Dual infection is infection with strains or pathogens derived from two different individuals, and can be categorized into co-infection and superinfection^[1]. Co-infection is defined as an infection with two heterologous strains or pathogens either simultaneously or within a brief period of time before an infection with the first strain or pathogen has been established and an immune response has developed^[2]. In the case of HIV, co-infection would occur within the first month of infection. Super-infection is defined as infection with a second strain after the initial infection and the immune response to it has been established.

  Human Immunodeficiency Virus (HIV) and Tuberculosis (TB) co-infections present an immense burden on health care systems and pose diagnostic and therapeutic challenges globally particularly in sub-Saharan Africa and Asia^[2]. The two diseases interfere and impact the pathogenesis of each other, leading to a typical presentations and diagnostic complications^[3,4]. These challenges have serious implications on the design, quality and continuity of care, monitoring and interpretation of control targets^[5,6].

  A number of mathematical models on co-infection have been formulated and analysed^{[7,8,9,10,11,12,13,14,15,16,17]}. The studies discussed the HIV-TB associated morbidity and mortality complications and ignored a possibility of simultaneous transmission of both HIV and TB pathogens (co-infection). For instance, Sharomi et. al formulated a deterministic model of TB and HIV co-infection with the aim of evaluating the impact of various treatment strategies in reducing the burden of the twin-epidemic. Corbett et.al reviewed TB epidemiology in Africa and policy implications of HIV/AIDS treatment scale-up. The study further investigated the dynamics of drug resistance and the effects of latent co-infection on intervention that targeted latent class. Long et.al developed a co-epidemic model to study the transmission dynamics of HIV/AIDS and TB. Castillo-Chavez^[18] and Song provided a detailed review transmission dynamics and control of TB. Colijn et.al developed a simple model of an infectious disease which incorporated a latent phase and compared and contrasted results of super infection and co-infection models. In this paper, we employ the idea introduced in^[19] to extend Long et.al co-infection model at population level by incorporating dual infection (co-infection and super infection). The aim of the study is to investigate the effects of simultaneous transmission of both HIV and TB pathogens on the disease dynamics.

  This paper is organized as follows: section 2, presents model formulation, model analysis is carried sections 3 (reproduction numbers, existence of equilibria) and 4 (stability analysis). In section 5, we perform some numerical analysis, and discuss and conclude the paper in section 6.

Model formulation
  We consider an SII x SI x SII problem in which the host population is divided into four mutually disconnected classes. The susceptible class, J_S, comprising individuals at risk of either HIV or TB or both (dual infection). TB-only infectious class, J_T, HIV-only infectious class, J_I, HIV-TB co-infected class, J_IT.

  The susceptible population is replenished through births at constant recruitment rate and is decrease through infection with TB, HIV and HIV-TB infection at rates λ_I, λ_T and λ_IT respectively. The TB, HIV and HIV-TB co-infection compartments are replenished through infection at rates λ_I, λ_T and λ_IT given by

λ_I = β_I (J_I + η_IJ_IT), λ_T = β_I (J_T + η_TJ_IT), and λ_IT = β_IT min(J_I, J_T)

  Where β_I, β_T and β_IT = κβ_I β_T are respectively the transmission coefficients for HIV, TB and HIV-TB. The parameter κ « 1, correspond to the assumption that the two pathogens are rarely transmitted simultaneously, while κ > 1 assumes high transmissibility of both pathogens. The modification parameters η_I ≥ 1 and η_T ≥ 1 account for the assumption that dually-infected individuals have higher transmission rates of HIV and TB respectively, compared to singly-infected individuals. Furthermore, the parameters φ_I ≥ 1 and φ_T ≥ 1 (also modification parameters) account for the level of risk of singly-infected individuals to another infection (super-infection). The host population is subjected to constant natural mortality rate μ with TB, HIV and HIV-TB populations subjected to an additional death associated to infections δ_I, δ_T and δ_IT respectively. Even though HIV does not cause death, we assume that individuals acquire opportunistic infections that lead to death. The description and assumptions above lead to the following autonomous system of differential equations:

ϳ_S = Λ - λ_IJ_S - λ_TJ_S - λ_ITJ_S - μJ_S
ϳ_I = λ_IJ_S - φ_Tλ_IJ_I - (μ + δ_I) J_I
ϳ_T = λ_TJ_S - φ_Iλ_IJ_T - (μ + δ_T)
ϳ_IT = λ_IT + φ_Tλ_TJ_I + φ_I λ_IJ_T - (μ + δ_IT)J_IT    (1)

with changes in the total population governed by
Ṅ(t) = Λ - μN - δ_IJ_I - δ_T,J_T - δ_IT J_IT,
Where N(t) = J_S + J_I + J_T + J_IT  (2)

Positivity of solutions and Invariant region
From equation (2), we have
Ṅ(t) = Λ - μN.
which upon integration yields
N(t) ≤ 1/μ [λ - Ae^-μt].   (3)
Taking the limit as t approaches infinity, we obtain
lim sup N(t) ≤ Λ/μ
t → ∞
Thus, the model represented by (1) can be analysed in the feasible region Ω = {J_S + J_I + J_T + J_IT) ∈ℜ₊⁴: N(t) ≤ Λ/μ}
result can be summarized with the following lemma.

