Modelling HIV/AIDS and monkeypox co-infection

Abstract

During the last two decades, reports on emerging human monkeypox outbreaks in Africa and North America have reminded us that beside the eradicated smallpox there are other pox viruses that have great potential to harm people. A deterministic model for the co-infection of HIV/AIDS and monkeypox is formulated and analysed. The endemic equilibria are shown to be locally and globally asymptotically stable using the Centre Manifold theory and the Lyapunov function approach, respectively. Analysis of the basic reproduction numbers and numerical simulations suggest that an increase in the number of monkeypox in the animal species results in an increase of the number of people having monkeypox. Threshold conditions that determine the competitive outcomes of the two diseases are provided. Furthermore, numerical simulations using a set of reasonable parameter values support the claim that HIV infection greatly enhances monkeypox infection and vice versa.

1. Introduction

Monkeypox virus is closely related to the small pox virus. It was first identified as the causal agent in two outbreaks of pox infection in cynomolgus monkeys originating from Singapore at Statens Serum institute, Coperhagen, Denmark [12], [15]. The virus was first described as causing a pox-like illness in monkeys in the late 1960 [9]. In monkeys, the disease is characterized by generalized skin eruptions which develop into papules on the trunk, face, palms and soles. Papules subsequently develop into vesicles and scabs which usually fall off after about 10 days after the rash has first appeared [9]. The severity of the disease varies with regard to host species; it is mild in cynomolgus monkeys, but more severe in orangutans [1], [16]. Epidemiological investigations have revealed that the monkeypox virus is endemic in squirrels in the tropical rain forest of Africa.

In the 1970s, human monkeypox was reported, for the first time in the Western and Central African countries [10]. The disease was discovered after smallpox had been eradicated in these regions. Investigations into the rash causing illnesses by the World Health Organisation from 1970 to 1986 revealed that monkeypox virus was the cause, and that the case fatality was 10–17%. The secondary attack rate (3%) was much lower than that of smallpox (up to 80% in non-immunized contacts). Large human monkeypox outbreaks were reported in the Democratic Republic of Congo in 1996/1997 and 2001–2004 [13]. In 1996/1997 the mortality was lower (1.5%) than the earlier epidemic, but secondary attack rate was high (upto 78%). This could have been the result of a reduced immunity due to the abolishment of mandatory smallpox vaccination [9]. That is, the mandatory smallpox vaccination may also have been contributing to the control of monkeypox. In 2005 the emergence of occasional human monkeypox virus infections were reported for the first time in Southern Sudan, an area ecological different from the tropical rain forest. An investigation by World Health Organisation found sporadic cases of monkeypox cases in the area supporting the argument of recurrent carryover from local animal reservoirs [8]. The outbreak of monkeypox in the USA in 2003 through imported exotic animals, such as prairie dogs from Ghana [7], reveals that monkeypox may show up in other parts of the world.

The mortality of monkeypox infection alone may be much higher in HIV infected individuals with a compromised immune system. Given the high prevalence of HIV/AIDS in Africa, and sporadic outbreaks of human monkeypox in West and Central Africa, there is need to look into the possible interaction of the two infections. Theoretical studies analyzing the co-dynamics of HIV/AIDS and other infectious diseases (malaria) ([3], [4] and the references therein) have recently provided insight into the dynamics of co-circulating infectious diseases. Deterministic monkeypox models exist [2], however, we are not aware of any previous study of the co-interaction of monkeypox and HIV. This paper mathematically analyzes the interaction of HIV/AIDS and monkeypox by formulating a mathematical model of their spread in a randomly variable population. We analyze both local and global stability of the equilibria. After illustrating their dynamics with numerical simulations using some reasonable parameter values, we conclude.

2. Model description

The model divides non-human primates including monkeys and wild rodents into susceptibles (S _n), infectives (I _n) and the recovered with permanent immunity (R _n). The total non-human population is given by

$$N_n(t)=S_n(t)+I_n(t)+R_n(t)\text{.}$$ (1)

Susceptible animals are recruited through births at a rate Λ _n and are infected with the monkeypox virus at a rate λ _n. We will assume the animal population is randomly mixing and the infection rate is given by

$${\lambda }_n=\frac{{\beta }_{n_1}{c}_{n_1}{I}_n}{N_n}\text{.}$$ (2)

Here ${\beta }_{n_1}$ is the probability of monkeys (animals) getting infected per contact with an infectious case and $c_{n_1}$ is the effective contact rate.

Once infected, susceptibles (S _n) progress to the infectious state (I _n). Animals in the I _n are capable of infecting other animals they are in close contact with, die due to disease at a rate d _n or recover with permanent immunity at a rate ρ _n into the recovered class, R _n. All animals experience natural death at a rate μ _n which is proportional to the number of animals in each class.

The total human population is divided into nine distinct subgroups that is the susceptibles (S), the monkeypox infected (I _m), the recovered from monkeypox with permanent immunity (R _m), HIV infected not in the AIDS stage of disease progression (I _h), HIV infected and in the AIDS stage of disease progression A _h, those dually infected with monkeypox and HIV not yet displaying AIDS symptoms (I _hm), those dually infected with monkeypox and HIV and in the AIDS stage of disease progression (A _hm), those who have recovered from monkeypox and HIV positive without AIDS symptoms (R _hm) and finally those who have recovered from monkeypox and are infected with HIV in the AIDS stage of disease progression (R _am). Thus, the total human population is given by

$$N_h(t)=S(t)+I_m(t)+R_m(t)+I_h(t)+A_h(t)+I_{\mathit{hm}}(t)+A_{\mathit{hm}}(t)+R_{\mathit{hm}}+R_{\mathit{am}}(t)\text{.}$$ (3)

Susceptibles humans enter the population, either through birth or migration, at a rate Λ _h and are infected with the monkeypox virus at rate λ _m. We assume a random mixing model and define

$${\lambda }_m=\frac{{\beta }_{\mathit{nm}}{c}_n{I}_n}{N_n}+\frac{{\beta }_m{c}_m(I_m+{\theta }_1{I}_{\mathit{hm}}+{\theta }_2{A}_{\mathit{hm}})}{N_h}\text{.}$$ (4)

where β _nm is the probability of getting infected with monkeypox virus through contact or by eating an infected animal carcass; c _n is the effective contact rate; β _m is the probability of infection via contact with an infected individual; c _m is the effective contact rate; (θ ₁,  θ ₂) > 1 models the fact that dually infected individuals are likely to be more infectious than their corresponding parts. It should be noted here that θ ₁  <  θ ₂ signifying that individuals in the AIDS stage are more infectious than those infected with HIV not yet in the AIDS stage. It is assumed that mortality of monkeys due to predation by human beings is negligible and ignored.

Susceptibles are infected with HIV at a rate λ _h where

$${\lambda }_h=\frac{{\beta }_h{c}_h\left(I_h+R_{\mathit{hm}}+{\varphi }_1{I}_{\mathit{hm}}+{\varphi }_2[A_h+R_{\mathit{am}}+{\varphi }_3{A}_{\mathit{hm}}]\right)}{N_h}$$ (5)

with β _h is the probability of getting infected with HIV per sexual contact with an infected partner; c _h is the effective contact rate; ϕ ₁  > 1 models the fact dually infected people not yet in the AIDS stage of disease is more infectious than their corresponding parts; ϕ ₂  > 1 models the fact people in the AIDS stage are more infectious than those not yet in the AIDS stage; ϕ ₃  > 1 models the fact that dually infected people in the AIDS stage are more infectious than those in the AIDS stage but not dually infected. For this model, we do not include vertical transmission of HIV. Susceptibles infected with the monkeypox virus progress to the monkeypox infected class (I _m) at a rate λ _m and those infected with HIV progress to HIV infected class only (I _h) at a rate λ _h. Individuals having monkeypox and are HIV negative recover into the recovered class (R _m) with permanent immunity at a rate ${\rho }_{m_1}$. Individuals in the I _m and R _m classes are infected with HIV at a rate λ _h to move into the I _hm and R _hm classes, respectively. HIV positive people not yet showing AIDS symptoms dually infected with monkeypox virus recover from the monkeypox infection at a ${\rho }_{m_2}({\rho }_{m_2}<{\rho }_{m_1})$. Human experience natural death at a rate μ _h which is proportional to the number of humans in each class. Since there is neither a vaccine nor a definte treatment for both diseases, the proposed model does not account for any preventive measures.

HIV positive only individuals (I _h) are infected with the monkeypox virus at a rate σ ₁ λ _m,σ ₁  > 1 to get into the I _hm class with σ ₁ accounting for the increased susceptibility to the monkeypox virus infection an HIV positive individual has. Individuals in I _h,I _hm and R _hm classes progress into their respective AIDS classes (A _h,  A _hm and R _am) at rates ${\rho }_{a_1}\text{,}{\rho }_{a_2}$ and ${\rho }_{a_1}$, respectively. AIDS patients, not yet infected with the monkeypox virus, are infected with the monkeypox virus at a rate σ ₂ λ _m, σ ₂  > 1 signifying the increased chances of a sick AIDS patient has when (s)he gets in contact with the monkeypox virus. All AIDS patients experience an additional disease induced death at a rate d _a and all monkeypox patients experience an additional disease induced death at a rate d _m. Unless stated otherwise, values for the parameters in the simulations are given in Table 1 .

Table 1.

Default model parameters used in the analysis and simulations.

