Understanding the transmission pathways of Lassa fever: A mathematical modeling approach

Abstract

The spread of Lassa fever infection is increasing in West Africa over the last decade. The impact of this can better be understood when considering the various possible transmission routes. We designed a mathematical model for the epidemiology of Lassa Fever using a system of nonlinear ordinary differential equations to determine the effect of transmission pathways toward the infection progression in humans and rodents including those usually neglected such as the environmental surface and aerosol routes. We analyzed the model and carried out numerical simulations to determine the impact of each transmission routes. Our results showed that the burden of Lassa fever infection is increased when all the transmission routes are incorporated and most single transmission routes are less harmful, but when in combination with other transmission routes, they increase the Lassa fever burden. It is therefore important to consider multiple transmission routes to better estimate the Lassa fever burden optimally and in turn determine control strategies targeted at the transmission pathways.

1. Introduction

Lassa fever (LF) is an acute viral zoonotic illness responsible for a severe hemorrhagic fever. It is an illness caused by Lassa virus which is a single-stranded RNA virus from the arenavirus family. It was first discovered in 1969 in a Nigerian town called Lassa in Borno State when two missionary nurses died from the illness (Richmond & Baglole, 2003). The animal vector of this virus is the multimammate rat (Mastomys natalensis) which is one of the most common rodents in West Africa (CDC, 2014; Gonzalez, 2020; NICD, 2020). The Mastomys rodents reproduce often and excrete the virus in urine for a very long period of time, and because they occupy human homes, especially where food is stored, they help spread the virus to humans. The transmission of this virus to humans can be through direct or indirect contact. Direct transmissions result from contact between humans and humans, rodents and humans, and rodents and rodents. The evidence available shows that human to human transmission occurs through contact with the body fluids, secretions, excretions, blood of the infectious individual, and sexual transmission (Newman, 2018; NICD, 2020). The infected Mastomys rodents are caught (as bush meat) and eaten as food in certain places which directly infects the individuals with the virus. Reports have also shown that there is both horizontal and vertical transmission in multimammate rats especially in seasons when these rodents are actively breeding (Fichet-Calvet et al., 2014; Tewogbola & Aung, ; Gibb et al., 2017). In this paper, however, vertical transmission from rodent to rodent is not covered. Rodents can deposit the virus through their urine and faeces on surfaces in the households such as walls and places where food is stored or even on surfaces where medical equipment is kept. Humans can indirectly acquire the virus when they come in contact with the virus on these contaminated surfaces. Rodents can also become infected because rats share garbage, food on surfaces contaminated with the excretions of infectious rodents yet do not die due to disease but carry the infection and continue to shed it throughout their lifetime (NICD, 2020; Obabiyi & Onifade, 2017). Another form of indirect transmission is through airborne (aerosol) transmission which occurs especially in health centres when people inhale air particles contaminated with the droppings of infected rodents especially during activities like sweeping and other wind activities (CDC, 2014; Gonzalez, 2020; NICD, 2020). The natural history of Lassa fever reveals that its transmission pattern is driven by the frequency of exposure to infected individuals or through contact with infected rodents and contaminated environments (Akhmetzhanov et al., 2019; Sabeti, 2015). It has been shown that Lassa virus is stable as an aerosol, particularly at low relative humidity (30% RH) and the biological half-life at both 24 °C and 32 °C ranges from 10.1 to 54.6 min (CDC, 2014; Stephenson et al., 1984).

Frequent cases of Lassa virus infection have been seen in endemic regions such as Nigeria, Benin, Ghana, Guinea, Liberia, Mali, Sierra Leone and Togo. Surrounding regions like Central African Republic, Burkina Faso, Côte d’Ivoire, Mali, Senegal, Ghana among others are also at risk, because the rodents that transmit the virus are very common throughout West and East Africa. Hospital staff are also at risk for infection especially in areas with inadequate protective measures and improper sterilization methods (Gibb et al., 2017; NICD, 2020; WHO, 2017). After contracting the virus, humans show symptoms between 1 and 3 weeks. The presence of virus in the blood is known to peak four to nine days after the onset of symptoms. In most cases, 80% of people infected show no observable or mild symptoms. For these individuals, they show mild signs like slight fever, general malaise and weakness, and headache but do not die due to the infection. Recovery can take place eight to ten days after inception. The remaining 20% of infected individuals can show more severe symptoms like bleeding in the gums, eyes, or nose, respiratory distress, repeated vomiting, facial swelling, pain in the chest, back, and abdomen, shock, and failure in body organs such as liver, spleen and kidneys. The virus in this group of people may also lead to complications such as hearing loss, tremors, encephalitis or even death within 2 weeks after the onset of symptoms due to multiple organ failure (CDC, 2014; WHO, 2017; Yun & Walker, 2012). The number of Lassa virus infections per year in West Africa is estimated at 100,000 to 300,000, with approximately 5000 deaths. In some places in Liberia and Sierra Leone, the virus led to 10%–16% of people admitted to hospitals every year (ACDC, 2018; Gonzalez, 2020; Newman, 2018; NICD, 2020; Richmond & Baglole, 2003). In 26 out of 36 Nigerian states, a case fatality ratio of 14.8% was recorded between January 1 to February 9, 2020 (WHO, 2020).

Several studies have laid a basis to understand the dynamics of Lassa fever. Some of the studies on Lassa Fever (Ojo et al., 2021; Onah et al., 2020; Onuorah et al., 2016) only considered the basic transmission pathways namely, the human-to-human and the rodent-to-human. Because a great percentage of people show little or no symptoms of Lassa fever, Peter, et al. (Peter et al., 2020) described Lassa fever transmission dynamics using a deterministic model integrating the exposed human and rat compartment instead of the usual SIR compartmental structure. Some other studies have tried to establish the time-dependent nature of the transmission dynamics of Lassa fever. For example, Ibrahim and Dénes (Ibrahim & Dénes, 2021) used a compartmental model with time-dependent parameters where the infectious class was partitioned into symptomatically infected, mildly infected and treated individuals alongside with the carrying capacity of the rodent because of the periodic change of weather. Factors like quarantine, hospitalization of infected individuals were also used in (Ibrahim & Dénes, 2021; Musa et al., 2020) to comprehend the transmission variability of Lassa fever. The association between the reproduction number and local rainfall was used to investigate the epidemiological features of Lassa fever on a large scale (Zhao et al., 2020). Abdulhamid et al. (Abdulhamid et al., 2022) incorporated the effect of quarantine and the environment to show that in poor resource countries, Lassa fever transmission is driven by environmental contribution. Differential infectivity has also been deployed as a technique to analyze the complex nature of Lassa fever dynamics (Musa et al., 2022). In order to understand the epidemiology of the disease, it is important to look at a number of possible transmission pathways through which the virus can be contracted. In this work, we focus on the effects of multiple transmission pathways of Lassa Fever towards the progression of the infection in the human and rodent population. The use of multiple transmission routes gives us a better understanding of the epidemiological structure of Lassa fever. Our study extends the work of Peter et al. (Peter et al., 2020) and Ibrahim and Dénes (Ibrahim & Dénes, 2021) by:

• ⋅Introducing the environment to human transmission pathway. We define the environment as the surfaces, walls and any other equipment where the virus is deposited.
• ⋅Introducing the aerosol to human route of transmission. By aerosol, we refer to air particles where the virus is concentrated through human and wind activities.

These two pathways are not usually considered as major drivers of infection on a large scale. In endemic areas like Nigeria, a major route of infection is the contact with infected rodents through harvesting, however, recent reports (Abdulhamid et al., 2022) have also shown that the environmental pathway contribute to the burden of Lassa fever and should not be neglected. Thus, our study captures (i) human to human transmission (ii) rodent to human transmission (iii) rodent to rodent transmission (iv) environment to human transmission (v) aerosol to human transmission (vi) environment to rodent transmission. The aforementioned studies form the basic fabric for our work and the understanding gained from them will help us build and analyze a more comprehensive study with more transmission pathways. The remaining part of this work will be arranged thus: Section 2 will be the formulation of the basic model with basic analysis; In Section 3, we will perform our numerical simulations, and discussion and recommendations will be presented in Section 4.

2. Model formulation

The total human population, given as N_H(t), is subdivided into five classes which consists of humans susceptible to the virus, S_H(t), humans that have Lassa virus but are not infectious, E_H(t), infectious humans that are asymptomatic, I_HA(t), infectious humans that are symptomatic, I_HS(t), and humans who have recovered from Lassa fever, R_H(t) so that

$$N_H(t)=S_H(t)+E_H(t)+I_{HA}(t)+I_{HS}(t)+R_H(t).$$ (1)

The total rodent population, given as N_R(t), is subdivided into three classes consisting of rodents susceptible to the virus, S_R(t), rodents that have Lassa virus but are not infectious, E_R(t), and infected rodents that can transmit the virus, I_R(t) with

$$N_R(t)=S_R(t)+E_R(t)+I_R(t).$$ (2)

We consider the following direct transmission pathways: the human-to-human contact, the rodent-to-human contact, the rodent-to-rodent contact and the indirect transmission pathway such as the environment-to-human contact and the aerosol-to-human contact and the environment-to-rodent contact. To incorporate the indirect transmission pathways, we use V_S to describe the concentration of Lassa fever virus on the environmental surfaces and V_A, the concentration of Lassa virus in the air. The maximum carrying capacity of virus on environmental surfaces and in the air is given by K_V, where V_S, V_A ≤ K_V.

We assume that π₁ is the constant recruitment rate of susceptible humans. The susceptible humans move to the exposed class, E_H, through a force of infection

$${\lambda }_H={\beta }_H\left(\frac{I_R}{N_R}+\frac{{\eta }_1{I}_{HS}}{N_H}+\frac{{\eta }_2{I}_{HA}}{N_H}+\frac{{\eta }_3{V}_S}{K_V}+\frac{{\eta }_4{V}_A}{K_V}\right).$$ (3)

Here, β_H is the effective contact rate between susceptible humans and infected rodents, susceptible humans and infectious humans, susceptible humans, the virus in the environment and the virus in the air, η₁ is the modification parameter which indicates that contact with I_HS is less infectious than with I_R. Similarly, η₂, η₃, η₄ are also modification parameters which account for level of infectiousness of contact with I_HA, V_S and V_A respectively. Evidence from (Bausch et al., 2010; Davies et al., 2019; Lehmann et al., 2017; Lo Iacono et al., 2015) ensures that the following inequality holds:

$${\eta }_4<{\eta }_3<{\eta }_1<{\eta }_2<1.$$

The exposed humans progress to the infectious compartment at the rate ψ₁, where the proportion of exposed individuals that become asymptomatic is νψ₁ and the proportion of exposed persons that become symptomatic is (1 − ν)ψ₁. Humans die naturally in all classes at the rate μ₁. Infectious symptomatic humans can die due to the disease at the rate δ but there are no cases of death due to infection for the infectious asymptomatic individuals. Infectious asymptomatic and infectious symptomatic humans recover at the rates ζ₁ and ζ₂, respectively.

The susceptible rodents are recruited at a constant rate π₂ and move to the exposed class E_R through a force of infection

$${\lambda }_R={\beta }_R\left(\frac{I_R}{N_R}+\frac{{\xi }_1{V}_S}{K_V}\right),$$ (4)

where β_R is the effective contact rate between susceptible rodents and infected rodents and between susceptible rodents and contaminated environment surfaces. ξ₁ is the modification parameter which shows that contact with V_S is less infectious than with I_R. The exposed rodents move to the infectious class at the rate ψ₂ and all rodents die naturally at a rate of μ₂. Rodents can also die at a rate ρ due to consumption by humans as food. Rodents do not die due to disease since infected rodents can continue to shed the virus throughout their lifetime. The Lassa fever virus is deposited into the environment at rates of φ₁, φ₂ and φ₃ by infectious asymptomatic humans, infectious symptomatic humans, and infected rodents respectively through activities such as urination, excretion of faeces, bleeding and fluid secretions. We further assume that the virus concentration on the environmental surfaces and in the air decays at the rate θ₂ while a proportion of the virus concentration moves into the air at the rate θ₃ through wind and human activities.

The Lassa fever model in Fig. 1 is expressed as a system of first order nonlinear ordinary differential equations as follows:

$$\begin{array}{lll}\frac{d{S}_H}{dt}& =& {\pi }_1-{\lambda }_H{S}_H-{\mu }_1{S}_H,\\ \frac{d{E}_H}{dt}& =& {\lambda }_H{S}_H-({\psi }_1+{\mu }_1)E_H,\\ \frac{d{I}_{HA}}{dt}& =& \nu {\psi }_1{E}_H-({\zeta }_1+{\mu }_1)I_{HA},\\ \frac{d{I}_{HS}}{dt}& =& (1-\nu ){\psi }_1{E}_H-(\delta +{\zeta }_2+{\mu }_1)I_{HS},\\ \frac{d{R}_H}{dt}& =& {\zeta }_1{I}_{HA}+{\zeta }_2{I}_{HS}-{\mu }_1{R}_H,\\ \frac{d{S}_R}{dt}& =& {\pi }_2-{\lambda }_R{S}_R-(\rho +{\mu }_2)S_R,\\ \frac{d{E}_R}{dt}& =& {\lambda }_R{S}_R-({\psi }_2+\rho +{\mu }_2)E_R,\\ \frac{d{I}_R}{dt}& =& {\psi }_2{E}_R-(\rho +{\mu }_2)I_R,\\ \frac{d{V}_S}{dt}& =& {\phi }_1{I}_{HA}+{\phi }_2{I}_{HS}+{\phi }_3{I}_R-({\theta }_2+{\theta }_3)V_S,\\ \frac{d{V}_A}{dt}& =& {\theta }_3{V}_S-{\theta }_2{V}_A,\end{array}$$ (5)

Fig. 1.

The Lassa fever schematic diagram for human, virus and rodent population.

which is subject to the following initial conditions:

$$\begin{array}{lll}{S}_H(0)& =& S_{H_0}>0,E_H(0)=E_{H_0}\ge 0,I_{HA}(0)=I_{{HA}_0}\ge 0,\\ I_{HS}(0)& =& I_{{HS}_0}\ge 0,R_H(0)=R_{H_0}\ge 0,S_R(0)=S_{R_0}>0,\\ E_R(0)& =& E_{R_0}\ge 0,I_R(0)=I_{R_0}\ge 0,V_S(0)=V_{S_0}\ge 0,\\ V_A(0)& =& V_{A_0}\ge 0,\forall t\ge 0.\end{array}$$ (6)

The variables and parameter descriptions and units are presented in Table 1.

Table 1.

Description of parameters and variables for model (5).

