Mathematical Model Analysis for Dynamics and Control of Yellow Fever and Malaria Disease Co-Infections

Abstract

Yellow fever (YF) and malaria co-infections are real public health concerns in Africa, especially in countries such as Nigeria, where mosquitoes carrying both pathogens (Aedes for YF, Anopheles for malaria) coexist. A mathematical model that considers the critical factors influencing the transmission dynamics and control interventions of YF and malaria co-infections is formulated and used to analyse the problem. The essential dynamical features of the model, such as the basic reproduction number and disease-free equilibrium, are determined and analysed. The qualitative analysis of the model illustrates the conditions under which the disease can be eradicated or persists. Further analysis, supported by numerical simulations, reveals the intrinsic dynamics of the model and the impact of control interventions such as yellow fever vaccination, use of insecticide-treated mosquito nets, treatment of malaria-infected humans, and use of insecticides. The results of the analysis demonstrate the impact of interventions; specifically, effective implementations of interventions such as yellow fever vaccination, use of insecticide-treated mosquito nets, and use of insecticides appear to have a significant impact in eradicating YF and malaria co-infections in endemic areas. Effective treatment of malaria-infected humans may lead to a decrease in infections but might not necessarily lead to eradicating infections in endemic areas. These findings are expected to aid in improving the management of YF and malaria co-infections in endemic regions for expeditious disease eradication.

1. Introduction

Yellow fever (YF) is a severe, mosquito-borne viral disease found in tropical regions of Africa and the Americas. Though no specific antiviral treatment exists, a vaccine that guarantees lifelong protection is available [1]. Most infections are mild, but 15% develop severe symptoms, including liver/kidney damage, bleeding, and potential death [1]. Prevention focuses on vaccination and the use of insecticide-treated mosquito nets to avoid mosquito bites. Globally, there are about 200,000 cases of yellow fever annually, resulting in approximately 30,000 deaths per year [1]. The majority of this disease burden (around 90%) occurs in Africa [2]. Yellow fever is transmitted to humans through the bite of infected mosquitoes, primarily Aedes and Haemagogus species [3]. It is a viral disease and does not spread person-to-person. The incubation period of yellow fever is 3–6 days [3].

Malaria is a serious parasitic disease transmitted through the bite of infected female Anopheles mosquitoes. Malaria symptoms include fever, chills, headache, muscle aches, and fatigue, and typically start a few weeks after an infected mosquito bite. The incubation period of malaria usually ranges from 7 to 30 days [4]. Globally in 2024, there were an estimated 265 million malaria cases and 579,000 malaria deaths in 80 countries [1]. Malaria is prevalent in tropical/subtropical regions (Sub-Saharan Africa, South America, Southeast Asia), with the African Region being home to 95% of malaria cases and 95% of malaria deaths, according to a report in 2024 [2]. Prevention of malaria involves using repellents or mosquito nets to avoid mosquito bites, and treatment using Artemisinin-based combination therapies (ACTs) [2].

YF and malaria co-infections are a major public health concern in Africa, especially Nigeria, where mosquitoes carrying both pathogens (Aedes for YF, Anopheles for malaria) coexist. Co-infection is a serious public health concern due to increased severity of symptoms and the diagnosis challenges involved [2]. This study is an attempt to use a mathematical model to investigate the transmission dynamics and control interventions of YF and malaria co-infections.

Mathematical models have been successfully used to study real-life problems, such as the spread of infectious diseases [5,6,7,8,9,10,11,12,13,14]. Some of the important research findings that utilized mathematical models to study YF infections are reviewed and systematically presented here. For instance, Handari et al. [15] use a new mathematical model for yellow fever to analyse the impact and optimal control of three different intervention strategies on the dynamics of yellow fever transmission: vaccination, hospitalization, and fumigation. Donkoh et al. [16] use a mathematical model to analyse the transmission dynamics of yellow fever, and the results of their studies reveal the significance of focusing on severe cases for controlling outbreaks. Martorano Raimundo et al. [17] use a mathematical dynamical model with vaccination to analyse the transmission dynamics of yellow fever. They discovered, using their model, that yellow fever can be eradicated from the human population if the vaccination rate is greater than a threshold; otherwise, the disease will persist and become endemic. Otoo et al. [18] propose a deterministic mathematical model in the form of nonlinear ordinary differential equations and use it to gain insight into the transmission dynamics of yellow fever for an unvaccinated community where the disease is endemic. Kalra & Ratti [19] formulated a nonlinear mathematical model for yellow fever and used it to analyse the impact of awareness programmes on disease transmission. Alsowait et al. [20] developed a dynamical system model that integrates vaccination, active case detection, and hospitalization, and use it to analyse the transmission dynamics of yellow fever infections and examined prevention strategies. Miller et al. [21] formulate a discrete-time delay mathematical model and use it to investigate the yellow fever transmission pattern. More important research findings that use a mathematical model to study yellow fever transmission dynamics and control interventions can be found in [22,23,24,25].