Parameter	Symbol	Value	Source
Human recruitment rate	Λh	0.029 yr−1 × 104	[2], [3], [4]
Natural human mortality rate	μh	0.02 yr−1	[2], [3], [4]
Natural human recovery rate from monkeypox infection	$({\rho }_{m_1}\text{,}{\rho }_{m_2}\text{,}{\rho }_{m_3})$	0.85 (0.83–0.9) yr−1	[9]
Disease induced human death rate	dm	0.15 (0.1–0.17) yr−1	[9]
Monkeys birth rate	Λn	2 yr−1 ∗103	[2]
Monkeys natural mortality rate	μn	1.5 yr−1	[2]
Monkeys natural recovery rate	ρn	0.6 yr−1	[2]
Monkeys disease induced death rate	dn	0.4 yr−1	[2]
Enhancement factors for transmission of monkeypox by dually infected people	(θ1, θ2)	(1.001, 1.00105)	Assumed
Natural rate of progression to AIDS	$({\rho }_{a_1}\text{,}{\rho }_{a_2})$	0.1 yr−1	[3]
Enhancement factor of monkeypox infection to HIV infectives	(σ1, σ2)	1.005	Assumed
AIDS related death rate	da	0.333 yr−1	[3]
Product of effective contact rate for HIV infection and probability of HIV transmission per contact	βhch	0.125 (0.011–0.95) yr−1	[3]
Product of effective contact rate for monkey pox infection and probability of monkey pox transmission per contact	$({\beta }_{n_1}{c}_{n_1}\text{,}{\beta }_{\mathit{nm}}{c}_n\text{,}{\beta }_m{c}_m)$	(3, 1.5, 0. 75) yr−1	Assumed
Modification parameters	(ϕ1, ϕ2, ϕ3)	1.00125	Assumed
Monkey pox induced basic reproduction number (non-humans)	${\mathcal{R}}_{0_n}$	1.2
Monkey pox induced basic reproduction number (humans)	${\mathcal{R}}_{0_m}$	0.9804
HIV/ AIDS induced basic reproduction number (humans)	${\mathcal{R}}_{0_h}$	1.3371

The schematic description of the model is shown in Fig. 1 and the model is represented by the following system of differential equations:

$$\begin{array}{cc}\hfill & S_n^{\prime }(t)={\Lambda }_n-({\mu }_n+{\lambda }_n)S_n\text{,}\hfill \\ \hfill & I_n^{\prime }(t)={\lambda }_n{S}_n-({\mu }_n+{\rho }_n+d_n)I_n\text{,}\hfill \\ \hfill & R_n^{\prime }(t)={\rho }_n{I}_n-{\mu }_n{R}_n\text{,}\hfill \\ \hfill & S^{\prime }(t)={\Lambda }_h-({\mu }_h+{\lambda }_m+{\lambda }_h)S\text{,}\hfill \\ \hfill & I_h^{\prime }(t)={\lambda }_{h}S-({\mu }_h+{\rho }_{a_1}+{\sigma }_1{\lambda }_m)I_h\text{,}\hfill \\ \hfill & A_h^{\prime }(t)={\rho }_{a_1}{I}_h-({\mu }_h+d_a+{\sigma }_2{\lambda }_m)A_h\text{,}\hfill \\ \hfill & I_m^{\prime }(t)={\lambda }_{m}S-({\mu }_h+d_m+{\rho }_{m_1}+{\lambda }_h)I_m\text{,}\hfill \\ \hfill & R_m^{\prime }(t)={\rho }_{m_1}{I}_m-({\mu }_h+{\lambda }_h)R_m\text{,}\hfill \\ \hfill & I_{\mathit{hm}}^{\prime }(t)={\lambda }_h{I}_m+{\sigma }_1{\lambda }_m{I}_h-({\mu }_h+d_m+{\rho }_{m_2}+{\rho }_{a_2})I_{\mathit{hm}}\text{,}\hfill \\ \hfill & A_{\mathit{hm}}^{\prime }(t)={\sigma }_2{\lambda }_m{A}_h+{\rho }_{a_2}{I}_{\mathit{hm}}-({\mu }_h+d_m+d_a+{\rho }_{m_3})A_{\mathit{hm}}\text{,}\hfill \\ \hfill & R_{\mathit{hm}}^{\prime }(t)={\rho }_{m_2}{I}_{\mathit{hm}}+{\lambda }_h{R}_m-({\mu }_h+{\rho }_{a_1})R_{\mathit{hm}}\text{,}\hfill \\ \hfill & R_{\mathit{am}}^{\prime }(t)={\rho }_{a_1}{R}_{\mathit{hm}}+{\rho }_{m_3}{A}_{\mathit{hm}}-({\mu }_h+d_a)R_{\mathit{am}}\text{.}\hfill \end{array}$$ (6)

All parameters and state variables for model system (6) are assumed to be non-negative to be consistent with human and animal populations. All feasible solutions of model system (6) are positive and eventually enter the invariant attracting region

$$\mathrm{\Omega }=\left\{\begin{array}{c}(S_n\text{,}{I}_n\text{,}{R}_n)\in {\mathbb{R}}_{+}^3:N_1⩽\frac{{\Lambda }_n}{{\mu }_n}\text{,}\hfill \\ (S\text{,}{I}_m\text{,}{R}_m\text{,}{I}_h\text{,}{A}_h\text{,}{I}_{\mathit{hm}}\text{,}{A}_{\mathit{hm}}\text{,}{R}_{\mathit{hm}}\text{,}{R}_{\mathit{am}})\in {\mathbb{R}}_{+}^9:N_h⩽\frac{{\Lambda }_h}{{\mu }_h}\text{.}\hfill \end{array}\right)$$ (7)

It is sufficient to consider solutions in Ω. Existence, uniqueness and continuation results for system (6) hold in this region and all solutions starting in Ω remain there for all t  ⩾ 0. Hence, (6) is mathematically and epidemiologically well-posed and it is sufficient to consider the dynamics of the flow generated by the model system(6) in Ω. Further, before analyzing the dynamics of the full model, it is instructive to analyze the sub-models first of all, since beginning with the simple case makes for a clearer exposition.

Fig. 1.

Structure of the model.

2.1. Disease-free equilibrium and stability analysis

The disease-free equilibrium of model system (6) is given by

$${\mathcal{E}}^0=\left(S_n^0\text{,}{I}_n^0\text{,}{R}_n^0\text{,}{S}^0\text{,}{I}_m^0\text{,}{R}_m^0\text{,}{I}_h^0\text{,}{A}_h^0\text{,}{I}_{\mathit{hm}}^0\text{,}{A}_{\mathit{hm}}^0\text{,}{R}_{\mathit{hm}}^0\text{,}{R}_{\mathit{am}}^0\right)=\left(\frac{{\Lambda }_n}{{\mu }_n}\text{,}0\text{,}0\text{,}\frac{{\Lambda }_h}{{\mu }_h}\text{,}0\text{,}0\text{,}0\text{,}0\text{,}0\text{,}0\text{,}0\text{,}0\right)\text{.}$$ (8)

Following van den Driessche and Watmough [14], the following dominant eigenvalues are obtained

$$\begin{array}{cc}\hfill & \text{Non human monkeypox}:{\mathcal{R}}_{0_n}=\frac{{\beta }_{n_1}{c}_{n_1}}{{\mu }_n+{\rho }_n+d_n}\text{,}\hfill \\ \hfill & \text{Human monkeypox}:{\mathcal{R}}_{0_m}=\frac{{\beta }_m{c}_m}{{\mu }_h+{\rho }_{m_1}+d_m}\text{,}\hfill \\ \hfill & \text{Human HIV infection}:{\mathcal{R}}_{0_h}=\frac{{\beta }_h{c}_h(d_a+{\mu }_h+{\rho }_{a_1}{\varphi }_2)}{({\mu }_h+{\rho }_{a_1})(\mu +d_a)}\hfill \end{array}$$ (9)

with ${\mathcal{R}}_{0_n}\text{,}{\mathcal{R}}_{0_m}$ and ${\mathcal{R}}_{0_h}$ being the reproduction numbers for the monkeypox infection in non-humans, monkeypox infection in humans and HIV infection in humans, respectively. Thus, the reproduction numbers for the model are given by

$${\mathcal{R}}_{0_n}\text{and}{\mathcal{R}}_0=\max \left\{{\mathcal{R}}_{0_m}\text{,}{\mathcal{R}}_{0_h}\right\}\text{.}$$ (10)

The following Theorem 1 follows from Theorem 2 of Van den Driessche and Watmough [14].

Theorem 1

The disease-free equilibrium${\mathcal{E}}^0$is locally asymptotically stable whenever${\mathcal{R}}_0<1$and${\mathcal{R}}_{0_n}<1$and unstable otherwise.

2.1.1. Analysis of the basic reproduction number ${\mathcal{R}}_0$

In this section we analyse the effect of the effect monkeypox basic reproduction number for non-human primates and some rodents on the monkeypox basic reproduction number for humans and the possible effects monkeypox has on HIV/AIDS.

Case 1: There is no HIV infection.