Variables	Description	Unit
SH	Susceptible human population	people
EH	Exposed human population	people
IHA	Infectious asymptomatic human population	people
IHS	Infectious symptomatic human population	people
RH	Recovered human population	people
SR	Susceptible rodent population	rodents
ER	Exposed rodent population	rodents
IR	Infected rodent population	rodents
VS	Concentration of Lassa virus in the environmental surfaces	virus
VA	Concentration of Lassa virus in the air	virus
Parameters	Description	Unit
βH	Contact rate between SH and IR, IHA, IHS, VS, VA	day−1
η1	Modification parameter	nil
η2	Modification parameter	nil
η3	Modification parameter	nil
η4	Modification parameter	nil
βR	Contact rate between SR and IR, VS	day−1
ξ1	Modification parameter	nil
ψ1	Rate of progression of humans from EH to IHA and IHS	day−1
ψ2	Rate of progression of rodents from ER to IR	day−1
θ2	Rate of decay of virus in VS	day−1
θ3	Rate of progression of virus from VS to VA	day−1
φ1	Rate at which virus is shed in VS by IHA	virus/people × day
φ2	Rate at which virus is shed in VS by IHS	virus/people × day
φ3	Rate at which virus is shed in VS by IR	virus/rodents × day
μ1	Natural death rate of humans	day−1
μ2	Natural death rate of rodents	day−1
ρ	Death rate of rodents due to consumption by humans	day−1
ζ1	Recovery rate of IHA	day−1
ζ2	Recovery rate of IHS	day−1
π1	Recruitment rate of humans	people/day
π2	Recruitment rate of rodents	rodents/day
KV	Maximum carrying capacity of virus	virus
δ	Disease-induced death rate of humans	day−1
ν	Proportion of humans progressing to IHA	nil

2.1. Model analysis

Here, we show that our model is mathematically and biologically meaningful. We shall also compute the basic reproduction number and carry out the stability analysis of the steady states.

2.1.1. Feasible region

We assume that all parameters are non-negative for time t and prove that in the proposed region, Ω, the solutions remain non-negative and bounded. We will analyze the Lassa fever transmission model in the region given as:

$$\mathrm{\Omega }=\left\{(S_H,E_H,I_{HA},I_{HS},R_H,S_R,E_R,I_R,V_S,V_A)\in {\mathfrak{R}}_{+}^{10}:N_H\le \frac{{\pi }_1}{{\mu }_1},N_R\le \frac{{\pi }_2}{\rho +{\mu }_2},V_S\le M_S,V_A\le M_A\right\},$$

where

$$M_S=\frac{({\phi }_1+{\phi }_2){\pi }_1}{{\mu }_1({\theta }_2+{\theta }_3)}+\frac{{\phi }_3{\pi }_2}{{\mu }_2({\theta }_2+{\theta }_3)},M_A=\frac{{\theta }_3{\pi }_1({\phi }_1+{\phi }_2)}{{\mu }_1{\theta }_2({\theta }_2+{\theta }_3)}+\frac{{\theta }_3{\phi }_3{\pi }_2}{{\mu }_2{\theta }_2({\theta }_2+{\theta }_3)}.$$

To show that the region Ω is positively invariant, we consider the first equation of (5)

$$\frac{d{S}_H}{dt}={\pi }_1-({\lambda }_H+{\mu }_1)S_H,$$

which is solved to obtain

$$S_H(t)=S_{H_0}{e}^{-{\int }_0^t({\mu }_1+{\lambda }_H(s))ds}+\left[\frac{{\pi }_1}{{\mu }_1}{e}^{-{\int }_0^t({\mu }_1+{\lambda }_H(s))ds}\times {\int }_0^t{e}^{{\int }_0^s({\mu }_1+{\lambda }_H(r))dr}ds\right]>0.$$

Also, for the solution component of E_H(t), we suppose that there exist a first time t₁ such that $E_H(t_1)=0,E_H^{\prime }(t_1)<0$ and the rest of the variables are non-negative for 0 < t₁ < t. The second equation of system (5) gives

$$\frac{d{E}_H}{dt}{|}_{t=t_1}={\lambda }_H(t_1)S_H(t_1)>0,$$

which is a contradiction, so E_H(t) ≥ 0, ∀ t ≥ 0.

Using a similar approach, it is easy to show that I_HA, I_HS, R_H, S_R, E_R, I_R, V_S, V_A are non-negative. Hence, all solutions of (5) are non-negative in Ω.

Now, we show that all solutions with non-negative initial conditions are bounded in the set Ω. It is easy to see that

$$\frac{d{N}_H}{dt}\le {\pi }_1-{\mu }_1{N}_H,\frac{d{N}_R}{dt}={\pi }_2-(\rho +{\mu }_2)N_R.$$ (6)

Solving the differential inequality and equation in (6), we use the standard comparison theorem (Lakshmikantham et al., 1989) and the integrating factor to show that as t → ∞, we have that $0\le N_H(t)\le \frac{{\pi }_1}{{\mu }_1}$ and $0\le N_R(t)=\frac{{\pi }_2}{\rho +{\mu }_2}$. Similarly the differential inequalities

$$\frac{d{V}_S}{dt}\le ({\phi }_1+{\phi }_2)\frac{{\pi }_1}{{\mu }_1}+{\phi }_3\frac{{\pi }_2}{{\mu }_2}-({\theta }_2+{\theta }_3)V_S,\frac{d{V}_A}{dt}\le \frac{{\theta }_3{\pi }_1({\phi }_1+{\phi }_2)}{{\mu }_1{\theta }_2}+\frac{{\theta }_3{\phi }_3{\pi }_2}{{\mu }_2{\theta }_2}-{\theta }_2{V}_A,$$

yield $V_S\le \frac{({\phi }_1+{\phi }_2){\pi }_1}{{\mu }_1({\theta }_2+{\theta }_3)}+\frac{{\phi }_3{\pi }_2}{{\mu }_2({\theta }_2+{\theta }_3)}=M_S,V_A\le \frac{{\theta }_3{\pi }_1({\phi }_1+{\phi }_2)}{{\mu }_1{\theta }_2({\theta }_2+{\theta }_3)}+\frac{{\theta }_3{\phi }_3{\pi }_2}{{\mu }_2{\theta }_2({\theta }_2+{\theta }_3)}=M_A$.

Thus, all possible solutions of (5) enter the region Ω and stay inside it. Hence, the region Ω is positively invariant and attracting and therefore a feasible region.

2.1.2. Reproduction number and equilibria stability analysis

We explore the existence of the equilibrium points of model (5). To obtain the disease free equilibrium (DFE), we equate the right hand side of model (5) to zero and solve when E_H = I_HA = I_HS = E_R = I_R = V_S = V_A = 0 to get:

$$E^0=\left(\frac{{\pi }_1}{{\mu }_1},0,0,0,0,\frac{{\pi }_2}{\rho +{\mu }_2},0,0,0,0\right).$$

The basic reproduction number R₀ of model (5) is the dominant eigenvalue of the matrix FV⁻¹ using the next generation matrix approach (Van den Driessche & Watmough, 2002). Here,

$$\mathcal{F}=\left(\begin{array}{c}{\lambda }_H{S}_H\\ 0\\ 0\\ {\lambda }_R{S}_R\\ 0\\ 0\\ 0\end{array}\right),\mathcal{V}=\left(\begin{array}{c}({\psi }_1+{\mu }_1)E_H\\ -\nu {\psi }_1{E}_H+({\zeta }_1+{\mu }_1)I_{HA}\\ -(1-\nu ){\psi }_1{E}_H+(\delta +{\zeta }_2+{\mu }_1)I_{HS}\\ ({\psi }_2+\rho +{\mu }_2)E_R\\ -{\psi }_2{E}_R+(\rho +{\mu }_2)I_R\\ -{\phi }_1{I}_{HA}-{\phi }_2{I}_{HS}-{\phi }_3{I}_R+({\theta }_2+{\theta }_3)V_S\\ -{\theta }_3{V}_S+{\theta }_2{V}_A\end{array}\right).$$

Computing the Jacobian of $\mathcal{F}$ and $\mathcal{V}$ evaluated at E⁰ we get

$$F=\left(\begin{array}{ccccccc}0& {\beta }_H{\eta }_2& {\beta }_H{\eta }_1& 0& \frac{{\beta }_H{\mu }_2{\pi }_1}{{\mu }_1{\pi }_2}& \frac{{\beta }_H{\eta }_3{\pi }_1}{{\mu }_1{K}_V}& \frac{{\beta }_H{\eta }_4{\pi }_1}{{\mu }_1{K}_V}\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& {\beta }_R& \frac{{\beta }_R{\xi }_1{\pi }_2}{{\mu }_2{K}_V}& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right),$$

$$V=\left(\begin{array}{ccccccc}{\psi }_1+{\mu }_1& 0& 0& 0& 0& 0& 0\\ -\nu {\psi }_1& {\zeta }_1+{\mu }_1& 0& 0& 0& 0& 0\\ -(1-\nu ){\psi }_1& 0& \delta +{\zeta }_2+{\mu }_1& 0& 0& 0& 0\\ 0& 0& 0& {\psi }_2+\rho +{\mu }_2& 0& 0& 0\\ 0& 0& 0& -{\psi }_2& \rho +{\mu }_2& 0& 0\\ 0& -{\phi }_1& -{\phi }_2& 0& -{\phi }_3& ({\theta }_2+{\theta }_3)& 0\\ 0& 0& 0& 0& 0& -{\theta }_3& {\theta }_2\end{array}\right).$$

The inverse of V is given as

$$V^{-1}=\left(\begin{array}{ccccccc}\frac{1}{{\mu }_1+{\psi }_1}& 0& 0& 0& 0& 0& 0\\ \frac{\nu {\psi }_1}{\left({\zeta }_1+{\mu }_1\right)\left({\mu }_1+{\psi }_1\right)}& \frac{1}{{\zeta }_1+{\mu }_1}& 0& 0& 0& 0& 0\\ -\frac{(-1+\nu ){\psi }_1}{\left(\delta +{\zeta }_2+{\mu }_1\right)\left({\mu }_1+{\psi }_1\right)}& 0& \frac{1}{\delta +{\zeta }_2+{\mu }_1}& 0& 0& 0& 0\\ 0& 0& 0& \frac{1}{{\mu }_2+\rho +{\psi }_2}& 0& 0& 0\\ 0& 0& 0& \frac{{\psi }_2}{\left(\rho +{\psi }_2\right)\left({\mu }_2+\rho +{\psi }_2\right)}& \frac{1}{\rho +{\psi }_2}& 0& 0\\ v_{61}& \frac{{\phi }_1}{({\theta }_2+{\theta }_3)\left({\zeta }_1+{\mu }_1\right)}& v_{63}& v_{64}& \frac{{\phi }_3}{({\theta }_2+{\theta }_3)\left(\rho +{\psi }_2\right)}& \frac{1}{({\theta }_2+{\theta }_3)}& 0\\ v_{71}& \frac{{\theta }_3{\phi }_1}{{\theta }_2({\theta }_2+{\theta }_3)\left({\zeta }_1+{\mu }_1\right)}& v_{73}& v_{74}& \frac{{\theta }_3{\phi }_3}{{\theta }_2({\theta }_2+{\theta }_3)\left(\rho +{\psi }_2\right)}& \frac{{\theta }_3}{{\theta }_2({\theta }_2+{\theta }_3)}& \frac{1}{{\theta }_2}\end{array}\right)$$

where

$$\begin{array}{lll}{v}_{61}& =& -\frac{-\left(\left(\delta +{\zeta }_2+{\mu }_1\right)\nu {\psi }_1{\phi }_1\right)+(-1+\nu )\left({\zeta }_1+{\mu }_1\right){\phi }_2{\psi }_1}{({\theta }_2+{\theta }_3)\left({\zeta }_1+{\mu }_1\right)\left(\delta +{\zeta }_2+{\mu }_1\right)\left({\mu }_1+{\psi }_1\right)},\\ v_{63}& =& \frac{{\phi }_2}{({\theta }_2+{\theta }_3)\left(\delta +{\zeta }_2+{\mu }_1\right)},\\ v_{64}& =& \frac{{\phi }_3{\psi }_2}{({\theta }_2+{\theta }_3)\left(\rho +{\psi }_2\right)\left({\mu }_2+\rho +{\psi }_2\right)},\\ v_{71}& =& -\frac{{\theta }_3\left(-\left(\left(\delta +{\zeta }_2+{\mu }_1\right)\nu {\psi }_1{\phi }_1\right)+(-1+\nu )\left({\zeta }_1+{\mu }_1\right){\phi }_2{\psi }_1\right)}{{\theta }_2({\theta }_2+{\theta }_3)\left({\zeta }_1+{\mu }_1\right)\left(\delta +{\zeta }_2+{\mu }_1\right)\left({\mu }_1+{\psi }_1\right)},\\ v_{73}& =& \frac{{\theta }_3{\phi }_2}{{\theta }_2({\theta }_2+{\theta }_3)\left(\delta +{\zeta }_2+{\mu }_1\right)},\\ v_{74}& =& \frac{{\theta }_3{\phi }_3{\psi }_2}{{\theta }_2({\theta }_2+{\theta }_3)\left(\rho +{\psi }_2\right)\left({\mu }_2+\rho +{\psi }_2\right)}.\end{array}$$ (7a)