Lemma 2.1Ñ
  All solutions of the system (1) starting in ℜ₊⁴ area bounded and consequently enter the attracting set Ω within the first octant.

Model analysis
The model reproduction number, R₀

  The basic reproduction number R₀ is defined as the number of secondary infections produced by a single infectious individual introduced in a wholly susceptible population during his or her entire infectious period^[20,21]. This quantity plays a pivotal role in characterizing the epidemic and the design of control programs. Using the next generation operator by^[20,21], we have decompose system (1) into a matrix of generation of new infections and other transitions as,

  Noting that the infected classes are J_I, J_T and J_IT (m = 3), we evaluate the derivatives of F and V at the disease-free equilibrium to get

  From which, we obtain FV^-1 and compute the reproduction number of the model, as the spectral radius or the dominant Eigen value given by
ρ(FV^-1) = R₀ = {R_I, R_T, R_IT},
Where
R_0I = (Λ/μ)(β_I/μ + δ_I), R_0T = (Λ/μ)(β_T / μ + δ_T) and
R_0IT = (Λ/μ)(β_IT /μ + δ_IT).
The threshold parameters R_0I , R_0T and R_0IT are defined as the basic reproduction numbers due to HIV, TB and HIV-TB respectively.

Theorem 3.1 The disease-free equilibrium, E₀ is locally asymptotically stable when R₀ < 1 and unstable whenever R₀ > 1.
To illustrate Theorem 3.1, we line arise of system (1) around the disease-free equilibrium and obtain


The eigen values of the Jacobian matrix J_E0 are λ₁ = -μ, λ₂ = -(μ + δ_I)(1 - R_0I), λ₃ = -(μ + δ_T)(1 - R_0T) and λ₄ = -(μ + δ_IT)(1 - R_0IT). All eigen values λ₁, λ₂, λ₃ and λ₄ have negative real parts only if R_0I < 1, R_0T < 1 and R_0IT < 1. Thus, establishing Theorem 3.1.

Steady State solution
  To determine the equilibria of system (1) we set the right hand side of the system to zero and obtain in terms of λ*_I, λ*_T and λ*_IT.

  (6)

  Where Φ₀ = (μ + δ_IT)(1 - R_ITJ*_S) and R_IT = β_IT / μ + δ_IT. We observe that the existence of equilibria is governed by the condition R_ITJ*_S < 1. The threshold parameter R_IT is define as the average number of new co-infections generated by a co-infected individual introduced in a wholly susceptible population.
 Substituting J*_S, J*_I, J*_T and J*_IT into the expressions for λ_T, λ_T, and λ_IT, we obtain

Where φ_T = φ_T / μ + δ_I, φ_I = φ_I / μ + δ_T, B₁ = μ + φ_Tλ*_T and B₁ = μ + φ_Iλ*_I, and .
  The threshold parameters R^I_IT is defined as the average number of new dual infections due to an HIV infective introduced into a TB infected population, while and R^T_IT is the average number of new dual infections due to a TB infective introduced in an HIV infected population.

The solutions (7) and (8) lead to the following results
λ*_I = 0 or F₁(λ*_I, λ*_T) = 1 and λ*_I = 0 or F₂(λ*_I, λ*_T) = 1,   (10)
With

  Due to non-linearity of the pair of equations (11), it is not easy to obtain the analytical solution for the interior equilibrium point resulting from the intersection of F₁ and F₂. However, numerically we were able to demonstrate existence and non-existence of the interior point (results not included).

Disease-free equilibrium point
The solutions λ*_I = 0 and λ*_T = 0, in results (10) lead to the disease-free equilibrium given by
E₀ = (Λ / μ, 0,0,0).

TB-state
The case λ*_I = 0 and λ*_T ≠ 0 , lead to the TB-state given by

HIV-state
The case λ*_I ≠ 0 and λ*_T = 0, lead to the HIV-state given by

Dual infection (full model)
  The full dual infection model is complex to obtain solutions in compact form. Simple numerical simulations are carried out in section 5, to provide insight in the transmission dynamics of dual infection.

Global stability
Theorem 4.1: The disease-free equilibrium of the HIV and TB dual-infection model (1), is globally asymptotically stable whenever R₀ < 1 and unstable when R₀ > 1.

We construct a Lyapunov function of the form V(J_S, J_I, J_T, J_IT) = J_S - J_S0 - J_S0 ln(J_S / J_S0) + J_I + J_T + J_IT .