Furthermore, some essential research findings that used a mathematical model to study malaria transmissions are reviewed here. Collins & Duffy [9] formulate a mathematical model that incorporates drug resistance, treatment, and the use of mosquito nets as preventive strategies, and use it to analyse the transmission dynamics of malaria in Nigeria. Herdicho et al. [26] construct a mathematical model that incorporates seasonal factors, insecticide, and treatment, and use it to analyse the optimal control of malaria transmission. Okuneye & Gumel [12] develop a non-autonomous mathematical model and use it to analyse the impact of variability in temperature and rainfall on malaria transmission dynamics. Kim et al. [27] develop a malaria transmission model that incorporates climate-dependent parameters and use it to analyse the impact of climate change on malaria transmission. They also investigate the potential risk of malaria outbreaks through the computation of the seasonal reproduction number and vectorial capacity. Agusto et al. [10] formulate a mathematical model for malaria and use it to investigate the impact of bed nets. More important research findings that use mathematical models to study the transmission dynamics and control interventions of malaria can be found in [28,29,30,31,32].

An experimental study conducted by Ogwezzy et al. [33] using patients clinically diagnosed with malaria in Delta State, Nigeria, as a case study revealed the existence of co-infections of yellow fever and malaria with a prevalence of 1.1%. Another experimental study by Rugarabamu et al. [34] use febrile patients seeking health care in Tanzania as a case study, revealing the existence of co-infections of viral haemorrhagic fevers and malaria in Tanzania, and consequently discovering that the co-infection affects more older people than younger people. There is no doubt that these research findings have made significant contributions to the transmission dynamics and control interventions of yellow fever, malaria, and their co-infections. However, none of these studies used a mathematical model to analyse the transmission dynamics and control interventions of yellow fever and malaria co-infections. The aim of this study is to fill this gap in the literature by formulating a mathematical model for the co-infections of yellow fever and malaria, and consequently using the model to study the transmission dynamics and control interventions of yellow fever and malaria. The results of this study are expected to assist in improving the management of yellow fever and malaria co-infections in endemic regions for expeditious disease eradication. The remaining part of this paper is organised as follows: Model Development is presented in Section 2, Analyses and Results are presented in Section 3, Numerical Simulations are captured in Section 4, and lastly, the Discussion is presented in Section 5.

2. Model Development

Yellow fever is a viral infection transmitted through the bite of infected Aedes or Haemogogus mosquitoes. Currently, the disease has no cure, but a single dose of the yellow fever vaccine can provide lifelong protection against infection. On the other hand, malaria is a parasitic infection transmitted through the bite of an infected female Anopheles mosquito. Malaria is treatable using Artemisinin-based combination therapy, but there is no approved vaccination for prevention. The co-infection of these two diseases is possible, especially in areas where these two diseases are endemic.

A mathematical model for yellow fever and malaria disease co-infection is formulated based on the following assumptions. The model comprises the human population, the Aedes or Haemogogus mosquito population, and the female Anopheles mosquito population. The total human population $N\left(t\right)$ at time (t) is partitioned into five sub-populations, namely, the susceptible human population ($S\left(t\right)$), the human population vaccinated with the yellow fever vaccine ($V\left(t\right)$), the human population infected with yellow fever ($I_a\left(t\right)$), the human population infected with malaria ($I_b\left(t\right)$), and the human population co-infected with yellow fever and malaria ($I_{ab}\left(t\right)$). The Aedes or Haemogogus mosquito populations that host/transmit the yellow fever virus is partitioned into two sub-populations: susceptible mosquitoes ($X_a\left(t\right)$) and infected mosquitoes ($Y_a\left(t\right)$). Similarly, the female Anopheles mosquito population that hosts/transmits the malaria parasite is partitioned into two sub-populations: susceptible mosquitoes ($X_b\left(t\right)$) and infected mosquitoes ($Y_b\left(t\right)$).

