in Ξ h and Ξ v [23][24]. Therefore, the solutions of the system of differential equations that model the malaria-dengue fever coinfection are said to be well posed mathematically and epidemiologically well stated.

3.2. Co-Infection-Free Equilibrium

The disease free equilibrium was determined by considering the steady state of the spread dynamics [25]. At the steady state, the system is at its equilibrium point implying that the system does not change with respect to time [26]. Therefore,

d d t = 0 which implies that the left hand side of the system of differential equation in Equation (1) below is equated to zero to obtain.

0 = Λ − β * S I d − β * S I m − μ S + Ψ R c 0 = β * S I d − ( μ + η c + η + τ ) E d 0 = η E d + κ I d − ε M d − ( μ + δ d ) M d 0 = τ E d + ε M d − κ I d − ( μ + σ d ) I d − ε c I d − γ I d 0 = γ I d + γ c I c − μ R d − ω R d 0 = β * S I m − μ E m − λ E m − δ c E m (15) 0 = λ E m − ρ c I m − ( μ + σ m ) I m − α I m 0 = α I m + α c I c − Φ R m − μ R m 0 = δ c E m + ρ c I m + η c E d − ( μ + δ c ) I c − α c I c 0 = Φ R m + ω R d − μ R c − Ψ R c 0 = Π − μ v S v − β v * S v 0 = β v * S v − ξ 1 E v − ξ 2 E v − μ v E v 0 = ξ 2 E v − μ v I v m 0 = ξ 1 E v − μ v I v d

At the same time, at the disease free equilibrium, the disease does not exist in the community therefore, all the compartments associated with the disease are equated to zero. That is E d = 0 , M d = 0 , I d = 0 , R d = 0 , E m = 0 , I m = 0 , R m = 0 , I c = 0 , R c = 0 , E v = 0 , I v d = 0 , I v m = 0 , as a consequence, the system of differential Equation (15) reduces to

(16) 0 = Λ − μ S

Making S the subject and simplifying the equation we obtained

(17) S = Λ μ

Again, considering the equation

(18) 0 = Π − μ v S v

Making S v the subject of the equation we obtained

(19) S v = Π μ v

Therefore, the disease free equilibrium point can be summarized as follows

(20) ξ 03 = ( S * , E d * , M d * , I d * , R d * , E m * , I m * , R m * , I c * , R c * , S v * , E v * , I v d * , I v m * ) = ( Λ μ , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , Π μ v , 0 , 0 , 0 )

Therefore Equation (20) confirms the existence of the Co-infection-free equilibrium point of the system of differential Equation (1) that describe the malaria-dengue fever coinfection spread dynamics.

3.3. The Co-Infection Reproductive Number

The basic reproduction number is a critical dimensionless number that effectively describes the trajectory the coinfection will take in future [27]. It was calculated by separating the compartments into the infectious compartments and the non-infectious compartments. The non-infectious compartments in this study included ( S , R d , R m , R c , S v ) while the infectious compartments included ( E d , M d , I d , E m , I m , I c , E v , I v m , I v d ) . We then constructed the matrix of new infections F and the matrix of transfers V using the infectious compartments and obtained the results as follows

(21) F = [ 0 b h θ   Λ μ 0 0 0 0 0 T b h β h m Λ μ b h β h v d Λ μ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b h θ   Λ μ 0 0 0 0 0 T b h β h m Λ μ b h β h v d Λ μ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b h β v h d Π μ v 0 T b v β h v m Π μ v 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]

and

(22) V = [ k 1 0 0 0 0 0 0 0 0 − η k 2 − κ 0 0 0 0 0 0 − τ − ε k 3 0 0 0 0 0 0 0 0 0 k 4 0 0 0 0 0 0 0 0 − λ k 5 0 0 0 0 − η c 0 0 − δ c − ρ c k 6 0 0 0 0 0 0 0 0 0 k 7 0 0 0 0 0 0 0 0 − ξ 2 k 8 0 0 0 0 0 0 0 − ξ 1 0 k 9 ]

where

k 1 = μ + η c + η + τ ,   k 2 = ε + μ + δ d ,   k 3 = κ + μ + σ d + ε c + γ , k 4 = μ + λ + δ c ,   k 5 = ρ c + μ + σ m + α ,   k 6 = μ + δ c + α c , k 7 = ξ 1 + ξ 2 + μ ,   k 8 = μ v ,   k 9 = μ v