The time derivative of V(J_S, J_I, J_T, J_IT) along the solution path yields
dV/dt = Λ - λ_IJ_S - λ_TJ_S - λ_ITJ_S - J_S0/J_S(Λ - λ_IJ_S - λ_TJ_S - λ_ITJ_S) + λ_IJ_S - φ_Tλ_TJ_I - (μ + δ_I)J_I + λ_TJ_S - φ_Iλ_IJ_T - (μ + δ_T)J_T + φ_Tλ_TJ_I + φ_Iλ_IJ_T + λ_ITJ_S - (μ + δ_IT)J_IT

Evaluating the time derivative at the disease-free equilibrium level and we obtain
dV/dt = μJ_S0 - μJ_S - J_S0 / J_S(μJ_S0 - λ_IJ_S- λ_TJ_S - λ_ITJ_S) - (μ + δ_I)J_I - (μ + δ_T)J_T - (μ + δ_IT)J_IT,

= - {μ(J_S - J_S0/J_S) + κ(μ + δ_T)(1 - R_0T)} - {κμ(1 - R_0I)J_I + κ(μ + δ_IT) (1 - R_0IT)J_IT} ≤ 0
Provided R_0I ≤ 1, R_0T ≤ 1 and R_0IT ≤ 1.
If R₀ < 1, •V = 0 implies J_I = 0, J_T = 0 and J_IT = 0. It follows from system (1) that the largest invariant set where •V = 0 satisfies J_I = 0, J_T = 0, J_IT = 0, and J_S = Λ/μ = J_S0. By Lassalle’s invariance principle^[22], the disease-free equilibrium is globally asymptotically stable.

Numerical simulation
  In this section, we present numerical results to illustrate analytical results and to demonstrate results which could not be solved analytically, using published data from literature. We consider various scenarios to assess the impact of the infectivity rates in the transmission dynamics of the co-epidemic. The following parameter values are used in the simulations (Table 1).

  We consider five key modification parameters associated with co-infection (η_I,η_T,κ) and super-infection (φ_I, φ_T). We wish to address the question 'How does levels of infectivity of co-infected individuals affect the dynamics of HIV and TB epidemics?

  Figures 1(a) and 1(b) present variation in the magnitudes of η_I. Increasing the values of η_T we obtain drastic increase in the prevalence of HIV to maximum levels and settle at different levels. The results show marked increase in HIV-TB co-infection prevalence, that remain for some time at high levels before reducing drastically to low levels and settle at a common endemic state.

Table 1: parameter values for simulation.

  Increasing η_T (Figures 2(a) and 2(b)) on the other hand rapidly increases the prevalence of TB to the maximum level before reducing and settling at low levels. The prevalence of HIV TB drastically increases and settles at high levels for some time before drastically reducing and settling at low levels.

Figure 1: Variation of ηI with all other parameters fixed.
Λ = 0,29, δ_IT = 0,5, β_I = 0,5586, β_T = 0,31025, δ_I = 0,03, δ_T = 0,02, δ₁ = 0.01, β_IT = 0,6, φ_I= 1,1, φ_T= 1,03.

Figure 2: Variation of with all other parameters fixed.
Λ = 0,29, δ_IT = 0,5, β_I = 0,5586, β_T = 0,31025, δ_I = 0,03, δ_T = 0,02, δ₁ = 0.01, β_IT = 0,6, φ_I = 1,1, φ_T = 1,03.

Figure 3: (a) Variation of and (b) variation of with all other parameters fixed.
Λ = 0,29, δ_IT = 0,5, β_I = 0,5586, β_T = 0,31025, δ_I = 0,03, δ_T = 0,02, δ₁ = 0.01, β_IT = 0,6, φ_I = 1,1, φ_T = 1,03.

Figure 4: Variation of k to assess effects of simultaneous transmission of two pathogens.

  These results confirm findings from other studies which indicate that the two pathogens exhibit a synergistic relationship that is, each pathogen exacerbates the progression of the other^[16,23,24]. Increase effects on super-infection such as increased risk of TB infectives to HIV φ_I or increased risk of HIV infectives to TB φ_T has the effect of reducing the prevalence of singly-infected populations and increasing the dually infected population (Figures 3(a) and 3(b)). The suggests that the dual infection prevalence is not sensitive to increased effects in simultaneous transmission of pathogens (Figure 4).

Discussion

  A non-linear deterministic mathematical model of dual infection of HIV and TB is formulated and analysed. The aim of the study is to investigate the effects of simultaneous transmission of both HIV and TB pathogens on the disease dynamics. We assume a possibility of simultaneous transmission of both HIV and TB pathogens to susceptible individuals. We employ traditional analytical method of analysis to determine the steady states and their stability. The study showed that the disease-free equilibrium exists for all values of the reproduction number $R_0$ and is locally and globally asymptotically stable if R₀ < 1 and unstable if R₀ > 1. Numerical simulations were used to confirm analytical results. Analytically, we determined additional threshold parameters which govern super-infection. Our model was highly simplified but still led to a complex and very difficult problem to solve analytically. The symmetry of solution equations seem to suggest that techniques in advanced linear algebra (co-planar systems) or advanced vector calculus may provide insights conditions for existence of the interior solution (co-existence). Further studies are required to systematically compute the reproduction number for super-infection.

Acknowledge:
  The authors are grateful to anonymous referees and the managing editor for having read the paper and for their valuable comments, which constitute to the betterment of the manuscript.

References