The interactions among the human, Aedes or Haemogogus mosquitoes, and female Anopheles mosquito sub-populations that give rise to yellow fever and malaria co-infection dynamics are described as follows. Humans are recruited into the $S\left(t\right)$ through birth at a rate ${\mathsf{\Lambda }}^{*}$ and leave the class as they get infected with yellow fever or malaria at a rate ${\beta }_a^{*}$, ${\beta }_b^{*}$, respectively. The $S\left(t\right)$ gets vaccinated with the yellow fever vaccine at a rate $\varphi$ with a vaccine whose efficacy is $\epsilon$. Since yellow fever has no treatment, those infected with it die due to the infection at a rate $\delta$. On the other hand, malaria has a cure, so those infected with malaria can recover due to effective treatment at a rate $\gamma$, and those not properly treated can die due to the disease at a rate ${\delta }_b^{*}$ or ${\delta }_{ab}^{*}$ for $I_a\left(t\right)$ or $I_{ab}\left(t\right)$, respectively. Natural death occurs in all human compartments at a rate $\mu$.

For the Aedes or Haemogogus mosquitoes, $X_a\left(t\right)$ increase through recruitment at a rate ${\mathsf{\Lambda }}_a^{*}$ and decrease as they get infected with infected $I_a\left(t\right)$ or $I_{ab}\left(t\right)$ at a rate ${\alpha }_a^{*}$. Both $X_a\left(t\right)$ and $Y_a\left(t\right)$ die naturally at a rate ${\xi }_a^{*}$. Using appropriate insecticides or fumigation (control interventions) also kill $X_a$ and $Y_a$ at a rate ${\sigma }_a^{*}$. Similarly, the female Anopheles mosquito $X_b$ increase through recruitment at a rate ${\mathsf{\Lambda }}_b^{*}$ and decrease as they get infected with $I_b\left(t\right)$, or $I_{ab}\left(t\right)$ at a rate ${\alpha }_b^{*}$. Both $X_b\left(t\right)$ and $Y_b\left(t\right)$ die naturally at a rate ${\xi }_b^{*}$. Applying insecticides or fumigation (control interventions) also kills $X_b\left(t\right)$ and $Y_b\left(t\right)$ at a rate ${\sigma }_b^{*}$. One of the methods of preventing mosquito bites is by using an insecticide-treated mosquito net. This is considered by assuming that the use of an insecticide-treated net reduces yellow fever or malaria transmission by $\kappa$. Based on these formulations, the following yellow fever and malaria co-infection model with control interventions is obtained:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dS\left(t\right)}{dt}}& =& {\mathsf{\Lambda }}^{*}-(1-\kappa ){\beta }_a^{*}S\left(t\right)Y_a\left(t\right)-(1-\kappa ){\beta }_b^{*}S\left(t\right)Y_b\left(t\right)-({\varphi }^{*}+\mu )S\left(t\right)+{\gamma }^{*}{I}_b\left(t\right),\hfill \\ \hfill {\displaystyle \frac{dV\left(t\right)}{dt}}& =& {\varphi }^{*}S\left(t\right)-(1-\kappa )(1-\epsilon ){\beta }_a^{*}V\left(t\right)Y_a\left(t\right)-(1-\kappa ){\beta }_b^{*}V\left(t\right)Y_b\left(t\right)-\mu V\left(t\right),\hfill \\ \hfill {\displaystyle \frac{d{I}_a\left(t\right)}{dt}}& =& (1-\kappa ){\beta }_a^{*}S\left(t\right)Y_a\left(t\right)+(1-\kappa )(1-\epsilon ){\beta }_a^{*}V\left(t\right)Y_a\left(t\right)+{\gamma }^{*}{I}_{ab}\left(t\right)-(1-\kappa ){\beta }_{ab}^{*}{I}_a\left(t\right)Y_b\left(t\right)-(\mu +{\delta }_a^{*})I_a\left(t\right),\hfill \\ \hfill {\displaystyle \frac{d{I}_b\left(t\right)}{dt}}& =& (1-\kappa ){\beta }_b^{*}(S\left(t\right)+V\left(t\right))Y_b\left(t\right)-(1-\kappa ){\beta }_{ba}^{*}{I}_b\left(t\right)Y_a\left(t\right)-(\mu +{\delta }_b^{*}+{\gamma }^{*})I_b\left(t\right),\hfill \\ \hfill {\displaystyle \frac{d{I}_{ab}\left(t\right)}{dt}}& =& (1-\kappa ){\beta }_{ab}^{*}{I}_a\left(t\right)Y_b\left(t\right)+(1-\kappa ){\beta }_{ba}^{*}{I}_b\left(t\right)Y_a\left(t\right)-(\mu +{\delta }_{ab}^{*}+{\gamma }^{*})I_{ab}\left(t\right),\hfill \\ \hfill {\displaystyle \frac{d{X}_a\left(t\right)}{dt}}& =& {\mathsf{\Lambda }}_a^{*}-(1-\kappa ){\alpha }_a^{*}{X}_a\left(t\right)(I_a\left(t\right)+I_{ab}\left(t\right))-({\xi }_a^{*}+{\sigma }_a^{*})X_a\left(t\right),\hfill \\ \hfill {\displaystyle \frac{d{Y}_a\left(t\right)}{dt}}& =& (1-\kappa ){\alpha }_a^{*}{X}_a\left(t\right)(I_a\left(t\right)+I_{ab}\left(t\right))-({\xi }_a^{*}+{\sigma }_a^{*})X_a\left(t\right),\hfill \\ \hfill {\displaystyle \frac{d{X}_b\left(t\right)}{dt}}& =& {\mathsf{\Lambda }}_b^{*}-(1-\kappa ){\alpha }_b^{*}{X}_b\left(t\right)(I_b\left(t\right)+I_{ab}\left(t\right))-({\xi }_b^{*}+{\sigma }_b^{*})X_b\left(t\right),\hfill \\ \hfill {\displaystyle \frac{d{Y}_b\left(t\right)}{dt}}& =& (1-\kappa ){\alpha }_b^{*}{X}_b\left(t\right)(I_b\left(t\right)+I_{ab}\left(t\right))-({\xi }_b^{*}+{\sigma }_b^{*})Y_b\left(t\right).\hfill \end{array}$$