The next generation matrix was determined from the matrix N = F V − 1 which used to determine the basic reproduction number R 0 c . We first determined the inverse of V in Equation (22). The inverse was determined as

(23) V − 1 = [ 1 k 1 0 0 0 0 0 0 0 0 − η k 3 + κ τ k 1 ( ϵ κ − k 3 k 2 ) − k 3 ϵ κ − k 3 k 2 − κ ϵ κ − k 3 k 2 0 0 0 0 0 0 − ϵ η + τ k 2 k 1 ( ϵ κ − k 3 k 2 ) − ϵ ϵ κ − k 3 k 2 − k 2 ϵ κ − k 3 k 2 0 0 0 0 0 0 0 0 0 1 k 4 0 0 0 0 0 0 0 0 λ k 4 k 5 1 k 5 0 0 0 0 η c k 1 k 6 0 0 ρ c λ + δ c k 5 k 5 k 4 k 6 ρ c k 5 k 6 1 k 6 0 0 0 0 0 0 0 0 0 1 k 7 0 0 0 0 0 0 0 0 ξ 2 k 7 k 8 1 k 8 0 0 0 0 0 0 0 ξ 1 k 7 k 9 0 1 k 9 ]

The matrix product F V − 1 was obtained as

(24) F V − 1 = [ − b h θ Λ ( η k 3 + κ τ ) μ k 1 ( ϵ κ − k 3 k 2 ) − b h θ Λ k 3 μ ( ϵ κ − k 3 k 2 ) − b h θ Λ κ μ ( ϵ κ − k 3 k 2 ) 0 0 0 T b h β h m Λ ξ 2 μ k 7 k 8 + b h β h v d Λ ξ 1 μ k 7 k 9 T b h β h m Λ μ k 8 b h β h v d Λ μ k 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − b h θ Λ ( η k 3 + κ τ ) μ k 1 ( ϵ κ − k 3 k 2 ) − b h θ Λ k 3 μ ( ϵ κ − k 3 k 2 ) − b h θ Λ κ μ ( ϵ κ − k 3 k 2 ) 0 0 0 T b h β h m Λ ξ 2 μ k 7 k 8 + b h β h v d Λ ξ 1 μ k 7 k 9 T b h β h m Λ μ k 8 b h β h v d Λ μ k 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − b h β v h d π ( ϵ η + τ k 2 ) μ v k 1 ( ϵ κ − k 3 k 2 ) − b h β v h d π ϵ μ v ( ϵ κ − k 3 k 2 ) − b h β v h d π k 2 μ v ( ϵ κ − k 3 k 2 ) T b v β h v m π λ μ v k 4 k 5 T b v β h v m π μ v k 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]

whose eigenvalues were determined as follows

(25) λ = [ 0 0 0 0 0 0 0 − b h θ Λ ( η k 3 + κ τ ) μ k 1 ( ϵ κ − k 2 k 3 ) + ( b h θ Λ ( η k 3 + κ τ ) μ k 1 ( ϵ κ − k 2 k 3 ) ) 2 + 4 ( T b h β h m Λ ξ 2 μ k 7 k 8 + b h β h v d Λ ξ 1 μ k 7 k 9 ) ( b h β v h d Π ( ϵ η + τ k 2 ) μ v k 1 ( ϵ κ − k 2 k 3 ) ) 2 − b h θ Λ ( η k 3 + κ τ ) μ k 1 ( ϵ κ − k 2 k 3 ) − ( b h θ Λ ( η k 3 + κ τ ) μ k 1 ( ϵ κ − k 2 k 3 ) ) 2 + 4 ( T b h β h m Λ ξ 2 μ k 7 k 8 + b h β h v d Λ ξ 1 μ k 7 k 9 ) ( b h β v h d Π ( ϵ η + τ k 2 ) μ v k 1 ( ϵ κ − k 2 k 3 ) ) 2 ]

The basic reproduction number is given by the spectral radius of the eigenvalues in Equation (25) [27]. Therefore, the basic reproduction number was determined as