Here we analyse the reproduction number when the human population is free of HIV/AIDS. It is worth noting here that ${\beta }_{n_1}{c}_{n_1}>{\beta }_{\mathit{nm}}{c}_{n_2}>{\beta }_m{c}_m$ given the higher rates of monkeypox virus infection among animals than among humans. We define the ratios of the transmission parameters as

$$r_1=\frac{{\beta }_{n_1}{c}_{n_1}}{{\beta }_{\mathit{nm}}{c}_{n_2}}\text{,}{r}_2=\frac{{\beta }_{n_1}{c}_{n_1}}{{\beta }_m{c}_m}\text{,}{r}_1>r_2>1\text{.}$$ (11)

The presence of monkeypox infections in humans depend on their availability in the non-human primates. Expressing the human and animal monkeypox basic reproduction numbers as a ratio, we have

$$\frac{{\mathcal{R}}_{0_m}}{{\mathcal{R}}_{0_n}}=\frac{{\mu }_n+{\rho }_n+d_n}{r_2({\mu }_h+{\rho }_{m_1}+d_{m_1})}\text{.}$$ (12)

For human outbreak of monkeypox, Eq. (12) must be less than unity (i.e ${\mathcal{R}}_{0_m}<{\mathcal{R}}_{0_n}$). This is basically so, because monkeypox virus transmission rates are higher from animal-to-animal than animal to human.

Case 2: When there is HIV in the population.

Here, we look into what happens when HIV infection exists in a population exposed to monkeypox. The long term effects of HIV on monkey-pox infections depend on the reproduction numbers of the two infections. The ratio $\frac{{\mathcal{R}}_{0_m}}{{\mathcal{R}}_{0_h}}$ measures the different scenarios the population experiences. If $\frac{{\mathcal{R}}_{0_m}}{{\mathcal{R}}_{0_h}}<1$, then, in as much as the two infections will co-exist in the population, HIV/AIDS will be the dominant one. If the inequality is reversed, then, monkeypox will be the dominant one. The above inequality provides threshold conditions that determine the competitive outcomes of the two diseases (which are determined by the relative magnitudes of ${\mathcal{R}}_{0_m}\text{,}$ and ${\mathcal{R}}_{0_h}$. The two diseases will be co-dominant if the two reproduction numbers are equal.

2.2. Endemic equilibrium and stability analysis

The model has seven possible endemic equilibria which are basically the

• (i)animal-only monkeypox infection,
• (ii)HIV/AIDS only,
• (iii)animal-only monkeypox and HIV/AIDS equilibrium,
• (iv)co-existence of monkeypox infections in both human and non-human animals,
• (v)co-existence of HIV/AIDS and monkeypox infections in human population together with monkeypox infections in non-human animals,
• (vi)human-only monkeypox infections,
• (vii)co-existence of monkeypox and HIV/AIDS in the human population.

Mathematically the endemic equilibria (vi) and (vii) exist, but in reality these do not exist naturally in nature as human monkeypox always exist in places where there is animal monkeypox. As such, these two trivial equilibria will not be discussed in this paper.

2.2.1. Animal-only monkeypox endemic equilibrium

This occurs when there is only animal to animal infections and no human to human infection and no animal to human infections and I _m  =  R _m  =  I _h  =  A _h  =  I _hm  =  A _hm  =  R _hm  =  R _am  = 0. In this case the endemic equilibrium is given by ${\mathcal{E}}_1^{\ast }=\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\text{,}{S}^0\text{,}0\text{,}0\text{,}0\text{,}0\text{,}0\text{,}0\text{,}0\text{,}0\right)\text{,}{S}^{\ast }=\frac{{\Lambda }_h}{{\mu }_h}$. The presence of S ⁰ shows that ${\mathcal{R}}_0<1$. We now carry out some manipulations to find the expressions of the remaining components of ${\mathcal{E}}_1^{\ast }$. The animal-only model is standard SIR model which at equilibrium we have

$$\begin{array}{cc}\hfill & {\Lambda }_n=\left({\mu }_n+{\lambda }_n^{\ast }\right)S_n^{\ast }\text{,}\hfill \\ \hfill & {\lambda }_n^{\ast }{S}_n^{\ast }=({\mu }_n+{\rho }_n+d_n)I_n^{\ast }\text{,}\hfill \\ \hfill & {\rho }_n{I}_n^{\ast }={\mu }_n{R}_n^{\ast }\text{,}{\lambda }_n=\frac{{\beta }_{n_1}{c}_{n_1}{I}_n}{N_1}\text{.}\hfill \end{array}$$ (13)

Adding the first two equations of system (13) yields

$${\Lambda }_n={\mu }_n{S}_n^{\ast }+({\mu }_n+{\rho }_n+d_n)I_n^{\ast }\Rightarrow S_n^{\ast }=\frac{{\Lambda }_n}{{\mu }_n}-\frac{{\mu }_n+{\rho }_n+d_n}{{\mu }_n}{I}_n^{\ast }$$ (14)

From the last equation in system (13) we have $R_n^{\ast }=\frac{{\rho }_n}{{\mu }_n}{I}_n^{\ast }$. Adding all the equations in system (13) we have

$${\Lambda }_n={\mu }_n{N}_1^{\ast }+d_n{I}_n^{\ast }\Rightarrow N_1^{\ast }=\frac{{\Lambda }_n}{{\mu }_n}-\frac{d_n}{{\mu }_n}{I}_n^{\ast }$$ (15)

then, the second equation in system (13) gives $I_n^{\ast }=0$ or

$${\beta }_{n_1}{c}_{n_1}{S}_n^{\ast }=({\mu }_n+{\rho }_n+d_n)N_1^{\ast }\text{,}\Rightarrow {\beta }_{n_1}{c}_{n_1}\left(\frac{{\Lambda }_n}{{\mu }_n}-\frac{{\mu }_n+{\rho }_n+d_n}{{\mu }_n}{I}_n^{\ast }\right)=\frac{{\mu }_n+{\rho }_n+d_n}{{\mu }_n}\left({\Lambda }_n-d_n{I}_n^{\ast }\right)\text{,}\Rightarrow I_n^{\ast }=\frac{{\Lambda }_n({\beta }_{n_1}{c}_{n_1}-({\mu }_n+{\rho }_n+d_n))}{({\beta }_{n_1}{c}_{n_1}-d_n)({\mu }_n+{\rho }_n+d_n)}\text{.}$$ (16)

Only solution with $S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{N}_1^{\ast }>0$ is if ${\beta }_{n_1}{c}_{n_1}>({\mu }_n+{\rho }_n+d_n)\Rightarrow {\mathcal{R}}_{0_n}>1$. This leads to Lemma 1.

Lemma 1

The endemic equilibrium${\mathcal{E}}_1^{\ast }$exists whenever${\mathcal{R}}_{0_n}>1$and${\mathcal{R}}_0<1$.

Now we have to check onto the stability of this endemic equilibrium. In order to investigate the global stability of the endemic equilibrium, we adopt the approach by Korobeinikov [11]. Assume that ${\mathcal{R}}_{0_n}>1$, then ${\mathcal{E}}_1^{\ast }$ exists for all S _n; I _n; R _n; S  >  ϵ, for some ϵ  > 0. Let λ _n S _n : =  g(S _n;  I _n;  R _n) be a positive and monotonic function, and define the following continuous function in ${\mathbb{R}}_{+}^3$ (for more details, see Korobeinikov [11]):

$$V(S_n\text{,}{I}_n\text{,}{R}_n)=S_n-{\int }_{ϵ}^{S_n}\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(\tau \text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}d\tau +I_n-{\int }_{ϵ}^{I_n}\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n^{\ast }\text{,}\tau \text{,}{R}_n^{\ast }\right)}d\tau +R_n-{\int }_{ϵ}^{R_n}\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S^{\ast }\text{,}{I}_n^{\ast }\text{,}\tau \right)}d\tau \text{.}$$ (17)

If g(S _n,  I _n,  R _n) is monotonic with respect to its variables, then the endemic state ${\mathcal{E}}_1^{\ast }$ is the only extremum and the global minimum of this function. Indeed

$$\frac{\partial V}{\partial S_n}=1-\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}\text{,}\frac{\partial V}{\partial I_n}=1-\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n^{\ast }\text{,}{I}_n\text{,}{R}_n^{\ast }\right)}\text{,}\frac{\partial V}{\partial R_n}=1-\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n\right)}\text{,}$$ (18)

grow monotonically, then the function g(S _n,  I _n,  R _n) has only one stationary point.

Furthermore, since

$$\begin{array}{cc}\hfill & \frac{{\partial }^{2}V}{\partial S_n^2}=\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{{\left[g\left(S_n\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)\right]}^2}·\frac{\partial g\left(S_n\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{\partial S_n}\text{,}\hfill \\ \hfill & \frac{{\partial }^{2}V}{\partial I_n^2}=\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{{\left[g\left(S_n^{\ast }\text{,}{I}_n\text{,}{R}_n^{\ast }\right)\right]}^2}·\frac{\partial g\left(S_n^{\ast }\text{,}{I}_n\text{,}{R}_n^{\ast }\right)}{\partial I_n}\text{,}\hfill \\ \hfill & \frac{{\partial }^{2}V}{\partial R_n^2}=\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{{\left[g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n\right)\right]}^2}·\frac{\partial g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n\right)}{\partial R_n}\text{,}\hfill \end{array}$$ (19)

are non-negative, then the point ${\mathcal{E}}_1^{\ast }$ is a minimum. That is, $V(S_n\text{,}{I}_n\text{,}{R}_n)⩾V\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)$ and hence, V is a Lyapunov function.