(1)

The description of variables and parameters are presented in

Table 1. Description for variables for model (1).

Variables	Description	Unit
$N_h\left(t\right)$	Total population of humans	Humans km−2
$S\left(t\right)$	Susceptible population of humans	Humans km−2
$V\left(t\right)$	Humans vaccinated with yellow fever vaccine	Humans km−2
$I_a\left(t\right)$	Humans infected with yellow fever	Humans km−2
$I_b\left(t\right)$	Humans infected with malaria	Humans km−2
$I_{ab}\left(t\right)$	Humans co-infected with yellow fever and malaria	Humans km−2
$M_a\left(t\right)$	Total population of mosquitoes that can host yellow fever virus	Vectors km−2
$X_a\left(t\right)$	Susceptible mosquitoes that can host yellow fever virus	Vectors km−2
$Y_a\left(t\right)$	Infected mosquitoes that transmit yellow fever virus	Vectors km−2
$M_b\left(t\right)$	Total population of mosquitoes that can host malaria parasite	Vectors km−2
$X_b\left(t\right)$	Susceptible mosquitoes that can host malaria parasite	Vectors km−2
$Y_b\left(t\right)$	Infected mosquitoes that transmit malaria parasite	Vectors km−2

and

Table 2. Description of parameters for model (1).

Parameters	Description	Unit
${\mathsf{\Lambda }}^{*}$	Recruitment rate of humans into $S\left(t\right)$	Humans km−2 Year−1
${\beta }_a^{*}$	Transmission rate from $Y_a\left(t\right)$ to $S\left(t\right)$	km−2 Vector−1 Year−1
${\beta }_b^{*}$	Transmission rate from $Y_b\left(t\right)$ to $S\left(t\right)$	km−2 Vector−1 Year−1
${\beta }_{ab}^{*}$	Transmission rate from $Y_b\left(t\right)$ to $I_a\left(t\right)$	km−2 Vector−1 Year−1
${\beta }_{ba}^{*}$	Transmission rate from $Y_a\left(t\right)$ to $I_b\left(t\right)$	km−2 Vector−1 Year−1
$\mu$	Natural mortality rate of humans	Year−1
${\delta }_a$	Disease induces mortality rate of $I_a\left(t\right)$	Year−1
${\delta }_b$	Disease induces mortality rate of $I_b\left(t\right)$	Year−1
${\delta }_{ab}$	Disease induces mortality rate of $I_{ab}\left(t\right)$	Year−1
${\varphi }^{*}$	Yellow fever vaccination rate	Year−1
$\epsilon$	Efficacy of the yellow fever vaccine	Dimensionless
${\gamma }^{*}$	Recovery rate due to malaria treatment	Year−1
${\mathsf{\Lambda }}_a^{*}$	Recruitment rate of $X_a\left(t\right)$	Vectors km−2 Year−1
${\mathsf{\Lambda }}_b^{*}$	Recruitment rate of $X_b\left(t\right)$	Vectors km−2 Year−1
${\alpha }_a^{*}$	Transmission rate from $Y_a\left(t\right)$ to $S\left(t\right)$	km−2 Human−1 Year−1
${\alpha }_b^{*}$	Transmission rate from $Y_b\left(t\right)$ to $S\left(t\right)$	km−2 Human−1 Year−1
${\xi }_a^{*}$	Natural death rate of $X_a\left(t\right)$ and $Y_a\left(t\right)$	Year−1
${\xi }_b^{*}$	Natural death rate of $X_b\left(t\right)$ and $Y_b\left(t\right)$	Year−1
${\sigma }_a^{*}$	Death rate of $X_a\left(t\right)$ and $Y_a\left(t\right)$ due to control interventions	Year−1
${\sigma }_b^{*}$	Death rate of $X_b\left(t\right)$ and $Y_b\left(t\right)$ due to control interventions	Year−1
$\kappa$	Reduction in transmission rate due to use of mosquito net	Dimensionless