(26) R 0 c = − b h θ Λ ( η k 3 + κ τ ) μ k 1 ( ϵ κ − k 2 k 3 ) + ( b h θ Λ ( η k 3 + κ τ ) μ k 1 ( ϵ κ − k 2 k 3 ) ) 2 + 4 ( T b h β h m Λ ξ 2 μ k 7 k 8 + b h β h v d Λ ξ 1 μ k 7 k 9 ) ( b h β v h d Π ( ϵ η + τ k 2 ) μ v k 1 ( ϵ κ − k 2 k 3 ) ) 2

3.4. Local Stability of the Co-Infection-Free Equilibrium

The stability of the disease free equilibrium is a vital analytical point in the spread dynamics of malaria and dengue coinfection. We utilized the jacobian approach to determine the stability.

Theorem 2.The Co-infection-Free Equilibrium ( ξ 03 ) is locally asymptotically stable point of the dynamical system 1 whenever R 0 c < 1 .

Proof. The jacobian approach required the determination of the partial differential of the system of differential Equation (1) with respect to each compartment [28]. The jacobian was determined to be

(27) J = [ − β h I d − I m β h − μ 0 0 − β h S 0 0 − β h S 0 0 Ψ 0 0 0 0 β h I d − μ − η c − η − τ 0 β h S 0 0 0 0 0 0 0 0 0 0 0 η − ϵ − μ − δ d κ 0 0 0 0 0 0 0 0 0 0 0 τ ϵ − κ − μ − σ d − ε c − γ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 γ 1 − μ − ω 0 0 0 γ c 0 0 0 0 0 I m β h 0 0 0 0 − λ − μ − δ c β h S 0 0 0 0 0 0 0 0 0 0 0 0 λ − ρ c − μ − σ m − α 0 0 0 0 0 0 0 0 0 0 0 0 0 α − μ − ϕ α c 0 0 0 0 0 0 η c 0 0 0 δ c ρ c 0 − μ − δ c − α c 0 0 0 0 0 0 0 0 0 ω 0 0 ϕ 0 − Ψ − μ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − β v − μ v 0 0 0 0 0 0 0 0 0 0 0 0 0 β v − μ v − ξ 2 0 0 0 0 0 0 0 0 0 0 0 0 0 ξ 2 − μ v 0 0 0 0 0 0 0 0 0 0 0 0 ξ 1 0 − μ v ]

where

k 1 = μ + η c + η + τ ,       k 2 = ε + μ + δ d ,       k 3 = κ + μ + σ d + ε c + γ ,   k 4 = μ + ω , k 5 = μ + λ + δ c ,       k 6 = ρ c + μ + σ m + α ,     k 7 = Φ + μ ,       k 8 = μ + δ c + α c ,       k 9 = μ + Ψ

Determining the jacobian at the disease free equilibrium J ξ 03 we obtained

(28) J ξ 03 = [ − μ 0 0 − β h Λ μ 0 0 − β h Λ μ 0 0 Ψ 0 0 0 0 0 − k 1 0 β h Λ μ 0 0 0 0 0 0 0 0 0 0 0 η − k 2 κ 0 0 0 0 0 0 0 0 0 0 0 τ ϵ − k 3 0 0 0 0 0 0 0 0 0 0 0 0 0 γ − k 4 0 0 0 γ c 0 0 0 0 0 0 0 0 0 0 − k 5 β h Λ μ 0 0 0 0 0 0 0 0 0 0 0 0 λ − k 6 0 0 0 0 0 0 0 0 0 0 0 0 0 α − k 7 α c 0 0 0 0 0 0 η c 0 0 0 δ c ρ c 0 − k 8 0 0 0 0 0 0 0 0 0 ω 0 0 ϕ 0 − k 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − μ v 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − μ v 0 0 0 0 0 0 0 0 0 0 0 0 0 ξ 2 − μ v 0 0 0 0 0 0 0 0 0 0 0 0 ξ 1 0 − μ v ]