The next generation matrix evaluated at DFE is

$$F{V}^{-1}=\left(\begin{array}{ccccccc}{b}_{11}& b_{12}& b_{13}& b_{14}& b_{15}& b_{16}& b_{17}\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ b_{41}& b_{42}& b_{43}& b_{44}& b_{45}& b_{46}& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right),$$

where

$$\begin{array}{lll}{b}_{11}& =& \frac{{\beta }_H\left(K_V{\theta }_2({\theta }_2+{\theta }_3){\mu }_1({\eta }_2(\delta +{\zeta }_2+{\mu }_1)\nu {\psi }_1+(1-\nu ){\eta }_1({\zeta }_1+{\mu }_1){\psi }_1)\right)}{K_V{\theta }_2({\theta }_2+{\theta }_3){\mu }_1({\zeta }_1+{\mu }_1)(\delta +{\zeta }_2+{\mu }_1)({\mu }_1+{\psi }_1)}\\ & +& \frac{{\beta }_H\left(({\pi }_1({\eta }_3{\theta }_2+{\eta }_4{\theta }_3)((\delta +{\zeta }_2+{\mu }_1)\nu {\psi }_1{\phi }_1+(1-\nu )({\zeta }_1+{\mu }_1){\phi }_2{\psi }_1))\right)}{K_V{\theta }_2({\theta }_2+{\theta }_3){\mu }_1({\zeta }_1+{\mu }_1)(\delta +{\zeta }_2+{\mu }_1)({\mu }_1+{\psi }_1)},\\ b_{12}& =& \frac{{\beta }_H{\eta }_2}{\left({\zeta }_1+{\mu }_1\right)}+\frac{{\pi }_1{\beta }_H{\eta }_3{\phi }_1}{K_V({\theta }_2+{\theta }_3){\mu }_1\left({\zeta }_1+{\mu }_1\right)}+\frac{{\pi }_1{\beta }_H{\eta }_4{\theta }_3{\phi }_1}{K_V{\theta }_2({\theta }_2+{\theta }_3){\mu }_1\left({\zeta }_1+{\mu }_1\right)},\\ b_{13}& =& \frac{{\beta }_H{\eta }_1}{\left(\delta +{\zeta }_2+{\mu }_1\right)}+\frac{{\pi }_1{\beta }_H{\eta }_3{\phi }_2}{K_V({\theta }_2+{\theta }_3){\mu }_1\left(\delta +{\zeta }_2+{\mu }_1\right)}+\frac{{\pi }_1{\beta }_H{\eta }_4{\theta }_3{\phi }_2}{K_V{\theta }_2({\theta }_2+{\theta }_3){\mu }_1\left(\delta +{\zeta }_2+{\mu }_1\right)},\\ b_{14}& =& \frac{{\pi }_1{\beta }_H\left(K_V{\theta }_2({\theta }_2+{\theta }_3){\mu }_2+{\pi }_2\left({\eta }_3{\theta }_2+{\eta }_4{\theta }_3\right){\phi }_3\right){\psi }_2}{K_V{\pi }_2{\theta }_2({\theta }_2+{\theta }_3){\mu }_1\left(\rho +{\psi }_2\right)\left({\mu }_2+\rho +{\psi }_2\right)},\\ b_{15}& =& \frac{{\pi }_1{\beta }_H{\mu }_2}{{\pi }_2{\mu }_1\left(\rho +{\psi }_2\right)}+\frac{{\pi }_1{\beta }_H{\eta }_3{\phi }_3}{K_V({\theta }_2+{\theta }_3){\mu }_1\left(\rho +{\psi }_2\right)}+\frac{{\pi }_1{\beta }_H{\eta }_4{\theta }_3{\phi }_3}{K_V{\theta }_2({\theta }_2+{\theta }_3){\mu }_1\left(\rho +{\psi }_2\right)},\\ b_{16}& =& \frac{{\pi }_1{\beta }_H{\eta }_3}{K_V({\theta }_2+{\theta }_3){\mu }_1}+\frac{{\pi }_1{\beta }_H{\eta }_4{\theta }_3}{K_V{\theta }_2({\theta }_2+{\theta }_3){\mu }_1},b_{17}=\frac{{\pi }_1{\beta }_H{\eta }_4}{K_V{\theta }_2{\mu }_1},\\ b_{41}& =& \frac{{\pi }_2{\beta }_R{\xi }_1\left(\left(\delta +{\zeta }_2+{\mu }_1\right)\nu {\psi }_1{\phi }_1+(1-\nu )\left({\zeta }_1+{\mu }_1\right){\phi }_2{\psi }_1\right)}{K_V({\theta }_2+{\theta }_3)\left({\zeta }_1+{\mu }_1\right)\left(\delta +{\zeta }_2+{\mu }_1\right){\mu }_2\left({\mu }_1+{\psi }_1\right)},\\ b_{42}& =& \frac{{\pi }_2{\beta }_R{\xi }_1{\phi }_1}{K_V({\theta }_2+{\theta }_3)\left({\zeta }_1+{\mu }_1\right){\mu }_2},b_{43}=\frac{{\pi }_2{\beta }_R{\xi }_1{\phi }_2}{K_V({\theta }_2+{\theta }_3)\left(\delta +{\zeta }_2+{\mu }_1\right){\mu }_2},\\ b_{44}& =& \frac{{\beta }_R\left(K_V({\theta }_2+{\theta }_3){\mu }_2+{\pi }_2{\xi }_1{\phi }_3\right){\psi }_2}{K_V({\theta }_2+{\theta }_3){\mu }_2\left(\rho +{\psi }_2\right)\left({\mu }_2+\rho +{\psi }_2\right)},\\ b_{45}& =& \frac{{\beta }_R}{\left(\rho +{\psi }_2\right)}+\frac{{\pi }_2{\beta }_R{\xi }_1{\phi }_3}{K_V({\theta }_2+{\theta }_3){\mu }_2\left(\rho +{\psi }_2\right)},b_{46}=\frac{{\pi }_2{\beta }_R{\xi }_1}{K_V({\theta }_2+{\theta }_3){\mu }_2}.\end{array}$$ (7b)

The basic reproduction number is given by

$$R_0=\frac{b_{11}+b_{44}}{2}+\frac{\sqrt{{(b_{11}-b_{44})}^2+4b_{14}{b}_{41}}}{2}.$$ (7c)

Remark 2.1.1

• (i)The terms contained in R₀ represent the direct and indirect transmission pathways. They are described thus: b₁₁ is the local reproduction number of infectious asymptomatic humans, infectious symptomatic humans, contaminated environmental surfaces and contaminated air particles in the progression of Lassa Fever virus in the human population only; b₄₄ is the local reproduction number of infected rodents and contaminated environmental surfaces in the progression of Lassa Fever virus in the rodent population only; b₁₄ is the local reproduction number of contaminated environmental surfaces and contaminated air particles in the progression of Lassa Fever virus in the human population only; and b₄₁ is the local reproduction number of contaminated environmental surfaces in the progression of Lassa Fever virus in the rodent population only.
• (ii)It is easy to see that

$${(b_{11}-b_{44})}^2+4b_{14}{b}_{41}={(b_{11}+b_{44})}^2+4(b_{14}{b}_{41}-b_{11}{b}_{44}).$$

Thus, b₁₄b₄₁ > b₁₁b₄₄ implies that R₀ > 0. This condition ensures that infection will be sustained across from rodents to humans and from humans to rodents via all the transmission pathways.

• (iii)The local stability of the disease free equilibrium point when R₀ < 1 is ensured by the hypothesis used in the computation of R₀ (Van den Driessche & Watmough, 2002).

Theorem 2.1.1

The disease free equilibrium point is globally asymptotically stable ifR₀ < 1.

Proof. It suffices to show that our model (5) satisfy conditions H1 and H2 of the global stability theorem by (Castillo-Chavez et al., 2002) when R₀ < 1.

Model (5) can be rewritten in the form

$$\begin{array}{cc}\frac{dX}{dt}& =F(X,Y),\\ \frac{dY}{dt}& =G(X,Y),G(X,\mathbf{0})=0.\end{array}$$

Here, $X={(S_H,S_R)}^T$, $Y={(E_H,I_{HA},I_{HS},E_R,I_R,V_S,V_A)}^T$, $X\in {\mathfrak{R}}_{+}^2$ represents the number of susceptible humans and rodents and $Y\in {\mathfrak{R}}_{+}^7$ represents the number of exposed humans, asymptomatic infectious humans, symptomatic infectious humans, exposed rodents, infected rodents, contaminated environment and contaminated air.

Our DFE is now written as E₀ = (X, 0) where $X_0=(\frac{{\pi }_1}{{\mu }_1},\frac{{\pi }_2}{\rho +{\mu }_2})$ and we show that the following conditions are satisfied:

• ⋅H1: For $\frac{dX}{d\tau }=F(X_0,\mathbf{0}),X_0$ is globally asymptotically stable.
• ⋅H2: $G(X,Y)=AY-\widehat{G}(X,Y)$, $\widehat{G}(X,Y)\ge 0$ for (X, Y) ∈ Ω where A = D_Y(G(X₀, 0)) is an M-matrix and Ω is the region where the model makes biological sense.

For the first condition H1, we have

$$\frac{d{S}_H}{dt}={\pi }_1-{\mu }_1{S}_H,\frac{d{S}_R}{dt}={\pi }_2-(\rho +{\mu }_2)S_R,$$ (8)

which can be solved to get

$$S_H(t)=\frac{{\pi }_1}{{\mu }_1}+\left(\frac{{\pi }_1}{{\mu }_1}-S_{H_0}\right)e^{-{\mu }_{1}t},S_R(t)=\frac{{\pi }_2}{\rho +{\mu }_2}+\left(\frac{{\pi }_2}{\rho +{\mu }_2}-S_{R_0}\right)e^{-(\rho +{\mu }_2)t}.$$

We take limits as t → ∞ to get,

$${\lim }_{t\to \infty }{S}_H(t)=\frac{{\pi }_1}{{\mu }_1},{\lim }_{t\to \infty }{S}_R(t)=\frac{{\pi }_2}{\rho +{\mu }_2}.$$

Hence, the solutions of equation (8) converge to X₀ regardless of the initial conditions. Therefore, X₀ is a globally asymptotically equilibrium point of (8).

For condition H2, we consider $F(X,0)=({\pi }_1-{\mu }_1{S}_H,{\pi }_2-(\rho +{\mu }_2)S_R),G(X,Y)=AY-\widehat{G}(X,Y)$

where

$$G(X,Y)=\left(\begin{array}{c}{\lambda }_H{S}_H-({\psi }_1+{\mu }_1)E_H\\ \nu {\psi }_1{E}_H-({\zeta }_1+{\mu }_1)I_{HA}\\ (1-\nu ){\psi }_1{E}_H-(\delta +{\zeta }_2+{\mu }_1)I_{HS}\\ {\lambda }_R{S}_R-({\psi }_2+\rho +{\mu }_2)E_R\\ {\psi }_2{E}_R-(\rho +{\mu }_2)I_R\\ {\phi }_1{I}_{HA}+{\phi }_2{I}_{HS}+{\phi }_3{I}_R-({\theta }_2+{\theta }_3)V_S\\ {\theta }_3{V}_S-{\theta }_2{V}_A\end{array}\right),$$

and

$$A=\left(\begin{array}{ccccccc}-({\psi }_1+{\mu }_1)& {\beta }_H{\eta }_2& {\beta }_H{\eta }_1& 0& \frac{{\beta }_H{\mu }_2{\pi }_1}{{\mu }_1{\pi }_2}& \frac{{\beta }_H{\eta }_3{\pi }_1}{{\mu }_1{K}_V}& \frac{{\beta }_H{\eta }_4{\pi }_1}{{\mu }_1{K}_V}\\ \nu {\psi }_1& -({\zeta }_1+{\mu }_1)& 0& 0& 0& 0& 0\\ (1-\nu ){\psi }_1& 0& -(\delta +{\zeta }_2+{\mu }_1)& 0& 0& 0& 0\\ 0& 0& 0& -({\psi }_2+\rho +{\mu }_2)& {\beta }_R& \frac{{\beta }_R{\xi }_1{\pi }_2}{{\mu }_2{K}_V}& 0\\ 0& 0& 0& {\psi }_2& -(\rho +{\mu }_2)& 0& 0\\ 0& {\phi }_1& {\phi }_2& 0& {\phi }_3& -({\theta }_2+{\theta }_3)& 0\\ 0& 0& 0& 0& 0& {\theta }_3& -{\theta }_2\end{array}\right),$$

which is an M-matrix and

$$\widehat{G}(X,Y)=\left(\begin{array}{c}{\beta }_H{\eta }_2(S_H^0-S_H)+{\beta }_H{\eta }_1(S_H^0-S_H)+\frac{{\beta }_H{\mu }_2}{{\pi }_2}(S_H^0-S_H)+\frac{{\beta }_H{\eta }_3}{K_V}(S_H^0-S_H)+\frac{{\beta }_H{\eta }_4}{K_V}(S_H^0-S_H)\\ 0\\ 0\\ {\beta }_R(S_R^0-S_R)+\frac{{\beta }_R{\xi }_1}{K_V}(S_R^0-S_R)\\ 0\\ 0\\ 0\end{array}\right).$$

Clearly, $\widehat{G}(X,Y)\ge 0$ since $0\le S_H\le S_H^0$ and $0\le S_R\le S_R^0$. □

The global stability of E⁰ thus follows and the Lassa fever virus can be eliminated from the human and rodent population over a period of time provided R₀ < 1.

The endemic equilibrium point of the model (5) is given by

$$E^{\ast }=(S_H^{\ast },E_H^{\ast },I_{HA}^{\ast },I_{HS}^{\ast },R_H^{\ast },S_R^{\ast },E_R^{\ast },I_R^{\ast },V_S^{\ast },V_A^{\ast }),$$

where

$$\begin{array}{lll}{S}_H^{\ast }& =& \frac{{\pi }_1}{{\lambda }_H^{\ast }+{\mu }_1},E_H^{\ast }=\frac{{\pi }_1{\lambda }_H^{\ast }}{({\psi }_1+{\mu }_1)({\lambda }_H^{\ast }+{\mu }_1)},I_{HA}^{\ast }=\frac{\nu {\psi }_1{\pi }_1{\lambda }_H^{\ast }}{({\zeta }_1+{\mu }_1)({\psi }_1+{\mu }_1)({\lambda }_H^{\ast }+{\mu }_1)},\\ I_{HS}^{\ast }& =& \frac{(1-\nu ){\psi }_1{\pi }_1{\lambda }_H^{\ast }}{(\delta +{\zeta }_2+{\mu }_1)({\psi }_1+{\mu }_1)({\lambda }_H^{\ast }+{\mu }_1)},\\ R_H^{\ast }& =& \frac{{\zeta }_1\nu {\psi }_1{\pi }_1{\lambda }_H^{\ast }}{{\mu }_1({\zeta }_1+{\mu }_1)({\psi }_1+{\mu }_1)({\lambda }_H^{\ast }+{\mu }_1)}+\frac{{\zeta }_2(1-\nu ){\psi }_1{\pi }_1{\lambda }_H^{\ast }}{{\mu }_1(\delta +{\zeta }_2+{\mu }_1)({\psi }_1+{\mu }_1)({\lambda }_H^{\ast }+{\mu }_1)},\\ S_R^{\ast }& =& \frac{{\pi }_2}{{\lambda }_R^{\ast }+\rho +{\mu }_2},E_R^{\ast }=\frac{{\pi }_2{\lambda }_R^{\ast }}{({\psi }_2+\rho +{\mu }_2)({\lambda }_R^{\ast }+\rho +{\mu }_2)},I_R^{\ast }=\frac{{\psi }_2{\pi }_2{\lambda }_R^{\ast }}{(\rho +{\mu }_2)({\psi }_2+\rho +{\mu }_2)({\lambda }_R^{\ast }+\rho +{\mu }_2)},\\ V_S^{\ast }& =& \frac{{\phi }_1\nu {\psi }_1{\pi }_1{\lambda }_H^{\ast }}{({\theta }_2+{\theta }_3)({\zeta }_1+{\mu }_1)({\psi }_1+{\mu }_1)({\lambda }_H^{\ast }+{\mu }_1)}+\frac{{\phi }_2(1-\nu ){\psi }_1{\pi }_1{\lambda }_H^{\ast }}{({\theta }_2+{\theta }_3)(\delta +{\zeta }_2+{\mu }_1)({\psi }_1+{\mu }_1)({\lambda }_H^{\ast }+{\mu }_1)}\\ & +& \frac{{\phi }_3{\psi }_2{\pi }_2{\lambda }_R^{\ast }}{({\theta }_2+{\theta }_3)(\rho +{\mu }_2)({\psi }_2+\rho +{\mu }_2)({\lambda }_R^{\ast }+\rho +{\mu }_2)},\\ V_A^{\ast }& =& \frac{{\theta }_3{\phi }_1\nu {\psi }_1{\pi }_1{\lambda }_H^{\ast }}{{\theta }_2({\theta }_2+{\theta }_3)({\zeta }_1+{\mu }_1)({\psi }_1+{\mu }_1)({\lambda }_H^{\ast }+{\mu }_1)}+\frac{{\theta }_3{\phi }_2(1-\nu ){\psi }_1{\pi }_1{\lambda }_H^{\ast }}{{\theta }_2({\theta }_2+{\theta }_3)(\delta +{\zeta }_2+{\mu }_1)({\psi }_1+{\mu }_1)({\lambda }_H^{\ast }+{\mu }_1)}\\ & +& \frac{{\theta }_3{\phi }_3{\psi }_2{\pi }_2{\lambda }_R^{\ast }}{{\theta }_2({\theta }_2+{\theta }_3)(\rho +{\mu }_2)({\psi }_2+\rho +{\mu }_2)({\lambda }_R^{\ast }+\rho +{\mu }_2)},\end{array}$$ (9)