In the case of our model system when β _nm  =  β _m  =  β _h  = 0 (animal to animal transmission only), then

$$\begin{array}{cc}\hfill & {\Lambda }_n=g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)+{\mu }_n{S}_n^{\ast }\text{,}\hfill \\ \hfill & ({\mu }_n+{\rho }_n+d_n)I_n^{\ast }=g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)\text{,}\hfill \\ \hfill & {\rho }_n{I}_n^{\ast }={\mu }_n{R}_n^{\ast }\text{.}\hfill \end{array}$$ (20)

The Lyapunov function (17) satisfies

$$\begin{array}{cc}\frac{\mathit{dV}}{\mathit{dt}}\hfill & =S_n^{\prime }-S_n^{\prime }\left(\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}\right)+I_n^{\prime }-I_n^{\prime }\left(\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n^{\ast }\text{,}{I}_n\text{,}{R}_n^{\ast }\right)}\right)\hfill \\ \hfill & +R_n^{\prime }-R_n^{\prime }\left(\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n\right)}\right)\hfill \\ \hfill & ={\mathrm{\Lambda }}_n-g(S_n\text{,}{I}_n\text{,}{R}_n)-{\mu }_n{S}_n-{\mathrm{\Lambda }}_n\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}+g(S_n\text{,}{I}_n\text{,}{R}_n)\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}\hfill \\ \hfill & +{\mu }_n{S}_n\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}+g(S_n\text{,}{I}_n\text{,}{R}_n)-({\mu }_n+{\rho }_n+d_n)I_n\hfill \\ \hfill & -g(S_n\text{,}{I}_n\text{,}{R}_n)\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n^{\ast }\text{,}{I}_n\text{,}{R}_n^{\ast }\right)}+({\mu }_n+{\rho }_n+d_n)I_n\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n^{\ast }\text{,}{I}_n\text{,}{R}_n^{\ast }\right)}+{\rho }_n{I}_n-\mu R_n\hfill \\ \hfill & ={\mu }_n{S}_n^{\ast }\left(1-\frac{S_n}{S_n^{\ast }}\right)\left(1-\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}\right)\hfill \\ \hfill & +g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)\left(1-\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}-\frac{g(S_n\text{,}{I}_n\text{,}{R}_n)}{g\left(S_n\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}\right)\hfill \\ \hfill & +g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)\left(-\frac{I_n}{I_n^{\ast }}+\frac{I_n}{I_n^{\ast }}\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n^{\ast }\text{,}{I}_n\text{,}{R}_n^{\ast }\right)}-\frac{g(S_n\text{,}{I}_n\text{,}{R}_n)}{g\left(S_n^{\ast }\text{,}{I}_n\text{,}{R}_n^{\ast }\right)}\right)\hfill \\ \hfill & +{\mu }_n{R}_n^{\ast }\left(\frac{I_n}{I_n^{\ast }}-\frac{R_n}{R_n^{\ast }}\right)\left(1-\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n\right)}\right)\hfill \\ \hfill & ={\mu }_n{S}_n^{\ast }\left(1-\frac{S_n}{S_n^{\ast }}\right)\left(1-\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}\right)\hfill \\ \hfill & +g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)\left(1-\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}\right)\left(1-\frac{g(S_n\text{,}{I}_n\text{,}{R}_n)}{g\left(S_n^{\ast }\text{,}{I}_n\text{,}{R}_n^{\ast }\right)}\right)\hfill \\ \hfill & +g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)\left(\frac{I_n}{I_n^{\ast }}-\frac{g(S_n\text{,}{I}_n\text{,}{R}_n)}{g\left(S_n\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}\right)\left(\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n^{\ast }\text{,}{I}_n\text{,}{R}_n^{\ast }\right)}-1\right)\hfill \\ \hfill & +{\mu }_n{R}_n^{\ast }\left(\frac{I_n}{I_n^{\ast }}-\frac{R_n}{R_n^{\ast }}\right)\left(1-\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n\right)}\right)\text{.}\hfill \end{array}$$ (21)

Since ${\mathcal{E}}_1^{\ast }>0$, the function g(S _n,  I _n,  R _n) is concave with respect to I _n, and $\frac{{\partial }^{2}g(S_n\text{,}{I}_n\text{,}{R}_n)}{\partial I_n^2}⩽0$, then $\frac{\mathit{dV}}{\mathit{dt}}⩽0$ for all S _n, I _n, R _n  > 0. Also, the monotonicity of g(S _n,  I _n,  R _n) with respect to S _n ensures that

$$\left(1-\frac{S_n}{S_n^{\ast }}\right)\left(1-\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}\right)⩽0$$ (22)

and

$$\left(1-\frac{g(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast })}{g\left(S_n\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}\right)\left(1-\frac{g(S_n\text{,}{I}_n\text{,}{R}_n)}{g\left(S_n^{\ast }\text{,}{I}_n\text{,}{R}_n^{\ast }\right)}\right)⩽0$$ (23)

holds for all S _n, I _n, R _n  > 0. Furthermore,

$$\left(\frac{I_n}{I_n^{\ast }}-\frac{g(S_n\text{,}{I}_n\text{,}{R}_n)}{g\left(S_n\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}\right)\left(\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n^{\ast }\text{,}{I}_n\text{,}{R}_n^{\ast }\right)}-1\right)⩽0$$

if

$$\begin{array}{cc}\hfill & \frac{g(S_n\text{,}{I}_n\text{,}{R}_n)}{g\left(S_n\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}⩾\frac{I_n}{I_n^{\ast }}\text{when}g\left(S_n^{\ast }\text{,}{I}_n\text{,}{R}_n^{\ast }\right)⩽g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)\text{and}\hfill \\ \hfill & \frac{g(S_n\text{,}{I}_n\text{,}{R}_n)}{g\left(S_n\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}⩽\frac{I_n}{I_n^{\ast }}\text{when}g\left(S^{\ast }\text{,}{I}_n\text{,}{R}_n^{\ast }\right)⩾g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)\text{.}\hfill \end{array}$$ (24)

holds for all S _n, I _n, R _n  > 0. Since g(S _n, I _n, R _n) is monotonic $g(S_n^{\ast }\text{,}{I}_n\text{,}{R}_n^{\ast })⩾g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)\Rightarrow I_n⩾I_n^{\ast }$ and $g\left(S_n^{\ast }\text{,}{I}_n\text{,}{R}_n^{\ast }\right)⩽g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)\Rightarrow I_n⩽I_n^{\ast }$. Also,

$$\left(\frac{I_n}{I_n^{\ast }}-\frac{R_n}{R_n^{\ast }}\right)\left(1-\frac{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)}{g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n\right)}\right)⩽0$$ (25)

if

$$\begin{array}{cc}\hfill & \frac{R_n}{R_n^{\ast }}⩾\frac{I_n}{I_n^{\ast }}\text{when}g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n\right)⩾g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)\text{and}\hfill \\ \hfill & \frac{R_n}{R_n^{\ast }}⩽\frac{I_n}{I_n^{\ast }}\text{when}g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n\right)⩽g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)\text{.}\hfill \end{array}$$ (26)

holds for all S _n, I _n, R _n  > 0. Since g(S _n,  I _n,  R _n) is a monotonic function $g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n\right)⩾g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)\Rightarrow R_n⩾R_n^{\ast }$ and $g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n\right)⩽g\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\right)\Rightarrow R_n⩽R_n^{\ast }$. Inequalities (24), (26) will hold for any concave function and are sufficient to ensure that $\frac{\mathit{dV}}{\mathit{dt}}⩽0$. Thus, we have established the following result:

Theorem 2

The unique endemic equilibrium${\mathcal{E}}_1^{\ast }$is globally asymptotically stable whenever conditions(24), (26)are satisfied.

2.2.2. HIV/AIDS-only endemic equilibrium

This occurs when there is no monkeypox infections in the animal and human population. Here, I _n  =  R _n  =  I _m  =  R _m  =  I _hm  =  R _hm  =  A _hm  =  R _am  = 0 so that this endemic equilibrium is given by ${\mathcal{E}}_2^{\ast }=\left(S_n^0\text{,}0\text{,}0\text{,}{S}^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\text{,}0\text{,}0\text{,}0\text{,}0\text{,}0\text{,}0\right)\text{,}{S}_n^0=\frac{{\Lambda }_n}{{\mu }_n}$. The presence of $S_n^0$ shows that ${\mathcal{R}}_{0_n}<1$. We now find the exact expressions the remaining components of ${\mathcal{E}}_2^{\ast }$. The HIV/AIDS-only sub-model at equilibrium is given by

$$\begin{array}{cc}\hfill & {\Lambda }_h=\left({\mu }_h+{\lambda }_h^{\ast }\right)S^{\ast }\text{,}\hfill \\ \hfill & {\lambda }_h^{\ast }{S}^{\ast }=({\mu }_h+{\rho }_{a_1})I_h^{\ast }\text{,}\hfill \\ \hfill & {\rho }_{a_1}{I}_h^{\ast }=({\mu }_h+d_a)A_h^{\ast }\text{,}{\lambda }_h=\frac{{\beta }_h{c}_h(I_h+{\varphi }_2{A}_h)}{N_h}\hfill \end{array}$$ (27)

Adding the first two equations of system (27) one gets

$${\Lambda }_h={\mu }_h{S}^{\ast }+({\mu }_h+{\rho }_{a_1})I_h^{\ast }\Rightarrow S^{\ast }=\frac{{\Lambda }_h}{{\mu }_h}-\frac{{\mu }_h+{\rho }_{a_1}}{{\mu }_h}{I}_h^{\ast }\text{.}$$ (28)