Analyzing the jacobian matrix at DFE to determine the condition under which the DFE is locally asymptotically stable. The disease free equilibrium is locally asymptotically stable if all the eigenvalues of the jacobian matrix are negative. In Equation (28), it is clear that the first five eigenvalues are strictly negative. Considering the sub blocks of the coinfection and using the Routh-Hurwitz criterion for characteristic polynomials, we determined that the characteristic polynomial was established to be P 2 ( λ ) = λ 2 + B 1 λ + B 0 = 0 where B 1 = k 5 + k 6 and B 0 = k 5 k 6 ( 1 − R 0 m ) . According to Routh-Hurwitz criterion for quadratic characteristic polynomials B 0 > 0 which is true since both k 5 and k 6 are positive while B 0 > 0 whose positivity is dependent on the value of ( 1 − R 0 m ) which is positive only when ( R 0 m < 1 ) for local stability to be achieved. Considering the dengue infectious compartments, we determined that the characteristic polynomial is given by P 1 ( λ ) = λ 3 + A 2 λ 2 + A 1 λ + A 0 = 0 . The third-order Ruoth-Huwirtz criterion requires that A 2 > 0 , A 0 > 0 , A 2 A 1 > A 0 . From the matrix A 2 = k 1 + k 2 + k 3 > 0 since k 1 , k 2 and k 3 are all positive. A 0 = k 1 ( k 2 k 3 − ε κ ) ( 1 − R 0 d ) which is positive only if R 0 d < 1 . The third condition is met by expansion which is proven based on the biological constraints. As a consequence the diseases free equilibrium is locally asymptotically stable when all the described conditions are met. Considering the vector population, all the diagonals are represented by the parameter − μ v thus making the eigenvalues of the vector population all negative thus making the disease free asymptotically stable.

3.5. Endemic Equilibrium

The endemic equilibrium is a steady state that exists when the disease is persistent in the community [29]. The system does not change with respect to time at

this steady state implying that d d t = 0 . Considering the system of differential Equation (1) the endemic equilibrium was obtained as

(29) ξ * * = ( S * * E d * * M d * * I d * * R d * * E m * * I m * * R m * * I c * * R c * * S v * * E v * * I v d * * I v m * * ) = ( S * * R 0 c * * β * S * * I d * * z k 1 η E d * + κ I d * * k 2 τ E d * * + ε M d * * k 3 γ I d * * + γ c I c * * k 4 β * S * I m * * k 5 λ E m * * k 6 α I m * * + α c I c * * k 7 δ c E m * * + ρ c I m * * + η c E d * * k 8 Φ R m * * + ω R d * * k 9 Π μ v + β v * β v * Π ( μ v + β v * ) ( ξ 1 + ξ 2 + μ v ) ξ 1 β v * Π μ v ( μ v + β v * ) ( ξ 1 + ξ 2 + μ v ) ξ 2 β v * Π μ v ( μ v + β v * ) ( ξ 1 + ξ 2 + μ v ) )

where

k 1 = μ + η c + η + τ ,         k 2 = ε + μ + δ d ,         k 3 = κ + μ + σ d + ε c + γ , k 4 = μ + ω ,         k 5 = μ + λ + δ c ,         k 6 = ρ c + μ + σ m + α , k 7 = Φ + μ ,         k 8 = μ + δ c + α c ,         k 9 = μ + Ψ

3.6. Numerical Simulations

In this section, the spread dynamics of malaria and dengue fever were further analysed numerically to determine the spread dynamics patterns.

3.7. Sensitivity Indices

This section focused on determining the most sensitive parameter of the basic reproduction number. It utilised the numerical values in Table 1 and normalized forwards sensitivity index formula to obtain the values of Table 2 below.

Table 2. Sensitivity indices of the model parameters.

Table 3

Parameter	Description	Sensitivity Index
Λ	Human recruitment rate	+1.000001
θ	Probability of infection at the misdiagnosed compartment	+1.000003
b h	Biting rate per human	+1.000000
κ	Infectious dengue moving to misdiagnosis	+0.503391
η	Exposed dengue moving to misdiagnosis	+0.473991
ξ 2	Incubation rate of malaria	+4.962018 × 10 −7
μ v	Natural death rate for vectors	+1.367197 × 10 −6
Π	Vector recruitment rate	−1.367197 × 10 −6
β v h d	Transmission rate (vector to human)	−1.367197 × 10 −6
β h v d	Transmission rate (human to vector)	−1.179910 × 10 −6
ξ 1	incubation rate of dengue fever	−4.964206 × 10 −7
T	the temperature coefficient of biting rate effectiveness	−1.872872 × 10 −7
β h m	Transmission rate for malaria	−1.872872 × 10 −7
σ d	Disease induced death rate (dengue)	−0.000185
δ d	Induced death rate (misdiagnosed)	−0.003129
ε c	Infectious dengue moving to co-infection	−0.013702
η c	Exposed dengue moving to co-infection	−0.026120
τ	Progression to infectious dengue	−0.447610
γ	Recovery from infectious dengue	−0.489365
ε	Misdiagnosed moving to infectious	−0.994788
μ	Natural death rate	−1.002485