and

$$\begin{array}{lll}{\lambda }_H^{\ast }& =& {\beta }_H\left(\frac{I_R^{\ast }}{N_R^{\ast }}+\frac{{\eta }_1{I}_{HS}^{\ast }}{N_H^{\ast }}+\frac{{\eta }_2{I}_{HA}^{\ast }}{N_H^{\ast }}+\frac{{\eta }_3{V}_S^{\ast }}{K_V^{\ast }}+\frac{{\eta }_4{V}_A^{\ast }}{K_V^{\ast }}\right),\end{array}$$ (9)

$$\begin{array}{lll}{\lambda }_R^{\ast }& =& {\beta }_R\left(\frac{I_R^{\ast }}{N_R^{\ast }}+\frac{{\xi }_1{V}_S^{\ast }}{K_V^{\ast }}\right).\end{array}$$ (10)

Equations (9), (9), (10) can be written explicitly as

$$\begin{array}{lll}{\lambda }_H^{\ast }& =& \frac{{\mu }_1{b}_{11}{\lambda }_H^{\ast }}{({\lambda }_H^{\ast }+{\mu }_1)}+\frac{{\mu }_1{\pi }_2{b}_{14}{\lambda }_R^{\ast }}{{\pi }_1({\lambda }_R^{\ast }+\rho +{\mu }_2)},\end{array}$$ (11)

$$\begin{array}{lll}{\lambda }_R^{\ast }& =& \frac{{\mu }_2{\pi }_1{b}_{41}{\lambda }_H^{\ast }}{{\pi }_2({\lambda }_H^{\ast }+{\mu }_1)}+\frac{{\mu }_2{b}_{44}{\lambda }_R^{\ast }}{({\lambda }_R^{\ast }+\rho +{\mu }_2)}.\end{array}$$ (12)

We see that the state variables are expressed in terms of ${\lambda }_H^{\ast }$ and ${\lambda }_R^{\ast }$. From here, we proceed by using the approach in (Moghadas et al., 2003; Velasco-Hernandez & Hsieh, 1994). Hence, we can obtain positive equilibrium points of the model by finding the fixed points of equations (11), (12) as

$$c({\lambda }_H,{\lambda }_R)=\left(\begin{array}{c}{c}_1({\lambda }_H,{\lambda }_R)\\ c_2({\lambda }_H,{\lambda }_R)\end{array}\right),$$

where

$$\left(\begin{array}{c}{c}_1({\lambda }_H,{\lambda }_R)\\ c_2({\lambda }_H,{\lambda }_R)\end{array}\right)$$

corresponds to the right hand sides of equations (11), (12).

Theorem 2.1.2

There exists a unique fixed point$({\lambda }_H^{\ast },{\lambda }_R^{\ast }),{\lambda }_H^{\ast }>0,{\lambda }_R^{\ast }>0$which satisfies

$$c({\lambda }_H^{\ast },{\lambda }_R^{\ast })=\left(\begin{array}{c}{c}_1({\lambda }_H^{\ast },{\lambda }_R^{\ast })\\ c_2({\lambda }_H^{\ast },{\lambda }_R^{\ast })\end{array}\right)$$

and corresponds to the endemic equilibrium point E∗ (Moghadas et al., 2003; Velasco-Hernandez & Hsieh, 1994).

Proof. From the first equation, we fix λ_R > 0 and look at the real-valued function depending on λ_H:

$$c_1^{{\lambda }_R}({\lambda }_H)=\frac{{\mu }_1{b}_{11}{\lambda }_H}{({\lambda }_H+{\mu }_1)}+\frac{{\mu }_1{\pi }_2{b}_{14}{\lambda }_R}{{\pi }_1({\lambda }_R+\rho +{\mu }_2)}.$$

We have that

$${\lim }_{{\lambda }_H\to 0}{c}_1^{{\lambda }_R}({\lambda }_H)=\frac{{\mu }_1{\pi }_2{b}_{14}{\lambda }_R}{{\pi }_1({\lambda }_R+\rho +{\mu }_2)}<\infty ,$$

and

$${\lim }_{{\lambda }_H\to \infty }{c}_1^{{\lambda }_R}({\lambda }_H)={\mu }_1{b}_{11}+\frac{{\mu }_1{\pi }_2{b}_{14}{\lambda }_R}{{\pi }_1({\lambda }_R+\rho +{\mu }_2)}<\infty .$$

It follows that $0<c_1^{{\lambda }_R}({\lambda }_H)<\infty$ which implies that $c_1^{{\lambda }_R}({\lambda }_H)$ is bounded for every fixed λ_R > 0.

Next,

$$\frac{\partial c_1^{{\lambda }_R}({\lambda }_H)}{\partial {\lambda }_H}=\frac{{\mu }_1^2{b}_{11}}{{({\lambda }_H+{\mu }_1)}^2},$$

and

$$\frac{{\partial }^2{c}_1^{{\lambda }_R}({\lambda }_H)}{{\partial }^2{\lambda }_H}=-\frac{2{\mu }_1^2{b}_{11}}{{({\lambda }_H+{\mu }_1)}^3}.$$

$c_1^{{\lambda }_R}({\lambda }_H)$ is an increasing concave down function since $\frac{\partial c_1^{{\lambda }_R}({\lambda }_H)}{\partial {\lambda }_H}>0$ and $\frac{{\partial }^2{c}_1^{{\lambda }_R}({\lambda }_H)}{{\partial }^2{\lambda }_H}<0$. Hence, there is no change in concavity of c₁ in the bounded domain. It follows that there exists a unique ${\lambda }_H^{\ast }>0$ which satisfies $c_1^{{\lambda }_R}({\lambda }_H^{\ast })={\lambda }_H^{\ast }$.

For this ${\lambda }_H^{\ast }>0$, we look at the real-valued function depending on λ_R:

$$c_2^{{\lambda }_H^{\ast }}({\lambda }_R)=\frac{{\mu }_2{\pi }_1{b}_{41}{\lambda }_H^{\ast }}{{\pi }_2({\lambda }_H^{\ast }+{\mu }_1)}+\frac{{\mu }_2{b}_{44}{\lambda }_R}{({\lambda }_R+\rho +{\mu }_2)}.$$

Then,

$${\lim }_{{\lambda }_R\to 0}{c}_2^{{\lambda }_H^{\ast }}({\lambda }_R)=\frac{{\mu }_2{\pi }_1{b}_{41}{\lambda }_H^{\ast }}{{\pi }_2({\lambda }_H^{\ast }+{\mu }_1)}<\infty ,$$

and

$${\lim }_{{\lambda }_R\to \infty }{c}_2^{{\lambda }_H^{\ast }}({\lambda }_R)={\mu }_2{b}_{44}+\frac{{\mu }_2{\pi }_1{b}_{41}{\lambda }_H^{\ast }}{{\pi }_2({\lambda }_H^{\ast }+{\mu }_1)}<\infty .$$

It follows that $0<c_2^{{\lambda }_H^{\ast }}({\lambda }_R)<\infty$ which implies that $c_2^{{\lambda }_H^{\ast }}({\lambda }_R)$ is bounded for every fixed ${\lambda }_H^{\ast }>0$.

Next,

$$\frac{\partial c_2^{{\lambda }_H^{\ast }}({\lambda }_R)}{\partial {\lambda }_R}=\frac{{\mu }_2{b}_{44}(\rho +{\mu }_2)}{{({\lambda }_R+\rho +{\mu }_2)}^2},$$

and

$$\frac{{\partial }^2{c}_2^{{\lambda }_H^{\ast }}({\lambda }_R)}{{\partial }^2{\lambda }_R}=-\frac{2{\mu }_2{b}_{44}(\rho +{\mu }_2)}{{({\lambda }_R+\rho +{\mu }_2)}^3}.$$

$c_2^{{\lambda }_H^{\ast }}({\lambda }_R)$ is an increasing concave down function since $\frac{\partial c_2^{{\lambda }_H^{\ast }}({\lambda }_R)}{\partial {\lambda }_R}>0$ and $\frac{{\partial }^2{c}_2^{{\lambda }_H^{\ast }}({\lambda }_R)}{{\partial }^2{\lambda }_R}<0$. Hence, there is no change in the concavity of c₂ in the positive domain. It follows that there exists a unique ${\lambda }_R^{\ast }>0$ which satisfies $c_2^{{\lambda }_H^{\ast }}({\lambda }_R^{\ast })={\lambda }_R^{\ast }$.

Therefore, there is a fixed point $({\lambda }_H^{\ast },{\lambda }_R^{\ast })$ which corresponds to the endemic equilibrium point E∗. □

We now investigate the stability of the equilibrium points using the stability of the fixed point system $({\lambda }_H^{\ast },{\lambda }_R^{\ast })$ corresponding to E∗. The Jacobian of the system is given by:

$$J({\lambda }_H^{\ast },{\lambda }_R^{\ast })=\left(\begin{array}{cc}\frac{\partial c_1}{\partial {\lambda }_H^{\ast }}& \frac{\partial c_1}{\partial {\lambda }_R^{\ast }}\\ \frac{\partial c_2}{\partial {\lambda }_H^{\ast }}& \frac{\partial c_2}{\partial {\lambda }_R^{\ast }}\end{array}\right),$$ (13)

where

$$\frac{\partial c_1}{\partial {\lambda }_H^{\ast }}=\frac{{\mu }_1^2{b}_{11}}{{({\lambda }_H+{\mu }_1)}^2},\frac{\partial c_1}{\partial {\lambda }_R^{\ast }}=\frac{{\mu }_1{\pi }_2{b}_{14}{(\rho +{\mu }_2)}^2}{{\pi }_1{\mu }_2{({\lambda }_R+\rho +{\mu }_2)}^2},\frac{\partial c_2}{\partial {\lambda }_H^{\ast }}=\frac{{\mu }_1{\mu }_2{\pi }_1{b}_{41}}{{\pi }_2{({\lambda }_H+{\mu }_1)}^2},\frac{\partial c_2}{\partial {\lambda }_R^{\ast }}=\frac{b_{44}{(\rho +{\mu }_2)}^2}{{({\lambda }_R+\rho +{\mu }_2)}^2}.$$

We note that $J({\lambda }_H^{\ast },{\lambda }_R^{\ast })$ evaluated at the fixed point, $({\lambda }_H^{\ast },{\lambda }_R^{\ast })=(0,0)$, is given by

$$J(0,0)=\left(\begin{array}{cc}{b}_{11}& \frac{{\mu }_1{\pi }_2{b}_{14}}{{\pi }_1{\mu }_2}\\ \frac{{\mu }_2{\pi }_1{b}_{41}}{{\pi }_2{\mu }_1}& b_{44}\end{array}\right),$$

and for stability, we require that |λ_i| < 1, where λ_i are the eigenvalues of J(0, 0), which corresponds to

$$R_0=\frac{b_{11}+b_{44}}{2}+\frac{\sqrt{{(b_{11}-b_{44})}^2+4b_{14}{b}_{41}}}{2}<1.$$ (14)

Hence, the stability of $({\lambda }_H^{\ast },{\lambda }_R^{\ast })=(0,0)$ is achieved when R₀ < 1. The point is unstable provided R₀ > 1. Thus, the stability of $({\lambda }_H^{\ast },{\lambda }_R^{\ast })=(0,0)$ corresponds to the stability of E⁰ when R₀ < 1. Now, for $({\lambda }_H^{\ast },{\lambda }_R^{\ast })\ne (0,0)$, we have

$$J({\lambda }_H^{\ast },{\lambda }_R^{\ast })=\left(\begin{array}{cc}{d}_{11}& d_{12}\\ d_{21}& d_{22}\end{array}\right),$$

and for stability, we require that |λ_i| < 1, that is,

$$|\frac{d_{11}+d_{22}}{2}+\frac{\sqrt{{(d_{11}-d_{22})}^2+4d_{12}{d}_{21}}}{2}|<1,$$

where

$$d_{11}=\frac{{\mu }_1^2{b}_{11}}{{({\lambda }_H^{\ast }+{\mu }_1)}^2},d_{12}=\frac{{\mu }_1{\pi }_2{b}_{14}{(\rho +{\mu }_2)}^2}{{\pi }_1{\mu }_2{({\lambda }_R^{\ast }+\rho +{\mu }_2)}^2},d_{21}=\frac{{\mu }_1{\mu }_2{\pi }_1{b}_{41}}{{\pi }_2{({\lambda }_H^{\ast }+{\mu }_1)}^2},d_{22}=\frac{b_{44}{(\rho +{\mu }_2)}^2}{{({\lambda }_R^{\ast }+\rho +{\mu }_2)}^2}.$$

The stability of the fixed point system is governed by the fact that the absolute value of the eigenvalues of the fixed point system is less than unity (Moghadas et al., 2003; Velasco-Hernandez & Hsieh, 1994). Hence, |λ_i| < 1 corresponds to

$$\frac{1+d_{11}{d}_{22}}{d_{11}+d_{22}+d_{12}{d}_{21}}>1.$$ (15)

Defining the left hand side of (15) as $R({\lambda }_H^{\ast },{\lambda }_R^{\ast })$, the fixed point $({\lambda }_H^{\ast },{\lambda }_R^{\ast })\ne (0,0)$ is stable when $R({\lambda }_H^{\ast },{\lambda }_R^{\ast })>1$.