From the last equation of system (27) one gets $A_h^{\ast }=\frac{{\rho }_{a_1}{I}_h^{\ast }}{{\mu }_h+d_a}$. Adding all equations in system (27), we obtain

$${\Lambda }_h={\mu }_h{N}_h^{\ast }+d_a{A}_h^{\ast }\Rightarrow N_h^{\ast }=\frac{{\Lambda }_h}{{\mu }_h}-\frac{d_a}{{\mu }_h}{A}_h^{\ast }\text{,}$$ (29)

then the second equation of system (27) gives $I_h^{\ast }=0$ or

$${\beta }_h{c}_h({\mu }_h+d_a+{\varphi }_2{\rho }_{a_1})({\Lambda }_h-({\mu }_h+{\rho }_{a_1})I_h^{\ast })=({\mu }_h+{\rho }_{a_1})({\Lambda }_h({\mu }_h+d_a)-d_a{\rho }_{a_1}{I}_h^{\ast })\Rightarrow I_h^{\ast }=\frac{{\Lambda }_h\left[{\beta }_h{c}_h({\mu }_h+d_a+{\varphi }_2{\rho }_{a_1})-({\mu }_h+{\rho }_{a_1})({\mu }_h+d_a)\right]}{({\mu }_h+{\rho }_{a_1})({\beta }_h{c}_h({\mu }_h+d_a+{\varphi }_2{\rho }_{a_1})-d_a{\rho }_{a_1})}$$ (30)

The only solution with $S^{\ast }\text{,}{I}_h^{\ast }\text{,}{N}_h^{\ast }>0$ is if ${\beta }_h{c}_h({\mu }_h+d_a+{\varphi }_2{\rho }_{a_1})-({\mu }_h+{\rho }_{a_1})({\mu }_h+d_a)>0\Rightarrow {\mathcal{R}}_{0_h}>1$. This result is summarized in Lemma 2.

Lemma 2

The endemic equilibrium${\mathcal{E}}_2^{\ast }$exists whenever${\mathcal{R}}_{0_h}>1\text{,}{\mathcal{R}}_{0_m}<1\text{and}{\mathcal{R}}_{0_n}<1$.

To investigate the global stability of the endemic equilibrium, we adopt the approach by Korobeinikov [11]. Assume that ${\mathcal{R}}_{0_h}>1$, then ${\mathcal{E}}_2^{\ast }$ exists for all S; I _h; A _h; S _n  >  ϵ, for some ϵ  > 0. Let λ _h S : =  g(S;  I _h;  A _h) be a positive and monotonic function, and define the following continuous function in ${\mathbb{R}}_{+}^3$ (for more details, see Korobeinikov [11])

$$V(S\text{,}{I}_h\text{,}{A}_h)=S-{\int }_{ϵ}^S\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(\tau \text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}d\tau +I_h-{\int }_{ϵ}^{I_h}\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S^{\ast }\text{,}\tau \text{,}{A}_h^{\ast }\right)}d\tau +A_h-{\int }_{ϵ}^{A_h}\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}\tau \right)}d\tau \text{.}$$ (31)

If g(S,  I _h,  A _h) is monotonic with respect to its variables, then the endemic state ${\mathcal{E}}_2^{\ast }$ is the only extremum and the global minimum of this function. Indeed

$$\frac{\partial V}{\partial S}=1-\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}\text{,}\frac{\partial V}{\partial I_h}=1-\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S^{\ast }\text{,}{I}_h\text{,}{A}_h^{\ast }\right)}\text{,}\frac{\partial V}{\partial A_h}=1-\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_n^{\ast }\right)}{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h\right)}\text{,}$$ (32)

grow monotonically, then the function g(S,  I _h,  A _h) has only one stationary point. Furthermore, since

$$\begin{array}{cc}\hfill & \frac{{\partial }^{2}V}{\partial S^2}=\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{{\left[g\left(S\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)\right]}^2}·\frac{\partial g\left(S\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{\partial S}\text{,}\hfill \\ \hfill & \frac{{\partial }^{2}V}{\partial I_h^2}=\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{{\left[g\left(S^{\ast }\text{,}{I}_h\text{,}{A}_h^{\ast }\right)\right]}^2}·\frac{\partial g\left(S^{\ast }\text{,}{I}_h\text{,}{A}_h^{\ast }\right)}{\partial I_h}\text{,}\hfill \\ \hfill & \frac{{\partial }^{2}V}{\partial A_h^2}=\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{{\left[g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h\right)\right]}^2}·\frac{\partial g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h\right)}{\partial A_h}\text{,}\hfill \end{array}$$ (33)

are non-negative, then the point ${\mathcal{E}}_2^{\ast }$ is a minimum. That is, $V(S\text{,}{I}_h\text{,}{A}_h)⩾V\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)$ and hence, V is a Lyapunov function. In the case of our model system when there is HIV transmission only, then

$$\begin{array}{cc}\hfill & {\Lambda }_h=g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)+{\mu }_h{S}^{\ast }\text{,}\hfill \\ \hfill & ({\mu }_h+{\rho }_{a_1})I_h^{\ast }=g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)\text{,}\hfill \\ \hfill & {\rho }_{a_1}{I}_h^{\ast }=({\mu }_h+d_a)A_h^{\ast }\text{.}\hfill \end{array}$$ (34)

The Lyapunov function (31) satisfies

$$\begin{array}{cc}\frac{\mathit{dV}}{\mathit{dt}}\hfill & =S^{\prime }-S^{\prime }\left(\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}\right)+I_h^{\prime }-I_h^{\prime }\left(\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S^{\ast }\text{,}{I}_h\text{,}{A}_h^{\ast }\right)}\right)\hfill \\ \hfill & +A_h^{\prime }-A_h^{\prime }\left(\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h\right)}\right)\hfill \\ \hfill & ={\mathrm{\Lambda }}_h-g(S\text{,}{I}_h\text{,}{A}_h)-{\mu }_{h}S-{\mathrm{\Lambda }}_h\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}+g(S\text{,}{I}_h\text{,}{A}_h)\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}\hfill \\ \hfill & +{\mu }_{h}S\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}+g(S\text{,}{I}_h\text{,}{A}_h)-({\mu }_h+{\rho }_{a_1})I_h\hfill \\ \hfill & -g(S\text{,}{I}_h\text{,}{A}_h)\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S^{\ast }\text{,}{I}_h\text{,}{A}_h^{\ast }\right)}+({\mu }_h+{\rho }_{a_1})I_h\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S^{\ast }\text{,}{I}_h\text{,}{A}_h^{\ast }\right)}+{\rho }_{a_1}{I}_h-({\mu }_h+d_a)A_h\hfill \\ \hfill & -{\rho }_{a_1}{I}_h\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h\right)}+({\mu }_h+d_a)A_h\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h\right)}\hfill \\ \hfill & ={\mu }_h{S}^{\ast }\left(1-\frac{S}{S^{\ast }}\right)\left(1-\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}\right)\hfill \\ \hfill & +g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)\left(1-\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}-\frac{g(S\text{,}{I}_h\text{,}{A}_h)}{g\left(S\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}\right)\hfill \\ \hfill & +g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)\left(-\frac{I_h}{I_h^{\ast }}+\frac{I_h}{I_h^{\ast }}\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S^{\ast }\text{,}{I}_h\text{,}{A}_h^{\ast }\right)}-\frac{g(S\text{,}{I}_h\text{,}{A}_h)}{g\left(S^{\ast }\text{,}{I}_h\text{,}{A}_h^{\ast }\right)}\right)\hfill \\ \hfill & +({\mu }_h+d_a)A_h^{\ast }\left(\frac{I_h}{I_h^{\ast }}-\frac{A_h}{A_h^{\ast }}\right)\left(1-\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h\right)}\right)\hfill \\ \hfill & ={\mu }_h{S}^{\ast }\left(1-\frac{S}{S^{\ast }}\right)\left(1-\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}\right)\hfill \\ \hfill & +g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)\left(1-\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}\right)\left(1-\frac{g(S\text{,}{I}_h\text{,}{A}_h)}{g\left(S^{\ast }\text{,}{I}_h\text{,}{A}_h^{\ast }\right)}\right)\hfill \\ \hfill & +g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)\left(\frac{I_h}{I_h^{\ast }}-\frac{g(S\text{,}{I}_h\text{,}{A}_h)}{g\left(S\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}\right)\left(\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S^{\ast }\text{,}{I}_h\text{,}{A}_h^{\ast }\right)}-1\right)\hfill \\ \hfill & +({\mu }_h+d_a)A_h^{\ast }\left(\frac{I_h}{I_h^{\ast }}-\frac{A_h}{A_h^{\ast }}\right)\left(1-\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h\right)}\right)\text{.}\hfill \end{array}$$ (35)