3.8. Numerical Coinfection Spread Dynamics

Table 1 above was utilized together with the following initial conditions S ( 0 ) = 99800 , E d ( 0 ) = 50 , M d ( 0 ) = 0 , I d ( 0 ) = 50 , R d ( 0 ) = 0 , E m ( 0 ) = 50 , I m ( 0 ) = 50 , R m ( 0 ) = 0 , I c ( 0 ) = 0 , R c ( 0 ) = 0 and the total population given by N ( t ) = S ( t ) + E d ( t ) + M d ( t ) + I d ( t ) + R d ( t ) + E m ( t ) + I m ( t ) + R m ( t ) + I c ( t ) + R c ( t ) , to generate graphs for the spread dynamics. A total population of 100,000 people was considered because it mirrors a closed community which is scalable to national level population [30]. The numerical analysis was conducted by utilization of the Maple software. The spread dynamics of each of the compartment was show in Figure 2 below.

Figure 2

Figure 2. Numerical simulation of the general spread dynamics of malaria dengue coinfection.

The susceptible population remains high in the first 50 days of the study ans then decreased rapidly in the next 50 days due to the diseases gaining traction in the community. The susceptible population stabilizes at around 40,000 people per day implying that about 40,000 never get to interact with the diseases during the break out. The total population suffers a gentle decline over the 730 days of the study due to exit of people through disease related attrition. The spread dynamics of the other compartments were better illustrated by Figure 3.

Figure 3 gives a closeup of the spread dynamics the other compartments. Some of the defining aspects of the spread dynamics include: The spread dynamics of dengue fever take a short period of about 100 days to gain traction in the community while the spread dynamics of malaria take a considerable long period to grain traction. The spread dynamics of dengue fever increase rapidly to reach a peak over a short period of time, while the decrease occurs gradually over a long period of time. The coinfected population raises to a stable population of about 3000 people per day. The population of coinfected individuals remains low but steady thus making it an point of concern for the containment of the diseases.

Figure 3

Figure 3. A close view of numerical simulation of the general spread dynamics of malaria dengue coinfection.

4. Conclusions and Recommendation

Malaria-dengue fever is an emerging public health concern among sub-Saharan Africa. This study has investigated the impact of misdiagnosis on the spread dynamics of the coinfection by developing and analysing a mathematical epidemiological model that describes the spread dynamics. The basic properties have been discussed and the model confirmed to be mathematically well stated and biologically well posed. The critical spread dynamics indicator such as the basic reproduction number has been determined. The behaviour of solutions has been determined through determining the existence of equilibrium points and their stabilities. The sensitivity indices have been determined to scout for the parameters that should be targeted for effective management of the spread dynamics. The study identified the biting rate, the human recruitment rate, the probability of infection by the misdiagnosed individuals, the natural death rate, and the rate of misdiagnosis as the most sensitive parameters. These parameters are the best candidates to develop measures against, for effective management of malaria-dengue fever spread dynamics.

The study recommends the development of an optimal control solution of the spread dynamics to explore the best strategies of controlling the diseases. It also recommends developing the most sensitive parameters into functions to determine the critical levels for which the parameters can be managed to in a resource scarce economy. It also recommends collection of raw data to investigate the extent to which the coinfection has invaded communities.

Author Contributions

Conceptualization, Ewesit EW; methodology, Ewesit EW; software, Ewesit EW; validation, Ewesit EW, Wainaina M and Maingi D; formal analysis, Ewesit EW; investigation, Ewesit EW; resources, Ewesit EW; data curation, Ewesit EW; writing-original draft preparation, Ewesit EW; writing-review and editing, Ewesit EW; visualization, Ewesit EW; supervision, Wainaina M and Maingi D; project administration, Wainaina M and Maingi D; funding acquisition, Ewesit EW. All authors have read and agreed to the published version of the manuscript.

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