2.1.3. Global stability of endemic equilibrium and bifurcation analysis

We first show that E∗ is globally asymptotically stable using the following theorem:

Theorem 2.1.3

The endemic equilibrium pointE∗ of model (5) is globally asymptotically stable when $\frac{{\mu }_2{\pi }_1{b}_{41}}{{\pi }_2(c+{\mu }_1)}+\frac{{\mu }_2{b}_{44}}{(c+\rho +{\mu }_2)}>1$.

Proof. We use the geometric approach in (Li & Muldowney, 1996). Let us consider the fixed point system (11) and (12). We convert it to a root finding problem to get

$$\begin{array}{lll}\widehat{f_1}& =& \frac{{\mu }_1{b}_{11}{\lambda }_H^{\ast }}{({\lambda }_H^{\ast }+{\mu }_1)}+\frac{{\mu }_1{\pi }_2{b}_{14}{\lambda }_R^{\ast }}{{\pi }_1({\lambda }_R^{\ast }+\rho +{\mu }_2)}-{\lambda }_H^{\ast },\\ \widehat{f_2}& =& \frac{{\mu }_2{\pi }_1{b}_{41}{\lambda }_H^{\ast }}{{\pi }_2({\lambda }_H^{\ast }+{\mu }_1)}+\frac{{\mu }_2{b}_{44}{\lambda }_R^{\ast }}{({\lambda }_R^{\ast }+\rho +{\mu }_2)}-{\lambda }_R^{\ast }.\end{array}$$ (16)

The Jacobian matrix corresponding to this system is

$$J=\left(\begin{array}{cc}{f}_{11}-1& f_{12}\\ f_{21}& f_{22}-1\end{array}\right),$$

where

$$f_{11}=\frac{{\mu }_1^2{b}_{11}}{{({\lambda }_H+{\mu }_1)}^2},f_{12}=\frac{{\mu }_1{\pi }_2{b}_{14}(\rho +{\mu }_2)}{{\pi }_1{({\lambda }_R+\rho +{\mu }_2)}^2},f_{21}=\frac{{\mu }_1{\mu }_2{\pi }_1{b}_{41}}{{\pi }_2{({\lambda }_H+{\mu }_1)}^2},f_{22}=\frac{{\mu }_2{b}_{44}(\rho +{\mu }_2)}{{({\lambda }_R+\rho +{\mu }_2)}^2}.$$

The second additive compound matrix (Muldowney, 1990) of J is

$$J^{[2]}=\left[\begin{array}{c}\frac{{\mu }_1^2{b}_{11}}{{({\lambda }_H+{\mu }_1)}^2}+\frac{{\mu }_2{b}_{44}(\rho +{\mu }_2)}{{({\lambda }_R+\rho +{\mu }_2)}^2}-2\end{array}\right]\equiv [\mathcal{C}].$$

We assume the function

$$Q=Q({\lambda }_H,{\lambda }_R)=\left[\begin{array}{cc}\frac{{\lambda }_H}{{\lambda }_R}& 0\\ 0& \frac{{\lambda }_H}{{\lambda }_R}\end{array}\right],$$

we have

$$Q^{-1}=\left[\begin{array}{cc}\frac{{\lambda }_R}{{\lambda }_H}& 0\\ 0& \frac{{\lambda }_R}{{\lambda }_H}\end{array}\right],Q_f=\left[\begin{array}{cc}\frac{{\lambda }_R\dot{{\lambda }_H}-{\lambda }_H\dot{{\lambda }_R}}{{\lambda }_H^2}& 0\\ 0& \frac{{\lambda }_R\dot{{\lambda }_H}-{\lambda }_H\dot{{\lambda }_R}}{{\lambda }_H^2}\end{array}\right],Q_f{Q}^{-1}=\left[\begin{array}{cc}\frac{\dot{{\lambda }_H}}{{\lambda }_H}-\frac{\dot{{\lambda }_R}}{{\lambda }_R}& 0\\ 0& \frac{\dot{{\lambda }_H}}{{\lambda }_H}-\frac{\dot{{\lambda }_R}}{{\lambda }_R}\end{array}\right],Q{J}^{[2]}{Q}^{-1}=\left[\begin{array}{cc}\mathcal{C}& 0\\ 0& \mathcal{C}\end{array}\right].$$

We define

$$\mathcal{B}=Q_f{Q}^{-1}+Q{J}^{[2]}{Q}^{-1}=\left[\begin{array}{cc}{B}_{11}& B_{12}\\ B_{21}& B_{22}\end{array}\right],$$

where $B_{11}=B_{22}=\frac{\dot{{\lambda }_H}}{{\lambda }_H}-\frac{\dot{{\lambda }_R}}{{\lambda }_R}+\mathcal{C},B_{12}=B_{21}=0$.

Using

$$\frac{\dot{{\lambda }_R}}{{\lambda }_R}=\frac{{\mu }_2{\pi }_1{b}_{41}{\lambda }_H}{{\pi }_2{\lambda }_R({\lambda }_H+{\mu }_1)}+\frac{{\mu }_2{b}_{44}}{({\lambda }_R+\rho +{\mu }_2)}-1,$$

Then

$$B_{11}=B_{22}=\frac{\dot{{\lambda }_H}}{{\lambda }_H}-\frac{{\mu }_2{\pi }_1{b}_{41}{\lambda }_H}{{\pi }_2{\lambda }_R({\lambda }_H+{\mu }_1)}-\frac{{\mu }_2{b}_{44}}{({\lambda }_R+\rho +{\mu }_2)}+1.$$

We follow the approach in (Li & Muldowney, 1996) to get

$$\nu (\mathcal{B})\le sup\{g_1,g_2\}\equiv sup\{{\nu }_1(B_{11})+|B_{12}|,{\nu }_1(B_{22})+|B_{21}|\},$$

where ν₁ denotes the Lozinskii measure with respect to the L¹ norm and |B₁₂|, |B₂₁| are matrix norms with respect to the L¹ vector norm. So,

$$\nu (\mathcal{B})\le \frac{\dot{{\lambda }_H}}{{\lambda }_H}-\frac{{\mu }_2{\pi }_1{b}_{41}{\lambda }_H}{{\pi }_2{\lambda }_R({\lambda }_H+{\mu }_1)}-\frac{{\mu }_2{b}_{44}}{({\lambda }_R+\rho +{\mu }_2)}+1.$$

Using Lemma 1 in (Srivastava et al., 2022) and the uniform persistence result in (Freedman et al., 1994), this gives us

$$\nu (\mathcal{B})\le \frac{\dot{{\lambda }_H}}{{\lambda }_H}-\left\{\frac{{\mu }_2{\pi }_1{b}_{41}}{{\pi }_2(c+{\mu }_1)}+\frac{{\mu }_2{b}_{44}}{(c+\rho +{\mu }_2)}-1\right\}.$$

We choose $\mathrm{\Delta }=\frac{{\mu }_2{\pi }_1{b}_{41}}{{\pi }_2(c+{\mu }_1)}+\frac{{\mu }_2{b}_{44}}{(c+\rho +{\mu }_2)}-1>0$, so

$$\nu (\mathcal{B})\le \frac{\dot{{\lambda }_H}}{{\lambda }_H}-\mathrm{\Delta }.$$

Integrating both sides, we get

$${\int }_0^t\nu (\mathcal{B})ds\le {\int }_0^t\frac{\dot{{\lambda }_H}}{{\lambda }_H}dt-{\int }_0^t\mathrm{\Delta }dt,$$

and it follows that

$$\frac{1}{t}{\int }_0^t\nu (\mathcal{B})ds\le \frac{1}{t}log\frac{{\lambda }_H(t)}{{\lambda }_H(0)}-\mathrm{\Delta },$$

or

$${\mathrm{lim\; sup}}_{t\to \infty }sup\frac{1}{t}{\int }_0^t\nu (\mathcal{B})ds\le -\mathrm{\Delta }\le 0,$$ (17)

as λ_H(t) is bounded and Δ >0

Hence, $\widehat{q_2}={\mathrm{lim\; sup}}_{t\to \infty }sup\frac{1}{t}{\int }_0^t\nu (\mathcal{B})ds<0$, if Δ > 0 or $\frac{{\mu }_2{\pi }_1{b}_{41}}{{\pi }_2(c+{\mu }_1)}+\frac{{\mu }_2{b}_{44}}{(c+\rho +{\mu }_2)}>1$.

Thus, system (16) is globally asymptotically stable for $\frac{{\mu }_2{\pi }_1{b}_{41}}{{\pi }_2(c+{\mu }_1)}+\frac{{\mu }_2{b}_{44}}{(c+\rho +{\mu }_2)}>1$, that is $({\lambda }_H,{\lambda }_R)\to ({\lambda }_H^{\ast },{\lambda }_R^{\ast })$ as t → ∞. □

Next, we investigate conditions on the parameter values in model (5) using Center Manifold Theory (Castillo-Chavez & Song, 2004).

Theorem 2.1.4

• (i)If$\frac{{\beta }_H{\mu }_2}{{\pi }_2}{w}_1{v}_2{w}_1<\frac{{\beta }_H{\pi }_1{\mu }_2^2}{{\pi }_2^2{\mu }_1}{w}_6{v}_2{w}_8$andb > 0, then the system (5) will undergo a forward bifurcation at R₀ = 1.
• (ii)If$B_{wv}<\frac{{\beta }_H{\mu }_2}{{\pi }_2}{w}_1{v}_2{w}_1-\frac{{\beta }_H{\pi }_1{\mu }_2^2}{{\pi }_2^2{\mu }_1}{w}_6{v}_2{w}_8$, andb > 0, then the system (5) will undergo a backward bifurcation at R₀ = 1.

(Castillo-Chavez & Song, 2004).

Proof. See Appendix A. □

3. Numerical simulations

3.1. Parameter estimation

It is crucial to estimate the model parameter values in order for us to perform numerical analysis. We consider the ecological niche where Lassa fever is endemic. We focus on three (3) states in Nigeria (Ondo, Edo and Ebonyi) where this virus has ravaged communities in the past few years based on Nigeria Centre for Disease Control (NCDC) reports (NCDC, 2021).

In the chosen region, we consider a few local sites in the three states of about 10000 persons, $\frac{{\pi }_1}{{\mu }_1}=10000$. The human natural death rate is ${\mu }_1=\frac{1}{55.12\times 365}$ day⁻¹ using the average human lifespan in Nigeria as 55.12 (Trends, 2021). The daily recruitment rate of humans is estimated as π₁ = 10000 × μ₁ = 0.497 day⁻¹. The value, ν = 0.8 since 80% of individuals are asymptomatic. The sample study of Lassa fever cases in Nigeria shows a case fatality ratio of 18.9% in the year 2021 (NCDC, 2021), we assume δ = 0.189 year⁻¹ which translates to δ = 0.0005 day⁻¹. Research conducted in these communities had reported a yearly rodent consumption rate of 29.9% in Edo State, 11% in Ebonyi State and 20.2% in Ondo State (Ossai et al., 2020; WHO, 2021). So, we use an average consumption rate of 20.4% per year giving us ρ = 0.0006 day⁻¹. The biological half-life of Lassa virus ranges from 10.1 to 54.6 min (Stephenson et al., 1984); so using 10.1 min implies that ${\theta }_3=\frac{2\times 10.1}{60\times 24}$ day⁻¹.

We consider a hypothetical average population of Mastomys rat to be $\frac{{\pi }_2}{{\mu }_2}=1000$, since there is no known quantified estimate of the rodents population. The average lifespan of a rodent is 1 year (Control, 2018), so we obtain ${\mu }_2=\frac{1}{1\times 365}$ day⁻¹. We estimate the rodent recruitment rate to be π₂ = 1000 × μ₂ = 2.74 day⁻¹. Some of the parameters cannot be found or estimated from literature, so we used model calibration to get ideal representation curves for all state variables to get approximate values. Thus ψ₁ = 0.0094 day⁻¹, ψ₂ = 0.048 day⁻¹, β_H = 0.00017 day⁻¹, β_R = 0.004 day⁻¹, ξ = 0.167, ${\phi }_1=\frac{1}{15}$, ${\phi }_2=\frac{1}{28}$, ${\phi }_3=\frac{3.7}{60\times 24}$ and the values of η₁, η₂, η₃, η₄, to lie in the interval (0, 1). For our simulation, we use the following initial conditions: S_H(0) = 10000, E_H(0) = 0, I_HA(0) = 324, I_HS(0) = 81, R_H(0) = 10, S_R(0) = 1000, E_R(0) = 0, I_R(0) = 100, V_S(0) = 1000, V_A(0) = 100. Table 2 contains the parameter values used in the simulations.

Table 2.

Parameter values and references.

Parameters	Value	Reference
π1	0.497	Calculated
π2	2.74	Calculated
μ1	0.0000497	Trends (2021)
μ2	0.00274	Control (2018)
ψ1	0.0094	Assumed
ψ2	0.048	Assumed
βH	0.00017	Assumed
βR	0.004	Assumed
ρ	0.0006	(Ossai et al., 2020; WHO, 2021)
ν	0.8	(CDC, 2014; WHO, 2017; Yun & Walker, 2012)
ζ1	0.0000476	Estimated
ζ2	0.0000323	Estimated
δ	0.0005	NCDC (2021)
θ2	0.01868	Estimated
θ3	0.00701	Stephenson et al. (1984)
φ1	0.0667	Assumed
φ2	0.0357	Assumed
φ3	0.02569	Assumed
ξ1	0.167	Assumed
η1	0.94	Assumed
η2	0.95	Assumed
η3	0.9	Assumed
η4	0.85	Assumed

3.2. Sensitivity analysis

Sensitivity analysis is a procedure used to determine the strength of model predictions to parameter values. It is crucial because there are usually flaws in assumed parameter values and generally in data collection. Sensitivity analysis shows the parameters that deserve the best numerical attention, reveals insensitive parameters that do not require much effort to estimate and shows which parameters should be targeted for intervention (Mikucki, 2012). Local sensitivity analysis is based on calculating the effect on the model output of small perturbations around a nominal parameter value. This perturbation is done on one parameter at a time using the first-order partial derivative of the model output with respect to the perturbed parameter. Here, we will investigate parameters that have a high impact on R₀, and should be targeted by intervention strategies. The global sensitivity analysis, on the other hand, seeks to explore the input parameters space across its range of variation and then quantify the input parameter importance based on a characterization of the resulting output response surface. It is a sampling-based method that investigates uncertainties for parameter values in the entire parameter range (Blower & Dowlatabadi, 1994; Marino et al., 2008; Saltelli et al., 2004, 2008; Turányi, 1990). We will perform both the local and global sensitivity analysis.