Since ${\mathcal{E}}_2^{\ast }>0$, the function g(S, I _h, A _h) is concave with respect to I _h (A _h), and $\frac{{\partial }^{2}g(S\text{,}{I}_h\text{,}{A}_h)}{\partial I_h^2}⩽0\left(\frac{{\partial }^{2}g(S\text{,}{I}_h\text{,}{A}_h)}{\partial A_h^2}⩽0\right)$, then $\frac{\mathit{dV}}{\mathit{dt}}⩽0$ for all S,I _h,A _h  > 0. Also, the monotonicity of g(S,  I _h,  A _h) with respect to S ensures that

$$\left(1-\frac{S}{S^{\ast }}\right)\left(1-\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}\right)⩽0$$ (36)

and

$$\left(1-\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}\right)\left(1-\frac{g(S\text{,}{I}_h\text{,}{A}_h)}{g\left(S^{\ast }\text{,}{I}_h\text{,}{A}_h^{\ast }\right)}\right)⩽0$$ (37)

holds for all S, I _h, A _h  > 0. Furthermore,

$$\left(\frac{I_h}{I_h^{\ast }}-\frac{g(S\text{,}{I}_h\text{,}{A}_h)}{g\left(S\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}\right)\left(\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g\left(S^{\ast }\text{,}{I}_h\text{,}{A}_h^{\ast }\right)}-1\right)⩽0$$

if

$$\begin{array}{cc}\hfill & \frac{g(S\text{,}{I}_h\text{,}{A}_h)}{g\left(S\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}⩾\frac{I_h}{I_h^{\ast }}\text{when}g\left(S^{\ast }\text{,}{I}_h\text{,}{A}_h^{\ast }\right)⩽g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)\text{and}\hfill \\ \hfill & \frac{g(S\text{,}{I}_h\text{,}{A}_h)}{g\left(S\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}⩽\frac{I_h}{I_h^{\ast }}\text{when}g\left(S^{\ast }\text{,}{I}_h\text{,}{A}_h^{\ast }\right)⩾g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)\text{.}\hfill \end{array}$$ (38)

holds for all S, I _h, A _h  > 0. Since g(S,  I _h,  A _h) is monotonic $g\left(S^{\ast }\text{,}{I}_h\text{,}{A}_h^{\ast }\right)⩾g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)\Rightarrow I_h⩾I_h^{\ast }$ and $g\left(S^{\ast }\text{,}{I}_h\text{,}{A}_h^{\ast }\right)⩽g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)\Rightarrow I_h⩽I_h^{\ast }$. Also,

$$\left(\frac{I_h}{I_h^{\ast }}-\frac{A_h}{A_h^{\ast }}\right)\left(1-\frac{g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)}{g(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h)}\right)⩽0$$ (39)

if

$$\begin{array}{cc}\hfill & \frac{A_h}{A_h^{\ast }}⩾\frac{I_h}{I_h^{\ast }}\text{when}g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h\right)⩾g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)\text{and}\hfill \\ \hfill & \frac{A_h}{A_h^{\ast }}⩽\frac{I_h}{I_h^{\ast }}\text{when}g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h\right)⩽g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)\text{.}\hfill \end{array}$$ (40)

holds for all S, I _h, A _h  > 0. Since g(S,  I _h,  A _h) is a monotonic function $g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h\right)⩾g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)\Rightarrow A_h⩾A_h^{\ast }$ and $g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h\right)⩽g\left(S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\right)\Rightarrow A_h⩽A_h^{\ast }$. Inequalities (38), (40) will hold for any concave function and are sufficient to ensure that $\frac{\mathit{dV}}{\mathit{dt}}⩽0$. Thus, we have established the following result:

Theorem 3

The unique endemic equilibrium${\mathcal{E}}_2^{\ast }$is globally asymptotically stable whenever conditions(38), (40)are satisfied.

2.2.3. HIV and animal monkeypox-only equilibrium

This occurs when there is HIV transmission among humans and monkeypox infections only among animals (there is no animal to human transmission of monkeypox). In this case I _m  =  R _m  =  I _hm  =  A _hm  =  R _hm  =  R _am  = 0. As such the this endemic equilibrium is given by ${\mathcal{E}}_3^{\ast }=\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\text{,}{S}^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\text{,}0\text{,}0\text{,}0\text{,}0\text{,}0\text{,}0\right)$ where $S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }$ are as defined in ${\mathcal{E}}_1^{\ast }$ (the animal-only monkeypox endemic equilibrium) and $S^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }$ are as defined in ${\mathcal{E}}_2^{\ast }$ (HIV-only endemic equilibrium). The following Lemma 3 which is a direct result of both Lemma 1, Lemma 2 result from the analysis of this equilibrium point.

Lemma 3

The endemic equilibrium${\mathcal{E}}_3^{\ast }$exists whenever${\mathcal{R}}_{0_h}>1\text{,}{\mathcal{R}}_{0_n}>1$and${\mathcal{R}}_{0_m}<1$.

To investigate the global asymptotic stability of ${\mathcal{E}}_3^{\ast }$, the Lyapunov function approach can be employed similar to the analysis of ${\mathcal{E}}_1^{\ast }\text{and}{\mathcal{E}}_2^{\ast }$ and the following result is obtained.

Theorem 4

The endemic equilibrium${\mathcal{E}}_3^{\ast }$is globally asymptotically stable if conditions(24), (26), (38), (40)are satisfied.

2.2.4. Co-existence of monkeypox in both human and animal populations endemic equilibrium

This occurs when there is monkeypox transmission in non-humans and humans and there is no HIV in the human population (λ _h  = 0). This endemic equilibrium is given by ${\mathcal{E}}_4^{\ast }$ where

$${\mathcal{E}}_4^{\ast }=\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\text{,}{S}^{\ast }\text{,}{I}_m^{\ast }\text{,}{R}_m^{\ast }\text{,}0\text{,}0\text{,}0\text{,}0\text{,}0\text{,}0\right)\text{,}$$ (41)

with $S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }$ as defined in ${\mathcal{E}}_1^{\ast }$ where they are shown to exist whenever ${\mathcal{R}}_{0_n}>1$. To find the remaining components it is necessary to consider the human subsystem in the absence of HIV which at equilibrium is given by

$$\begin{array}{cc}\hfill & {\Lambda }_h=\left({\mu }_h+{\lambda }_m^{\ast }\right)S^{\ast }\text{,}\hfill \\ \hfill & {\lambda }_m^{\ast }{S}^{\ast }=({\mu }_h+{\rho }_{m_1}+d_m)I_m^{\ast }\text{,}\hfill \\ \hfill & {\rho }_{m_1}{I}_m^{\ast }={\mu }_h{R}_m^{\ast }\text{,}{\lambda }_m=\frac{{\beta }_{\mathit{nm}}{c}_n{I}_n}{N_1}+\frac{{\beta }_m{c}_m{I}_m}{N_2}\text{.}\hfill \end{array}$$ (42)

Adding the first two equations of system (42) the following is obtained

$${\Lambda }_h={\mu }_h{S}^{\ast }+({\mu }_h+{\rho }_{m_1}+d_m)I_m^{\ast }\Rightarrow S^{\ast }=\frac{{\Lambda }_h}{{\mu }_h}-\frac{{\mu }_h+{\rho }_{m_1}+d_m}{{\mu }_h}{I}_m^{\ast }\text{.}$$ (43)

Adding all the equations of system (42) we obtain

$${\Lambda }_h={\mu }_h{N}_h^{\ast }+d_m^{\ast }{I}_m^{\ast }\Rightarrow N_h^{\ast }=\frac{{\Lambda }_h}{{\mu }_h}-\frac{d_m}{{\mu }_h}{I}_m^{\ast }\text{,}$$ (44)

then the second equation of system (42) gives

$$\left(\frac{{\beta }_{\mathit{nm}}{c}_n{I}_n^{\ast }}{N_1^{\ast }}+\frac{{\beta }_m{c}_m{I}_m^{\ast }}{N_h^{\ast }}\right)\left(\frac{{\Lambda }_h}{{\mu }_h}-\frac{{\mu }_h+{\rho }_{m_1}+d_m}{{\mu }_h}{I}_m^{\ast }\right)=({\mu }_h+{\rho }_{m_1}+d_m)I_m^{\ast }\text{.}$$ (45)

Set $x^{\ast }=\frac{1}{{\mu }_h}\frac{{\beta }_{\mathit{nm}}{c}_n{I}_n^{\ast }}{N_1^{\ast }}$ then (45) becomes

$$\left({\beta }_m{c}_m{I}_m^{\ast }+x^{\ast }\left({\Lambda }_h-d_m{I}_m^{\ast }\right)\right)\left(\frac{{\Lambda }_h}{{\mu }_h}-\frac{{\mu }_h+{\rho }_{m_1}+d_m}{{\mu }_h}{I}_m^{\ast }\right)=({\mu }_h+{\rho }_{m_1}+d_m)\left(\frac{{\Lambda }_h}{{\mu }_h}-\frac{d_m}{{\mu }_h}{I}_m^{\ast }\right)I_m^{\ast }\Rightarrow ({\beta }_m{c}_m-d_m(1+x^{\ast })){\left(I_m^{\ast }\right)}^2+{\Lambda }_h\left((1+x^{\ast })-\frac{{\beta }_m{c}_m-d_m{x}^{\ast }}{{\mu }_h+{\rho }_{m_1}+d_m}\right)I_m^{\ast }-\frac{{\Lambda }_h^2{x}^{\ast }}{{\mu }_h+{\rho }_{m_1}+d_m}=0$$ (46)

and this has a single positive root if β _m c _m  >  d _m(1 +  x ^∗).