3.2.1. Local sensitivity analysis of R₀

We calculate the local sensitivity indices of the parameters with respect to R₀ using the normalized forward index. These indices reveal the importance of each parameter to disease transmission and should be taken into consideration while defining our control strategies. According to (Chitnis et al., 2008), the normalized forward sensitivity index of a variable u that depends differentiably on a parameter ρ is defined as:

$${\mathrm{\Upsilon }}_{\rho }^u=\frac{\partial u}{\partial \rho }\times \frac{\rho }{u}.$$

For example, the sensitivity index of R₀ with respect to β_H will be

$${\mathrm{\Upsilon }}_{{\beta }_H}^{R_0}=\frac{\partial R_0}{\partial {\beta }_H}\times \frac{{\beta }_H}{R_0}.$$

When ${\mathrm{\Upsilon }}_{{\beta }_H}^{R_0}>0$, we say that β_H increases the value of R₀ as its value increases, while if ${\mathrm{\Upsilon }}_{{\beta }_H}^{R_0}<0$, then β_H decreases the value of R₀ as its value increases. The results of the sensitivity indices is shown in Table 3.

Table 3.

Sensitivity indices of R₀.

Parameters	Sensitivity indices of R0	Parameters	Sensitivity indices of R0
π1	0.444782	δ	−0.0272335
π2	0.0000581831	θ2	−0.439338
μ1	−0.946283	θ3	−0.00550217
μ2	−0.000168322	φ1	0.43505
ψ1	0.00524857	φ2	0.00973221
ψ2	−0.00190397	φ3	0.0000581831
βH	0.997936	ξ1	0.00199971
βR	0.0020637	η1	0.0219675
ρ	−0.0000495958	η2	0.531187
ν	0.839438	η3	0.32696
ζ1	−0.472691	η4	0.115881
ζ2	−0.00175928

We observe from Table 3 that parameters such as β_H, ν, η₂, π₁, φ₁, η₃, η₄, η₁, β_R, ξ₁ are positively correlated with R₀ thus increase in these parameters increase the reproduction number. The parameters μ₁, ζ₁, δ, θ₂ are negatively correlated with R₀ thus they decrease the value of R₀ as they are increased. There are parameters such as π₂, μ₂, ρ, φ₃ that are insensitive with respect to the reproduction number of Lassa fever in the population. These parameters do not require too much effort to estimate and will not cause much changes to R₀ when they are increased or decreased. All parameters associated with infection pathways have positive indices and thus, all the infection pathways have a potential to collectively or otherwise increase the infection. We also observe that β_H is the most sensitive parameter followed by ν, η₂, π₁, φ₁, η₃, η₄, η₁, β_R, ξ₁ respectively. Intervention strategies can be targeted at reducing the impact of parameters which increase R₀ whilst increasing those that reduce it.

3.2.2. Global sensitivity analysis

The global sensitivity analysis is carried out using the Latin Hypercube Sampling and Partial Rank Correlation Coefficients (PRCCs) (Marino et al., 2008). This is a robust sensitivity measure that combines uncertainty analysis with partial correlation on rank-transformed data to assess the sensitivity of our outcome variable to parameter variation. Fig. 2, Fig. 3 shows that the parameters β_H, β_R, φ₁, φ₃, θ₃, ψ₁, ν are positively correlated to C_H and thus increase the burden of Lassa fever infection in the human population; the parameters θ₂, μ₂, π₁ are negatively correlated to C_H and decrease the burden of Lassa virus in the human population when they are increased. There are also insensitive parameters ψ₂, μ₁, ψ₁, ζ₁, ζ₂, φ₂ with PRCCs very close to zero. The parameters associated with infection pathways though weakly correlated, remain positively correlated in the different populations. In the rodent population, the parameters β_H, β_R, ψ₁, ψ₂, μ₁, μ₂, θ₃ are positively correlated to C_R and increase the infection burden in rodents, parameters π₁, π₂, μ₂, φ₂, ρ, μ₂, θ₃ are negatively correlated to C_R while there are parameters with PRCCs close to zero such as δ, θ₃, ζ₁, ζ₂ and so on. From the PRCCs of the virus population, we see positively correlated parameters such as β_R, ψ₂, π₂, μ₁, ξ₁, negatively correlated parameters μ₂, ρ, π₁ and other parameters with PRCCs very close to zero. In all populations, we see that some parameters are positively or negatively correlated at certain time points but become insensitive at other time points and verse visa. Hence, the interplay and the exchange of sensitivity by different parameters on different variables alludes to the complexities brought about by the multiple transmission pathways which in turn suggest the importance of every pathway in the prognosis of Lassa fever.

Fig. 2.

Partial Rank Correlation Coefficient for the full range of parameters of model (5) in the cumulative cases of human and rodent population.

Fig. 3.

Partial Rank Correlation Coefficient for the full range of parameters of model (5) in the cumulative cases of virus population.

3.3. Simulation results

Fig. 4 shows the baseline graphs of system (5) without varying the system parameters. The simulations were done over a time period of 40000 days. The baseline graph is perceived to represent the ideal situation where Lassa fever persists in the system. We will illustrate the impact of the transmission pathways in the next subsection.

Fig. 4.

Model simulations for all the state variables in model (5) with R₀ greater than unity: R₀ = 3.1301.

3.3.1. Simulation of the transmission pathways

We now investigate the impact of the various transmission routes on the progression of Lassa fever in both human and rodent population as well as the growth of virus in the environment. We shall proceed using the following strategies:

• 1.
  Transmission pathways for the human population

  • (a)
    5 Single transmission pathways (see Fig. 5).
  • (b)
    10 combinations of two transmission pathways (see Fig. 6).
  • (c)
    10 combinations of three transmission pathways (see Fig. 7).
  • (d)
    5 combinations of four transmission pathways (see Fig. 8).
  • (e)
    1 combination of five transmission pathways (see Fig. 9).
• 2.
  Transmission pathways for rodent population

  • (a)
    Single transmission pathways (see Fig. 9).
  • (b)
    1 combination of two transmission pathways (see Fig. 10).

Fig. 5.

Graphical illustration of model (5) for single transmission routes on the cumulative cases of human, rodent and virus classes.

Fig. 6.

Graphical illustration of model (5) for possible combination of two transmission routes on the cumulative cases of human, rodent and virus classes.

Fig. 7.

Graphical illustration of model (5) for possible combination of three transmission routes on the cumulative cases of human, rodent and virus classes.

Fig. 8.

Graphical illustration of model (5) for possible combination of four transmission routes on the cumulative cases of human, rodent and virus classes.

Fig. 9.

Graphical illustration of model (5) for all possible combination of transmission routes alongside one transmission route on the cumulative cases of human, rodent and virus classes.

Fig. 10.

Graphical illustration of model (5) for possible combination of the two rodent transmission routes on all the classes.

By single transmission pathway, we use each of the single contact rates in the human force of infection and test its impact on the system while the entire rodent force of infection is operational. We do the same for two transmission routes and continue till we exhaust all other transmission pathways. We also investigate using each of the single contact rates in the rodent force of infection and the two transmission routes while keeping the entire human force of infection in use. We shall test these strategies using the cumulative cases (NCDC, 2021) in the humans, rodents and virus in the environment. To capture this, we will simulate the cumulative cases using the equations:

$$\begin{array}{lll}\frac{d{C}_H}{dt}& =& {\psi }_1{E}_H,\\ \frac{d{C}_R}{dt}& =& {\psi }_2{E}_R,\\ \frac{d{C}_{VS}}{dt}& =& {\phi }_1{I}_{HA}+{\phi }_2{I}_{HS}+{\phi }_3{I}_R,\end{array}$$ (19)

subject to the initial conditions $C_H(0)=0,C_R(0)=0,C_{V_S}(0)=505$, where C_H is the cumulative infection cases in the human population, C_R is the cumulative infection cases in the rodent population, and C_VS is the cumulative infection cases in the virus population.

Fig. 5 reveals that the effective contact rate between susceptible humans and infected rodents does the most damage with regards to the progression of infection. This is followed by the contact rate between susceptible humans and infectious asymptomatic humans which is less infectious and then by the contact with contaminated environment, contaminated air, and infectious symptomatic humans. We observe that every single route of transmission plays a role in driving the Lassa fever infection even though some are less significant than the others. In the rodent population, we also see a notable difference in the level of infectiousness of the transmission routes likewise in the virus population. This shows that some pathways are more deadly than others yet every pathway makes their own contribution. We see from Fig. 6 that a combination of two transmission pathways increases disease burden more than a single pathway. We also see that some combinations are more deadly than others. Any combination with the effective contact rate between susceptible humans and infected rodents produces a surge of infections followed by any combination with the effective contact rate between susceptible humans and infectious asymptomatic humans and then other pathways. Overall, we see that as the number of transmission routes increase, the burden of infection increases also (see Fig. 7, Fig. 8). Fig. 9 shows a combination of all the transmission routes plotted alongside the dominant single transmission pathway. The region between the two graphs accounts for the contribution of other pathways in combination with the effective contact rate between susceptible humans and infected rodents. This shows that even though the effective contact rate between susceptible humans and infected rodents is dominant, other pathways should not be neglected because when they work in combination, there is an additional increase in the burden of Lassa fever over a cumulative period of time. It is also important to note that horizontal transmissions between susceptible rodents and infected rodents also play a huge role in increasing the infection as well as contact rate between susceptible rodent and contaminated environmental surfaces (see Fig. 10).

3.4. Discussion of results

We investigated the transmission dynamics of Lassa fever infection incorporating multiple transmission routes to capture their impact on the progression of the infection. Using a deterministic model that accounts for Lassa fever infection, we were able to show how incorporating several transmission pathways affects the prevalence of the disease. We used some mathematical tools to establish the local stability of the endemic equilibrium and the global stability of the disease free equilibrium. From our analysis, we got mathematical expressions that shows the conditions for which the disease will persist or be controlled in the system and illustrated sensitivity of parameters changes as system dynamics progress.

From our model simulations, we see that every transmission pathway has an impact towards the progression of Lassa fever. However, there are some routes of transmission that contribute significantly more than others. Control measures should be targeted more on the contact rates between susceptible humans and infected rodents (especially in areas where rodent consumption is high), and contact rates between susceptible humans and infectious asymptomatic humans. A great challenge arises when dealing with susceptible and asymptomatic infected humans pathway because they are not easily identified through symptoms. This calls for control methods that can detect this category of people such as mass testings in endemic areas, vaccination and so on. It is also important not to neglect the contact rates between susceptible humans and contaminated air particles especially in health centres with recorded Lassa fever cases and the contact rates between susceptible humans and contaminated environment (especially in poorly sanitized areas) because they are further drivers of infection (CDC, 2014).

Most single transmission routes are less harmful, but when they operate in combination with other transmission routes, they contribute additional damage to the system. Studies (Ibrahim & Dénes, 2021; Ojo et al., 2021; Onah et al., 2020; Peter et al., 2020) that only concentrated on the human and rodent direct transmission routes have not captured valuable information on the environmental impact towards the progression of the infection. This work gives a more comprehensive breakdown of the transmission dynamics of Lassa fever as it integrates indirect transmission routes which are sometimes neglected but play a crucial role in increasing the infection statistics. Current reports show that about four medical doctors died and 38 health workers were infected during a recent Lassa fever outbreak in Nigeria (NCDC, 2022; Punch, 2022). This increase in the death of health workers tells of the fact that serious measures should be taken to curb the spread of the virus through the indirect transmission routes like the environmental surfaces and aerosol. This will help to reduce the impact of these indirect transmission routes on the infection chain. Public health agencies in Nigeria have done a lot in mitigating Lassa fever infection through administering Ribavirin, public awareness campaigns amongst others. A lot still needs to be done in endemic areas by adequate fumigation of the environment and provision of protective gears for health workers. The results from our work show that interventions on these areas should not be undermined during health policy making. Further studies can be targeted at.

• ⋅combination of multiple routes of transmission incorporating the effect of seasonality of infection,
• ⋅proper sanitation, intervention strategies and holistic control measures that integrate these multiple transmission pathways which can help public health reduce disease prevalence,
• ⋅optimizing cost of several control measures using Cost Effective Analysis so that individuals in endemic areas with issues of poverty can be properly assisted.

Access to real field data can also improve the predictive capacity of the current model. Vertical transmission of Lassa fever in rodents can also be incorporated in further studies. Alternative techniques like the scaling of the model can be used to help simplify the analysis where parameters are dimensionless and express ratios of physical effects rather than levels of individual effects(Ledder, 2017).

Declaration of competing interest

The Authors declare no conflict of interest in this work.

Handling Editor: Dr HE DAIHAI

Appendix A. Sample Appendix Section

Proof. Let $x_i={(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_{10})}^T={(S_H,E_H,I_{HA},I_{HS},R_H,S_R,E_R,I_R,V_S,V_A)}^T$.

Then, model (5) can be written in the form $\frac{d{x}_i}{dt}=g(x)$ as follows:

$$\begin{array}{lll}\frac{d{x}_1}{dt}& =& g_1={\pi }_1-{\lambda }_H{x}_1-{\mu }_1{x}_1,\\ \frac{d{x}_2}{dt}& =& g_2={\lambda }_H{x}_1-({\psi }_1+{\mu }_1)x_2,\\ \frac{d{x}_3}{dt}& =& g_3=\nu {\psi }_1{x}_2-({\zeta }_1+{\mu }_1)x_3,\\ \frac{d{x}_4}{dt}& =& g_4=(1-\nu ){\psi }_1{x}_2-(\delta +{\zeta }_2+{\mu }_1)x_4,\\ \frac{d{x}_5}{dt}& =& g_5={\zeta }_1{x}_3+{\zeta }_2{x}_4-{\mu }_1{x}_5,\\ \frac{d{x}_6}{dt}& =& g_6={\pi }_2-{\lambda }_R{x}_6-(\rho +{\mu }_2)x_6,\\ \frac{d{x}_7}{dt}& =& g_7={\lambda }_R{x}_6-({\psi }_2+\rho +{\mu }_2)x_7,\\ \frac{d{x}_8}{dt}& =& g_8={\psi }_2{x}_7-(\rho +{\mu }_2)x_8,\\ \frac{d{x}_9}{dt}& =& g_9={\phi }_1{x}_3+{\phi }_2{x}_4+{\phi }_3{x}_8-({\theta }_2+{\theta }_3)x_9,\\ \frac{d{x}_{10}}{dt}& =& g_{10}={\theta }_3{x}_9-{\theta }_2{x}_{10},\end{array}$$ (A.1)

where

$$\begin{array}{lll}{\lambda }_H& =& \frac{{\beta }_H{x}_8}{x_6+x_7+x_8}+\frac{{\beta }_H{\eta }_1{x}_4}{x_1+x_2+x_3+x_4+x_5}+\frac{{\beta }_H{\eta }_2{x}_3}{x_1+x_2+x_3+x_4+x_5}+\frac{{\beta }_H{\eta }_3{x}_9}{K_V}\\ & +& \frac{{\beta }_H{\eta }_4{x}_{10}}{K_V},\\ {\lambda }_R& =& {\beta }_R\left(\frac{x_8}{x_6+x_7+x_8}+\frac{{\xi }_1{x}_9}{K_V}\right).\end{array}$$

We choose β_H as the bifurcation parameter by setting R₀ = 1. To do this, we let β_R ∝ β_H which implies that β_R = τβ_H for some τ > 0. Then from the value of R₀ we get

$${\beta }_H={\beta }_H^{\ast }=\frac{2}{{\beta }_{H1}^{\ast }+{\beta }_{H4}^{\ast }+\sqrt{{({\beta }_{H1}^{\ast }-{\beta }_{H4}^{\ast })}^2+4{\beta }_{H2}^{\ast }{\beta }_{H3}^{\ast }}},$$ (A.2)

where

$$\begin{array}{lll}{\beta }_{H1}^{\ast }& =& \frac{K_V{\theta }_2({\theta }_2+{\theta }_3){\mu }_1({\eta }_2(\delta +{\zeta }_2+{\mu }_1)\nu {\psi }_1-(-1+\nu ){\eta }_1({\zeta }_1+{\mu }_1){\psi }_1)}{K_V{\theta }_2({\theta }_2+{\theta }_3){\mu }_1({\zeta }_1+{\mu }_1)(\delta +{\zeta }_2+{\mu }_1)({\mu }_1+{\psi }_1)}\\ & +& \frac{\left({\pi }_1({\eta }_3{\theta }_2+{\eta }_4{\theta }_3)((\delta +{\zeta }_2+{\mu }_1)\nu {\psi }_1{\phi }_1-(-1+\nu )({\zeta }_1+{\mu }_1){\phi }_2{\psi }_1)\right)}{K_V{\theta }_2({\theta }_2+{\theta }_3){\mu }_1({\zeta }_1+{\mu }_1)(\delta +{\zeta }_2+{\mu }_1)({\mu }_1+{\psi }_1)},\\ {\beta }_{H2}^{\ast }& =& \frac{{\pi }_1\left(K_V{\theta }_2({\theta }_2+{\theta }_3){\mu }_2+{\pi }_2\left({\eta }_3{\theta }_2+{\eta }_4{\theta }_3\right){\phi }_3\right){\psi }_2}{K_V{\pi }_2{\theta }_2({\theta }_2+{\theta }_3){\mu }_1\left(\rho +{\psi }_2\right)\left({\mu }_2+\rho +{\psi }_2\right)},\\ {\beta }_{H3}^{\ast }& =& \frac{{\pi }_2\tau {\xi }_1\left(\left(\delta +{\zeta }_2+{\mu }_1\right)\nu {\psi }_1{\phi }_1-(-1+\nu )\left({\zeta }_1+{\mu }_1\right){\phi }_2{\psi }_1\right)}{K_V({\theta }_2+{\theta }_3)\left({\zeta }_1+{\mu }_1\right)\left(\delta +{\zeta }_2+{\mu }_1\right){\mu }_2\left({\mu }_1+{\psi }_1\right)},\\ {\beta }_{H4}^{\ast }& =& \frac{\tau \left(K_V({\theta }_2+{\theta }_3){\mu }_2+{\pi }_2{\xi }_1{\phi }_3\right){\psi }_2}{K_V({\theta }_2+{\theta }_3){\mu }_2\left(\rho +{\psi }_2\right)\left({\mu }_2+\rho +{\psi }_2\right)}.\end{array}$$

The Jacobian of system (8) evaluated at the DFE E₀ with the bifurcation parameter ${\beta }_H^{\ast }$ donated by $J_{E_0}$ is given as

$$\left(\begin{array}{cccccccccc}-{\mu }_1& 0& -{\beta }_H^{\ast }{\eta }_2& -{\beta }_H^{\ast }{\eta }_1& 0& 0& 0& -\frac{{\beta }_H^{\ast }{\mu }_2{\pi }_1}{{\mu }_1{\pi }_2}& -\frac{{\beta }_H^{\ast }{\eta }_3{\pi }_1}{{\mu }_1{K}_V}& -\frac{{\beta }_H^{\ast }{\eta }_4{\pi }_1}{{\mu }_1{K}_V}\\ 0& j_{22}& {\beta }_H^{\ast }{\eta }_2& {\beta }_H^{\ast }{\eta }_1& 0& 0& 0& \frac{{\beta }_H^{\ast }{\mu }_2{\pi }_1}{{\mu }_1{\pi }_2}& \frac{{\beta }_H^{\ast }{\eta }_3{\pi }_1}{{\mu }_1{K}_V}& \frac{{\beta }_H^{\ast }{\eta }_4{\pi }_1}{{\mu }_1{K}_V}\\ 0& \nu {\psi }_1& j_{33}& 0& 0& 0& 0& 0& 0& 0\\ 0& (1-\nu ){\psi }_1& 0& j_{44}& 0& 0& 0& 0& 0& 0\\ 0& 0& {\zeta }_1& {\zeta }_2& -{\mu }_1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -(\rho +{\mu }_2)& 0& -\tau {\beta }_H^{\ast }& -\frac{\tau {\beta }_H^{\ast }{\xi }_1{\pi }_2}{{\mu }_2{K}_V}& 0\\ 0& 0& 0& 0& 0& 0& j_{77}& \tau {\beta }_H^{\ast }& \frac{\tau {\beta }_H^{\ast }{\xi }_1{\pi }_2}{{\mu }_2{K}_V}& 0\\ 0& 0& 0& 0& 0& 0& {\psi }_2& j_{88}& 0& 0\\ 0& 0& {\phi }_1& {\phi }_2& 0& 0& 0& {\phi }_3& -({\theta }_2+{\theta }_3)& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& {\theta }_3& -{\theta }_2\end{array}\right).$$

where

$$j_{22}=-({\psi }_1+{\mu }_1),j_{33}=-({\zeta }_1+{\mu }_1),j_{44}=-(\delta +{\zeta }_2+{\mu }_1),j_{77}=-({\psi }_2+\rho +{\mu }_2),j_{88}=-(\rho +{\mu }_2).$$

A right eigenvector associated with the zero eigenvalue is given by

$$w={(w_1,w_2,w_3,w_4,w_5,w_6,w_7,w_8,w_9,w_{10})}^T.$$

We get it from the following equations:

$$\begin{array}{lll}-{\mu }_1{w}_1-{\beta }_H^{\ast }{\eta }_2{w}_3-{\beta }_H^{\ast }{\eta }_1{w}_4-\frac{{\mu }_2{\pi }_1}{{\mu }_1{\pi }_2}{\beta }_H^{\ast }{w}_8-\frac{{\beta }_H^{\ast }{\eta }_3{\pi }_1}{{\mu }_1{K}_V}{w}_9-\frac{{\beta }_H^{\ast }{\eta }_4{\pi }_1}{{\mu }_1{K}_V}{w}_{10}=0& & \\ -({\psi }_1+{\mu }_1)w_2+{\beta }_H{\eta }_2{w}_3+{\beta }_H^{\ast }{\eta }_1{w}_4+\frac{{\mu }_2{\pi }_1}{{\mu }_1{\pi }_2}{\beta }_H^{\ast }{w}_8+\frac{{\beta }_H^{\ast }{\eta }_3{\pi }_1}{{\mu }_1{K}_V}{w}_9+\frac{{\beta }_H^{\ast }{\eta }_4{\pi }_1}{{\mu }_1{K}_V}{w}_{10}=0\nu {\psi }_1{w}_2-({\zeta }_1+{\mu }_1)w_3=0\\ (1-\nu ){\psi }_1{w}_2-(\delta +{\zeta }_2+{\mu }_1)w_4=0& & \\ {\zeta }_1{w}_3+{\zeta }_2{w}_4-{\mu }_1{w}_5=0-(\rho +{\mu }_2)w_6-\tau {\beta }_H^{\ast }{w}_8-\frac{\tau {\beta }_H^{\ast }{\xi }_1{\pi }_2}{{\mu }_2{K}_V}{w}_9=0-({\psi }_2+\rho +{\mu }_2)w_7+\tau {\beta }_H^{\ast }{w}_8+\frac{\tau {\beta }_H^{\ast }{\xi }_1{\pi }_2}{{\mu }_2{K}_V}{w}_9=0\\ {\psi }_2{w}_7-(\rho +{\mu }_2)w_8=0& & \\ {\phi }_1{w}_3+{\phi }_2{w}_4+{\phi }_3{w}_8-({\theta }_2+{\theta }_3)w_9=0& & \\ {\theta }_3{w}_9-{\theta }_2{w}_{10}=0& & \end{array}$$ (A.3)

The solution to (A.3) gives

$$\begin{array}{lll}{w}_1& =& -\frac{{\beta }_H^{\ast }{\eta }_2{w}_3}{{\mu }_1}-\frac{{\beta }_H^{\ast }{\eta }_1{w}_4}{{\mu }_1}-\frac{{\beta }_H^{\ast }{\mu }_2{\pi }_1}{{\mu }_1^2{\pi }_2}{w}_8-\frac{{\beta }_H^{\ast }{\eta }_3{\pi }_1}{{\mu }_1^2{K}_V}{w}_9-\frac{{\beta }_H^{\ast }{\eta }_4{\pi }_1}{{\mu }_1^2{K}_V}{w}_{10},\\ w_2& =& w_2>0,w_3=\frac{\nu {\psi }_1{w}_2}{{\zeta }_1+{\mu }_1},w_4=\frac{(1-\nu ){\psi }_1{w}_2}{\delta +{\zeta }_2+{\mu }_1},w_5=\frac{{\zeta }_1{w}_3+{\zeta }_2{w}_4}{{\mu }_1},\\ w_6& =& -\frac{\tau {\beta }_H^{\ast }{w}_8}{\rho +{\mu }_2}-\frac{\tau {\beta }_H^{\ast }{\xi }_1{\pi }_2}{{\mu }_2{K}_V(\rho +{\mu }_2)}{w}_9,w_7=\frac{\tau {\beta }_H^{\ast }{w}_8}{{\psi }_2+\rho +{\mu }_2}+\frac{\tau {\beta }_H^{\ast }{\xi }_1{\pi }_2}{{\mu }_2{K}_V({\psi }_2+\rho +{\mu }_2)}{w}_9,\\ w_8& =& \frac{{\psi }_2{w}_7}{\rho +{\mu }_2},w_9=\frac{{\phi }_1{w}_3+{\phi }_2{w}_4+{\phi }_3{w}_8}{({\theta }_2+{\theta }_3)},w_{10}=\frac{{\theta }_3{w}_9}{{\theta }_2}.\end{array}$$ (A.4)

Similarly, a left eigenvector (associated with the zero eigenvalue) given by

$$v={(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8,v_9,v_{10})}^T,$$

which satisfies v.w = 1 is obtained by the transpose of the matrix $J_{E_0}$ which is

$$\left(\begin{array}{cccccccccc}-{\mu }_1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& j_{22}^T& \nu {\psi }_1& (1-\nu ){\psi }_1& 0& 0& 0& 0& 0& 0\\ -{\beta }_H^{\ast }{\eta }_2& {\beta }_H^{\ast }{\eta }_2& j_{33}^T& 0& {\zeta }_1& 0& 0& 0& {\phi }_1& 0\\ -{\beta }_H^{\ast }{\eta }_1& {\beta }_H^{\ast }{\eta }_1& 0& j_{44}^T& {\zeta }_2& 0& 0& 0& {\phi }_2& 0\\ 0& 0& 0& 0& -{\mu }_1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -(\rho +{\mu }_2)& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& j_{77}^T& {\psi }_2& 0& 0\\ -\frac{{\beta }_H^{\ast }{\mu }_2{\pi }_1}{{\mu }_1{\pi }_2}& \frac{{\beta }_H^{\ast }{\mu }_2{\pi }_1}{{\mu }_1{\pi }_2}& 0& 0& 0& -\tau {\beta }_H^{\ast }& \tau {\beta }_H^{\ast }& j_{88}^T& {\phi }_3& 0\\ -\frac{{\beta }_H^{\ast }{\eta }_3{\pi }_1}{{\mu }_1{K}_V}& \frac{{\beta }_H^{\ast }{\eta }_3{\pi }_1}{{\mu }_1{K}_V}& 0& 0& 0& -\frac{\tau {\beta }_H^{\ast }{\xi }_1{\pi }_2}{{\mu }_2{K}_V}& \frac{\tau {\beta }_H^{\ast }{\xi }_1{\pi }_2}{{\mu }_2{K}_V}& 0& -({\theta }_2+{\theta }_3)& {\theta }_3\\ -\frac{{\beta }_H^{\ast }{\eta }_4{\pi }_1}{{\mu }_1{K}_V}& \frac{{\beta }_H^{\ast }{\eta }_4{\pi }_1}{{\mu }_1{K}_V}& 0& 0& 0& 0& 0& 0& 0& -{\theta }_2\end{array}\right).$$

where

$$j_{22}^T=-({\psi }_1+{\mu }_1),j_{33}^T=-({\zeta }_1+{\mu }_1),j_{44}^T=-(\delta +{\zeta }_2+{\mu }_1),j_{77}^T=-({\psi }_2+\rho +{\mu }_2),j_{88}^T=-(\rho +{\mu }_2).$$

The system of equations obtained is given by

$$\begin{array}{l}-{\mu }_1{v}_1=0-({\psi }_1+{\mu }_1)v_2+\nu {\psi }_1{v}_3+(1-\nu ){\psi }_1{v}_4\\ =0-{\beta }_H^{\ast }{\eta }_2{v}_1+{\beta }_H^{\ast }{\eta }_2{v}_2-({\zeta }_1+{\mu }_1)v_3+{\zeta }_1{v}_5+{\phi }_1{v}_9\\ =0-{\beta }_H^{\ast }{\eta }_1{v}_1+{\beta }_H^{\ast }{\eta }_1{v}_2-(\delta +{\zeta }_2+{\mu }_1)v_4+{\zeta }_2{v}_5+{\phi }_2{v}_9\\ =0-{\mu }_1{v}_5\\ =0-(\rho +{\mu }_2)v_6\\ =0-({\psi }_2+\rho +{\mu }_2)v_7+{\psi }_2{v}_8\\ =0-\frac{{\beta }_H^{\ast }{\mu }_2{\pi }_1}{{\mu }_1{\pi }_2}{v}_1+\frac{{\beta }_H^{\ast }{\mu }_2{\pi }_1}{{\mu }_1{\pi }_2}{v}_2-\tau {\beta }_H^{\ast }{v}_6+\tau {\beta }_H^{\ast }{v}_7-(\rho +{\mu }_2)v_8+{\phi }_3{v}_9\\ =0-\frac{{\beta }_H^{\ast }{\eta }_3{\pi }_1}{{\mu }_1{K}_V}{v}_1+\frac{{\beta }_H^{\ast }{\eta }_3{\pi }_1}{{\mu }_1{K}_V}{v}_2-\frac{\tau {\beta }_H^{\ast }{\xi }_1{\pi }_2}{{\mu }_2{K}_V}{v}_6+\frac{\tau {\beta }_H^{\ast }{\xi }_1{\pi }_2}{{\mu }_2{K}_V}{v}_7-({\theta }_2+{\theta }_3)v_9+{\theta }_3{v}_{10}\\ =0-\frac{{\beta }_H^{\ast }{\eta }_4{\pi }_1}{{\mu }_1{K}_V}{v}_1+\frac{{\beta }_H^{\ast }{\eta }_4{\pi }_1}{{\mu }_1{K}_V}{v}_2-{\theta }_2{v}_{10}\\ & =& 0\end{array}$$ (A.4)