To analyze the local stability of this equilibrium point we make use of the Centre Manifold Theory [5], as described in Theorem 4.1 of Castillo-Chavez and Song [6]. To establish the local asymptotic stability of the non smoking only endemic equilibrium, we first make the change of variables S _n  =  x ₁,I _n  =  x ₂,R _n  =  x ₃,S  =  x ₄, I _m  =  x ₅,R _m  =  x ₆, so that $N_1(t)={\sum }_{n=1}^3{x}_n$ and $N_h(t)={\sum }_{n=1}^3{x}_{n+3}$. Using the vector notation X  = (x ₁,x ₂,x ₃,x ₄,x ₅,x ₆)^T, the system (6) can be written in the form $\frac{\mathit{dX}}{\mathit{dt}}=(f_1\text{,}{f}_2\text{,}{f}_3\text{,}{f}_4\text{,}{f}_5\text{,}{f}_6)$, where

$$\begin{array}{cc}\hfill & x_1^{\prime }(t)=f_1={\Lambda }_n-\frac{{\beta }_{n_1}{c}_{n_1}{x}_2{x}_1}{{\sum }_{n=1}^3{x}_n}-{\mu }_n{x}_1\text{,}\hfill \\ \hfill & x_2^{\prime }(t)=f_2=\frac{{\beta }_{n_1}{c}_{n_1}{x}_2{x}_1}{{\sum }_{n=1}^3{x}_n}-({\mu }_n+{\rho }_n+d_n)x_2\text{,}\hfill \\ \hfill & x_3^{\prime }(t)=f_3={\rho }_n{x}_2-{\mu }_n{x}_3\text{,}\hfill \\ \hfill & x_4^{\prime }(t)=f_4={\Lambda }_h-\frac{{\beta }_m{c}_m{x}_5{x}_4}{{\sum }_{n=1}^3{x}_{n+3}}-\frac{{\beta }_{\mathit{nm}}{c}_{n_2}{x}_2{x}_4}{{\sum }_{n=1}^3{x}_n}-{\mu }_h{x}_4\text{,}\hfill \\ \hfill & x_5^{\prime }(t)=f_5=\frac{{\beta }_m{c}_m{x}_5{x}_4}{{\sum }_{n=1}^3{x}_{n+3}}+\frac{{\beta }_{\mathit{nm}}{c}_{n_2}{x}_2{x}_4}{{\sum }_{n=1}^3{x}_n}-({\mu }_h+{\rho }_{m_1}+d_m)x_5\text{,}\hfill \\ \hfill & x_6^{\prime }(t)=f_6={\rho }_{m_1}{x}_5-{\mu }_h{x}_6\text{.}\hfill \end{array}$$ (47)

The Jacobian matrix of system (47) at ${\mathcal{E}}^0$ is given by

$$J({\mathcal{E}}^0)=\left[\begin{array}{cccccc}\hfill -{\mu }_n\hfill & \hfill -{\beta }_{n_1}{c}_{n_1}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\beta }_{n_1}{c}_{n_1}-({\mu }_n+{\rho }_n+d_n)\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\rho }_n\hfill & \hfill -{\mu }_n\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -\frac{{\beta }_{\mathit{nm}}{c}_{n_2}{\Lambda }_h{\mu }_n}{{\Lambda }_n{\mu }_h}\hfill & \hfill 0\hfill & \hfill -{\mu }_h\hfill & \hfill -{\beta }_m\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{{\beta }_{\mathit{nm}}{c}_{n_2}{\Lambda }_h{\mu }_n}{{\Lambda }_n{\mu }_h}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_m-({\mu }_h+d_m+{\rho }_{m_1})\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\rho }_{m_1}\hfill & \hfill -{\mu }_h\hfill \end{array}\right]\text{.}$$ (48)

From Eq. (48) it follows we have the following reproduction numbers, ${\mathcal{R}}_{0_n}$ and ${\mathcal{R}}_{0_m}$ as defined in Eq. (9). Consider the case when ${\mathcal{R}}_{0_m}=1$, and solve for β _m c _m to obtain

$${\beta }_m^{\ast }={\beta }_m{c}_m={\mu }_h+{\rho }_{m_1}+d_m\text{.}$$ (49)

The linearized system of the transformed Eq. (47) has a simple zero eigenvalue, hence the Centre Manifold theory [5], can be used to analyze the bifurcation dynamics of (47) near ${\beta }_m^{\ast }$. The Jacobian of (47) at ${\beta }_m^{\ast }$ has a right eigenvector associated with the zero eigenvalue given by w  = [w ₁,  w ₂,  w ₃,  w ₄,  w ₅,  w ₆]^T where,

$$w_1=w_2=w_3=0\text{,}{w}_4=-\frac{{\beta }_m^{\ast }{w}_5}{{\mu }_h}\text{,}{w}_5>0\text{,}{w}_6=\frac{{\rho }_{m_1}{w}_5}{{\mu }_h}\text{.}$$ (50)

The left eigenvector of $J({\mathcal{E}}^0)$ associated with the eigenvalue at β  =  β ^∗ is given by z  = [z ₁,  z ₂,  z ₃,  z ₄,  z ₅,  z ₆]^T where,

$$z_1=z_3=z_4=z_6=0\text{,}{z}_5>0\text{,}{z}_2=-\frac{{\beta }_{\mathit{nm}}^{\ast }{\Lambda }_h{\mu }_n{z}_5}{{\Lambda }_n{\mu }_h\left({\beta }_{n_1}^{\ast }-{\mu }_n-{\rho }_n-d_n\right)}\text{.}$$ (51)

The transmission rates β _nm and ${\beta }_{n_1}$ are as given in (11). To establish the conditions for the existence of backward bifurcation, we use Theorem 4.1 of Castillo–Chavez and Song [6], stated in the Appendix for convenience.

Computations of the bifurcation parameters a and b

For system (47), the non-zero partial derivatives of F associated with the bifurcation coefficient b are

$$\frac{{\partial }^2{f}_2}{\partial x_2\partial {\beta }_m^{\ast }}=D_1\text{,}\frac{{\partial }^2{f}_5}{\partial x_4\partial {\beta }_m^{\ast }}=\frac{{\mathcal{D}}_1{\Lambda }_h{\mu }_n}{{\mathcal{D}}_2{\Lambda }_n{\mu }_h}\text{,}\frac{{\partial }^2{f}_5}{\partial x_5\partial {\beta }_m^{\ast }}=1\text{.}$$ (52)

It follows from (52) that

$$b=z_5{w}_5>0\text{.}$$ (53)

Since w ₁  =  w ₂  =  w ₃  = 0, then, for system (47), the non-zero partial derivatives of F associated with a at the disease-free equilibrium are

$$\frac{{\partial }^2{f}_5}{\partial x_5^2}=-\frac{2{\beta }_m^{\ast }{\mu }_h}{{\Lambda }_h}\text{,}\frac{{\partial }^2{f}_5}{\partial x_5\partial x_5}=-\frac{{\beta }_m^{\ast }{\mu }_h}{{\Lambda }_h}\text{.}$$ (54)

It follows from (54) that

$$a=-\frac{2({\mu }_h+{\rho }_{m_1}){\beta }_m^{\ast }{\mu }_h}{{\mu }_h{\Lambda }_h}{z}_5{w}_5^2<0\text{.}$$ (55)

So, a  < 0 and b  > 0. Using Theorem A-1 item (iv), we establish Theorem 5.

Theorem 5

The unique endemic equilibrium${\mathcal{E}}_4^{\ast }$is locally asymptotically stable for${\mathcal{R}}_{0_m}>1$but close to 1,${\mathcal{R}}_{0_n}>1$and${\mathcal{R}}_{0_h}<1$.

We note however the local stability of ${\mathcal{E}}_4^{\ast }$ implies its global stability, thus the disease may persist.

2.2.5. Co-existence of monkeypox and HIV/AIDS endemic equilibrium

This occurs when monkeypox and HIV/AIDS co-exist in the population and in terms of the forces of infection this is given by

$${\mathcal{E}}_5^{\ast }=\left(S_n^{\ast }\text{,}{I}_n^{\ast }\text{,}{R}_n^{\ast }\text{,}{S}^{\ast }\text{,}{I}_m^{\ast }\text{,}{R}_m^{\ast }\text{,}{I}_h^{\ast }\text{,}{A}_h^{\ast }\text{,}{I}_{\mathit{hm}}^{\ast }\text{,}{A}_{\mathit{hm}}^{\ast }\text{,}{R}_{\mathit{hm}}^{\ast }\text{,}{R}_{\mathit{am}}^{\ast }\right)$$ (56)