Solving (A.4) gives

$$\begin{array}{lll}{v}_1& =& 0,v_2=v_2>0,v_3=\frac{{\beta }_H^{\ast }{\eta }_2{v}_2+{\phi }_1{v}_9}{{\zeta }_1+{\mu }_1},v_4=\frac{{\beta }_H^{\ast }{\eta }_1{v}_2+{\phi }_2{v}_9}{\delta +{\zeta }_2+{\mu }_1},v_5=0,v_6=0,v_7=\frac{{\psi }_2{v}_8}{{\psi }_2+\rho +{\mu }_2},\\ v_8& =& \frac{\tau {\beta }_H^{\ast }{v}_7}{\rho +{\mu }_2}+\frac{{\beta }_H^{\ast }{\mu }_2{\pi }_1{v}_2}{{\mu }_1{\pi }_2(\rho +{\mu }_2)}+\frac{{\phi }_3{v}_9}{\rho +{\mu }_2},v_9=\frac{{\beta }_H^{\ast }{\eta }_3{\pi }_1{v}_2}{{\mu }_1{K}_V({\theta }_2+{\theta }_3)}+\frac{\tau {\beta }_H^{\ast }{\xi }_1{\pi }_2{v}_7}{{\mu }_2{K}_V({\theta }_2+{\theta }_3)}+\frac{{\theta }_3{v}_{10}}{{\theta }_2+{\theta }_3},v_{10}=\frac{{\beta }_H^{\ast }{\eta }_4{\pi }_1{v}_2}{{\theta }_2{\mu }_1{K}_V}.\end{array}$$

We use the property v.w = 1 to get

$$v_1{w}_1+v_2{w}_2+v_3{w}_3+v_4{w}_4+v_5{w}_5+v_6{w}_6+v_7{w}_7+v_8{w}_8+v_9{w}_9+v_{10}{w}_{10}=1.$$

Choosing w₂ = 1 without loss of generality gives us

$$v_2=\frac{1}{1+(A_3{w}_3+A_4{w}_4+A_7{w}_7+A_8{w}_8+A_9{w}_9+A_{10}{w}_{10})}>0,$$

where

$$\begin{array}{lll}{A}_3& =& \frac{{\beta }_H\left({\eta }_2+\frac{{\pi }_1{\mu }_2{\phi }_1\left(\tau {\beta }_H{\theta }_2{\xi }_1{\psi }_2+{\eta }_3{\theta }_2\left({\rho }^2+{\mu }_2^2+\left(\rho -\tau {\beta }_H\right){\psi }_2+{\mu }_2\left(2\rho +{\psi }_2\right)\right)+{\eta }_4{\theta }_3\left({\rho }^2+{\mu }_2^2+\left(\rho -\tau {\beta }_H\right){\psi }_2+{\mu }_2\left(2\rho +{\psi }_2\right)\right)\right)}{{\theta }_2{\mu }_1\left(-\tau {\pi }_2{\beta }_H{\xi }_1{\phi }_3{\psi }_2+K_V\left({\theta }_2+{\theta }_3\right){\mu }_2\left({\rho }^2+{\mu }_2^2+\left(\rho -\tau {\beta }_H\right){\psi }_2+{\mu }_2\left(2\rho +{\psi }_2\right)\right)\right)}\right)}{{\zeta }_1+{\mu }_1},\\ A_4& =& \frac{{\beta }_H\left({\eta }_1+\frac{{\pi }_1{\mu }_2{\phi }_2\left(\tau {\beta }_H{\theta }_2{\xi }_1{\psi }_2+{\eta }_3{\theta }_2\left({\rho }^2+{\mu }_2^2+\left(\rho -\tau {\beta }_H\right){\psi }_2+{\mu }_2\left(2\rho +{\psi }_2\right)\right)+{\eta }_4{\theta }_3\left({\rho }^2+{\mu }_2^2+\left(\rho -\tau {\beta }_H\right){\psi }_2+{\mu }_2\left(2\rho +{\psi }_2\right)\right)\right)}{{\theta }_2{\mu }_1\left(-\tau {\pi }_2{\beta }_H{\xi }_1{\phi }_3{\psi }_2+K_V\left({\theta }_2+{\theta }_3\right){\mu }_2\left({\rho }^2+{\mu }_2^2+\left(\rho -\tau {\beta }_H\right){\psi }_2+{\mu }_2\left(2\rho +{\psi }_2\right)\right)\right)}\right)}{\delta +{\zeta }_2+{\mu }_1},\\ A_7& =& -\frac{{\pi }_1{\beta }_H{\mu }_2\left(K_V{\theta }_2\left({\theta }_2+{\theta }_3\right){\mu }_2+{\pi }_2\left({\eta }_3{\theta }_2+{\eta }_4{\theta }_3\right){\phi }_3\right){\psi }_2}{{\pi }_2{\theta }_2{\mu }_1\left(\tau {\pi }_2{\beta }_H{\xi }_1{\phi }_3{\psi }_2-K_V\left({\theta }_2+{\theta }_3\right){\mu }_2\left({\rho }^2+{\mu }_2^2+\left(\rho -\tau {\beta }_H\right){\psi }_2+{\mu }_2\left(2\rho +{\psi }_2\right)\right)\right)},\\ A_8& =& -\frac{{\pi }_1{\beta }_H{\mu }_2\left(K_V{\theta }_2\left({\theta }_2+{\theta }_3\right){\mu }_2+{\pi }_2\left({\eta }_3{\theta }_2+{\eta }_4{\theta }_3\right){\phi }_3\right)\left(\rho +{\mu }_2+{\psi }_2\right)}{{\pi }_2{\theta }_2{\mu }_1\left(\tau {\pi }_2{\beta }_H{\xi }_1{\phi }_3{\psi }_2-K_V\left({\theta }_2+{\theta }_3\right){\mu }_2\left({\rho }^2+{\mu }_2^2+\left(\rho -\tau {\beta }_H\right){\psi }_2+{\mu }_2\left(2\rho +{\psi }_2\right)\right)\right)},A_{10}=\frac{\left({\beta }_H{\eta }_4{\pi }_1\right)}{\left({\theta }_2{\mu }_1{K}_V\right)},\\ A_9& =& \frac{{\pi }_1{\beta }_H{\mu }_2\left(\tau {\beta }_H{\theta }_2{\xi }_1{\psi }_2+{\eta }_3{\theta }_2\left({\rho }^2+{\mu }_2^2+\left(\rho -\tau {\beta }_H\right){\psi }_2+{\mu }_2\left(2\rho +{\psi }_2\right)\right)+{\eta }_4{\theta }_3\left({\rho }^2+{\mu }_2^2+\left(\rho -\tau {\beta }_H\right){\psi }_2+{\mu }_2\left(2\rho +{\psi }_2\right)\right)\right)}{{\theta }_2{\mu }_1\left(-\tau {\pi }_2{\beta }_H{\xi }_1{\phi }_3{\psi }_2+K_V\left({\theta }_2+{\theta }_3\right){\mu }_2\left({\rho }^2+{\mu }_2^2+\left(\rho -\tau {\beta }_H\right){\psi }_2+{\mu }_2\left(2\rho +{\psi }_2\right)\right)\right)}.\end{array}$$

This value of v₂ and w₂ satisfies the given property. We now calculate the second order partial derivatives of g_i at the disease free equilibrium E₀ to get

$$\begin{array}{lll}\frac{{\partial }^2{g}_2}{\partial x_1\partial x_8}& =& \frac{{\beta }_H{\mu }_2}{{\pi }_2},\frac{{\partial }^2{g}_2}{\partial x_1\partial x_9}=\frac{{\beta }_H{\eta }_3}{K_V},\frac{{\partial }^2{g}_2}{\partial x_1\partial x_{10}}=\frac{{\beta }_H{\eta }_4}{K_V},\frac{{\partial }^2{g}_2}{\partial x_2\partial x_3}=\frac{-{\beta }_H{\eta }_2{\mu }_1}{{\pi }_1},\\ \frac{{\partial }^2{g}_2}{\partial x_2\partial x_4}& =& \frac{-{\beta }_H{\eta }_1{\mu }_1}{{\pi }_1},\frac{{\partial }^2{g}_2}{\partial x_3\partial x_4}=\frac{-{\beta }_H{\mu }_1({\eta }_1+{\eta }_2)}{{\pi }_1},\frac{{\partial }^2{g}_2}{\partial x_3\partial x_5}=\frac{-{\beta }_H{\eta }_2{\mu }_1}{{\pi }_1},\frac{{\partial }^2{g}_2}{\partial x_3\partial x_3}=\frac{-2{\beta }_H{\eta }_2{\mu }_1}{{\pi }_1},\\ \frac{{\partial }^2{g}_2}{\partial x_4\partial x_4}& =& \frac{-2{\beta }_H{\eta }_1{\mu }_1}{{\pi }_1},\frac{{\partial }^2{g}_2}{\partial x_4\partial x_5}=\frac{-{\beta }_H{\eta }_1{\mu }_1}{{\pi }_1},\frac{{\partial }^2{g}_2}{\partial x_6\partial x_8}=\frac{-{\beta }_H{\pi }_1{\mu }_2^2}{{\pi }_2^2{\mu }_1},\frac{{\partial }^2{g}_2}{\partial x_7\partial x_8}=\frac{-{\beta }_H{\pi }_1{\mu }_2^2}{{\pi }_2^2{\mu }_1},\\ \frac{{\partial }^2{g}_2}{\partial x_8\partial x_8}& =& \frac{-2{\beta }_H{\pi }_1{\mu }_2^2}{{\pi }_2^2{\mu }_1},\frac{{\partial }^2{g}_7}{\partial x_7\partial x_8}=\frac{-{\beta }_H{\mu }_2}{{\pi }_2},\frac{{\partial }^2{g}_7}{\partial x_8\partial x_8}=\frac{-2{\beta }_H{\mu }_2}{{\pi }_2},\frac{{\partial }^2{g}_7}{\partial x_6\partial x_9}=\frac{{\beta }_H{\xi }_1}{K_V}.\end{array}$$

We now compute the values of a and b to get

$$\begin{array}{lll}a& =& \sum _{k,i,j=1}^{10}{v}_k{w}_i{w}_j\frac{{\partial }^2{g}_k(0,0)}{\partial x_i\partial x_j}\\ & =& \left[\frac{{\beta }_H{\eta }_3}{K_V}{w}_9+\frac{{\beta }_H{\eta }_4}{K_V}{w}_{10}\right]v_2{w}_1+\left(\frac{{\beta }_H{\xi }_1}{K_V}\right)v_7{w}_6{w}_9,\\ & -& \frac{{\beta }_H{\eta }_2{\mu }_1}{{\pi }_1}\left[w_2+w_5+2w_3+w_4\right]v_2{w}_3-\frac{{\beta }_H{\mu }_2}{{\pi }_2}\left[w_7+2w_8\right]v_7{w}_8,\\ & -& \left[w_7+2w_8\right]v_2{w}_8-\frac{{\beta }_H{\eta }_1{\mu }_1}{{\pi }_1}\left[w_2+w_3+2w_4+w_5\right]v_2{w}_4,\\ & +& \frac{{\beta }_H{\mu }_2}{{\pi }_2}{w}_1{v}_2{w}_1-\frac{{\beta }_H{\pi }_1{\mu }_2^2}{{\pi }_2^2{\mu }_1}{w}_6{v}_2{w}_8,\end{array}$$ (A.5)

and

$$\begin{array}{lll}b& =& \sum _{k,i=1}^{10}{v}_k{w}_i\frac{{\partial }^2{g}_k(0,0)}{\partial x_i\partial {\beta }_H}\\ & =& v_2{w}_3{\eta }_2+v_2{w}_4{\eta }_1+v_2{w}_8\frac{{\pi }_1{\mu }_2}{{\pi }_2{\mu }_1}+v_2{w}_9\frac{{\pi }_1{\eta }_3}{{\mu }_1{K}_V}+v_2{w}_{10}\frac{{\pi }_1{\eta }_4}{{\mu }_1{K}_V}+v_7{w}_8+v_7{w}_9\frac{{\pi }_2{\xi }_1}{{\mu }_2{K}_V}>0.\end{array}$$ (A.6)

• (i)If $\frac{{\beta }_H{\mu }_2}{{\pi }_2}{w}_1{v}_2{w}_1<\frac{{\beta }_H{\pi }_1{\mu }_2^2}{{\pi }_2^2{\mu }_1}{w}_6{v}_2{w}_8$ and b > 0, then system (5) will undergo a forward bifurcation at R₀ = 1.
• (ii)If $B_{wv}<\frac{{\beta }_H{\mu }_2}{{\pi }_2}{w}_1{v}_2{w}_1-\frac{{\beta }_H{\pi }_1{\mu }_2^2}{{\pi }_2^2{\mu }_1}{w}_6{v}_2{w}_8$, and b > 0, then system (5) will undergo a backward bifurcation at R₀ = 1, where

$$\begin{array}{lll}{B}_{wv}& =& \left[\frac{{\beta }_H{\eta }_3}{K_V}{w}_9+\frac{{\beta }_H{\eta }_4}{K_V}{w}_{10}\right]v_2{w}_1+\left(\frac{{\beta }_H{\xi }_1}{K_V}\right)v_7{w}_6{w}_9-\frac{{\beta }_H{\eta }_2{\mu }_1}{{\pi }_1}\left[w_2+w_5+2w_3+w_4\right]v_2{w}_3,\\ & -& \frac{{\beta }_H{\mu }_2}{{\pi }_2}\left[w_7+2w_8\right]v_7{w}_8-\left[w_7+2w_8\right]v_2{w}_8-\frac{{\beta }_H{\eta }_1{\mu }_1}{{\pi }_1}\left[w_2+w_3+2w_4+w_5\right]v_2{w}_4.\end{array}$$ (A.6)