where,

$$\begin{array}{c}{S}_n^{\ast }=\frac{{\Lambda }_n}{{\mu }_n+{\lambda }_n^{\ast }}\text{,}{I}_n^{\ast }=\frac{{\lambda }_n^{\ast }{S}_n^{\ast }}{{\mu }_n+{\rho }_n+d_n}\text{,}{R}_n^{\ast }=\frac{{\rho }_n{I}_n^{\ast }}{{\mu }_n}\text{,}\hfill \\ \hfill \\ S^{\ast }=\frac{{\Lambda }_h}{{\mu }_h+{\lambda }_m^{\ast }+{\lambda }_h^{\ast }}\text{,}{I}_h^{\ast }=\frac{{\lambda }_h^{\ast }{S}^{\ast }}{{\mu }_h+{\rho }_{a_1}+{\sigma }_1{\lambda }_m^{\ast }}\text{,}{A}_h^{\ast }=\frac{{\rho }_{a_1}{I}_h^{\ast }}{{\mu }_h+d_a+{\sigma }_2{\lambda }_m^{\ast }}\text{,}\hfill \\ I_m^{\ast }=\frac{{\lambda }_m^{\ast }{S}^{\ast }}{{\mu }_h+d_m+{\rho }_{m_1}+{\lambda }_h^{\ast }}\text{,}{R}_m^{\ast }=\frac{{\rho }_{m_1}{I}_m^{\ast }}{{\mu }_h+{\lambda }_h^{\ast }}\text{,}{I}_{\mathit{hm}}^{\ast }=\frac{{\lambda }_h^{\ast }{I}_m^{\ast }+{\sigma }_1{\lambda }_m^{\ast }{I}_h^{\ast }}{{\mu }_h+d_m+{\rho }_{m_2}+{\rho }_{a_2}}\text{,}\hfill \\ A_{\mathit{hm}}^{\ast }=\frac{{\sigma }_2{\lambda }_m^{\ast }{A}_h^{\ast }+{\rho }_{a_2}{I}_{\mathit{hm}}^{\ast }}{{\mu }_h+d_m+d_a+{\rho }_{m_3}}\text{,}{R}_{\mathit{hm}}^{\ast }=\frac{{\rho }_{m_2}{I}_{\mathit{hm}}^{\ast }+{\lambda }_h^{\ast }{R}_m^{\ast }}{{\mu }_h+{\rho }_{a_1}}\text{,}{R}_{\mathit{am}}^{\ast }=\frac{{\rho }_{m_3}{A}_{\mathit{hm}}^{\ast }+{\rho }_{a_1}{R}_{\mathit{hm}}^{\ast }}{{\mu }_h+d_a}\text{.}\hfill \end{array}$$ (57)

Using the Centre Manifold theory, similar to the analysis of ${\mathcal{E}}_4^{\ast }$, the following Theorem 6 holds for ${\mathcal{E}}_5^{\ast }$.

Theorem 6

For${\mathcal{R}}_{0_n}>1\text{,}{\mathcal{R}}_{0_m}>1\text{and}{\mathcal{R}}_{0_h}>1$, the endemic equilibrium${\mathcal{E}}_5^{\ast }$is locally asymptotically stable.

3. Numerical simulations

We now investigate how solutions of the model (6) can help the medical/scientific community understand and anticipate the spread of impact and co-dynamics of HIV/AIDS and monkeypox. We will use the parameter values given in Table 1, unless explicitly stated otherwise. Although there is little data to estimate many of these parameter values and validate our model for the co-infection of HIV/AIDS and monkeypox, the hope is that the qualitative implications of these simulations can help identify the relative effectiveness of different approaches for bringing the spread of these diseases under control. The model can be used to help identify which parameters are most important in predicting the spread of the diseases and act as a template for more detailed and realistic models of co-circulating human-zoonotic diseases. In Fig. 2 , we consider the impact of co-circulating monkeypox virus in a population where HIV/AIDS is endemic by varying the animal-to-animal monkeypox infection rate.

Fig. 2.

Simulations of model system (6) showing plots of I_m, R_m, I_h, A_h, I_m, R_m, I_hm, A_hm, R_hm and R_am with varying ${\beta }_{n_1}{c}_{n_1}$ among non-human primates and rodents. The direction of the arrow shows an increase in ${\beta }_{n_1}{c}_{n_1}$ among non-humans from 2.0 with a step size of 0.5. Parameters values used are as in Table 1. The direction of the arrow shows the direction of increase.

Fig. 2(a) and (b) increase in monkey pox infection rates result in an increase of monkey pox related sickness in non-humans which in turn leads to an increase of non-humans recovering from the infection. As noted in Fig. 2(c) the monkey pox infected and recovered humans first increase with increase in monkey pox infection rates before declining to asymptotically low levels. Monkey pox virus infected people decline to asymptotically low levels later due to high natural recovery rates accompanied by low animal to human disease transmissibility, as well as HIV infection. A decrease in monkey pox infected individuals translate to decrease in the number of the recovered as well.

An increase in the susceptibles being infected with monkey pox virus, in turn means less will be available to be infected with HIV only in the long run. Less individuals in I _h translate to less AIDS only patients as depicted in Fig. 2(d). Owing to increased susceptibility to monkey pox virus infection for the HIV infected individuals (I _h  and  A _h), this results in an increase of the dually infected individuals with increase in non-human monkey pox infection rates as noted in Fig. 2(e). High monkey pox recovery rates for the HIV negative monkey pox virus infected individuals in the first months coupled with high HIV infection rates makes the monkey pox recovered HIV positive R _hm increase in the first months and then decline. However, they will not decline to asymptotically lower state than those in R _m as some individuals in I _hm recover and enters this class as well. The monkey pox recovered people in the AIDS follow the same trend as R _hm though they will be less in terms of population size as noted in Fig. 2(f).

Results from Fig. 2 suggest infection by HIV greatly enhances monkey pox infections, as in the absence of HIV/ AIDS most people will naturally recover as noted but that is not the case for the dually infected.

4. Conclusion

A deterministic mathematical model for the co-infection of HIV and monkeypox virus in a homogeneously mixing population of people and animals is presented and analysed. Using the Centre Manifold theory and the Lyapunov function approach the endemic equilibria are proved to be locally and globally asymptotically stable, respectively. When co-infection of monkeypox and HIV increases the transmissibility of HIV among humans, then the increasing the basic reproduction number of the monkeypox in the animal population will have a direct impact on the HIV disease transmission in the human population. That is, when the number of animal suffering from monkeypox increases, then the number of humans infected with monkeypox increases. Since some of the monkeypox infected humans are also infected with HIV, then the infectiousness of HIV in the human population increases. Our analysis of the basic reproduction numbers provides conditions under which monkeypox virus and HIV infection are correlated and the threshold for competitive exclusion.

In Central and West Africa due to poverty, people hunt monkeys and rodents for their meat, resulting in an increase in contact rates (which is further increased with urbanization and deforestation) and consequently an increase in the number of monkeypox cases, especially in HIV infected individuals. Thus, alternative source of protein as well as creation of jobs should be provided as incentives to peasants living is such regions in order to encourage them to halt hunting via educational campaigns. However, campaigning for a halt to hunting alone may not be enough, but improving the general well being of people in these communities may play a significant role on controlling some zoonotic diseases such as monkeypox. Given the high prevalence of HIV/AIDS in West and Central Africa, and difficulties these people have in accessing antiretroviral therapy, it may be necessary to re-introduce chicken pox vaccination as it was known to have some positive impact in the control of monkeypox spread. This may in turn lessen the impact monkeypox infections have on HIV/AIDS epidemics.

This study provides the first in-depth mathematical analysis of a model for the transmission dynamics of HIV and monkeypox at the population level. Although the model is simple, it still gave insight into how the diseases are correlated and must be considered together when the viruses are co-circulating. Some of the assumptions in the model, such as the homogenous random mixing, need to be improved upon before the quantitative model predictions are used in practice. The model can also be extended to incorporate preventive and therapeutic measures for HIV such as the use of anti-retroviral therapy, condom use, voluntary HIV testing and screening, and monkeypox such as contact reduction, potential treatment and vaccination strategies.

Appendix A.

Theorem A-1 Castillo–Chavez and Song [6] —

Consider the following general system of ordinary differential equation equations with a parameter ϕ

$$\frac{\mathit{dx}}{\mathit{dt}}=f(x\text{,}\varphi )\text{,}f:{\mathbb{R}}^n\times \mathbb{R}\to \mathbb{R}\text{and}f\in {\mathbb{C}}^2({\mathbb{R}}^n\times \mathbb{R})\text{,}$$ (A-1)

where 0 is an equilibrium of the system, that is f(0,   ϕ)   =   0 for all ϕ, and assume

• A1: $A=D_{x}f(0\text{,}0)=\left(\frac{\partial f_i}{\partial x_j}(0\text{,}0)\right)$is the linearization of system(A-1)around the equilibrium 0 with ϕ evaluated at 0. Zero is a simple eigenvalue of A and other eigenvalues of A have negative real parts;

• A2: Matrix A has a right eigenvector u and a left eigenvector v corresponding to the zero eigenvalue.

Let f _k be the k ^th component of f and

$$\begin{array}{cc}\hfill & a={\displaystyle \sum _{k\text{,}i\text{,}j=1}^n}{v}_k{u}_i{u}_j\frac{{\partial }^2{f}_k}{\partial x_i\partial x_j}(0\text{,}0)\text{,}\hfill \\ \hfill & b={\displaystyle \sum _{k\text{,}i=1}^n}{v}_k{u}_i\frac{{\partial }^2{f}_k}{\partial x_i\partial \varphi }(0\text{,}0)\text{.}\hfill \end{array}$$ (A-2)

The local dynamics of (A-1) around 0 are totally governed by a and b.

• i.a > 0,b > 0. When ϕ < 0 with ∣ϕ∣ ≪  1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium; when 0  <  ϕ  ≪  1, 0 is unstable and there exists a negative and locally asymptotically stable equilibrium;
• ii.a < 0,b < 0. When ϕ < 0 with ∣ϕ∣ ≪  1, 0 unstable; when 0  <  ϕ  ≪  1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium;
• iii.a > 0,b < 0. When ϕ < 0 with ∣ϕ∣ ≪  1, 0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when 0  <  ϕ  ≪  1, 0 is stable, and a positive unstable equilibrium appears;
• iv.a < 0,b > 0. When ϕ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes positive and locally asymptotically stable.