. A schematic representation of model (1) is given in Figure 1.

3. Analyses and Results

3.1. Model Transformation

The mathematical model (1) comprises both the human population and mosquito vector populations, each having different measurement units. Therefore, to conduct a more accurate analysis of the model, it is crucial to non-dimensionalise it by converting all variables and parameters into dimensionless quantities. This aids in revealing hidden relationships and dependencies between variables and parameters, thereby providing deeper insight into the model dynamics [35,36]. Rescaling model (1) such that $\tau =\mu t,s={\displaystyle \frac{S}{N_0}},v={\displaystyle \frac{V}{N_0}},i_a={\displaystyle \frac{I_a}{N_0}},i_b={\displaystyle \frac{I_b}{N_0}},i_{ab}={\displaystyle \frac{I_{ab}}{N_0}},x_a={\displaystyle \frac{X_a}{M_a^0}},y_a={\displaystyle \frac{Y_a}{M_b^0}},x_b={\displaystyle \frac{X_b}{M_b^0}},y_b={\displaystyle \frac{Y_b}{M_b^0}},{\beta }_a={\displaystyle \frac{{\beta }_a^{*}{M}_a^0}{\mu }},{\beta }_b={\displaystyle \frac{{\beta }_b^{*}{M}_b^0}{\mu }},{\beta }_{ab}={\displaystyle \frac{{\beta }_{ab}^{*}{M}_b^0}{\mu }},{\beta }_{ba}={\displaystyle \frac{{\beta }_{ba}^{*}{M}_a^0}{\mu }},{\alpha }_a={\displaystyle \frac{{\alpha }_a^{*}{N}_0}{\mu }},{\alpha }_b={\displaystyle \frac{{\alpha }_b^{*}{N}_0}{\mu }},\mathsf{\Lambda }={\displaystyle \frac{{\mathsf{\Lambda }}^{*}}{\mu N_0}},{\mathsf{\Lambda }}_a={\displaystyle \frac{{\mathsf{\Lambda }}_a^{*}}{\mu M_a^0}},{\mathsf{\Lambda }}_b={\displaystyle \frac{{\mathsf{\Lambda }}_b^{*}}{\mu M_b^0}},$ ${\xi }_a={\displaystyle \frac{{\xi }_a^{*}}{\mu }}$, ${\xi }_b={\displaystyle \frac{{\xi }_b^{*}}{\mu }}$, $\varphi ={\displaystyle \frac{{\varphi }^{*}}{\mu }},\gamma ={\displaystyle \frac{{\gamma }^{*}}{\mu }},{\sigma }_a={\displaystyle \frac{{\sigma }_a^{*}}{\mu }},{\sigma }_b={\displaystyle \frac{{\sigma }_b^{*}}{\mu }},M_a^0=X_a\left(0\right)+Y_a\left(0\right),M_b^0=X_b\left(0\right)+Y_b\left(0\right),N_0=S\left(0\right)+V\left(0\right)+I_a\left(0\right)+I_b\left(0\right)+I_{ab}\left(0\right)$, and if ${\mathsf{\Lambda }}^{*}=\mu N_0,{\mathsf{\Lambda }}_a^{*}=\mu M_a^0,{\mathsf{\Lambda }}_b^{*}=\mu M_b^0$, the dimensionless version of model (1) is obtained as follows:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{ds\left(\tau \right)}{d\tau }}& =& 1-(1-\kappa ){\beta }_{a}s\left(\tau \right)y_a\left(\tau \right)-(1-\kappa ){\beta }_{b}s\left(\tau \right)y_b\left(\tau \right)-(\varphi +1)s\left(\tau \right)+\gamma i_b\left(\tau \right),\hfill \\ \hfill {\displaystyle \frac{dv\left(\tau \right)}{d\tau }}& =& \varphi s\left(\tau \right)-(1-\kappa )(1-\epsilon ){\beta }_{a}v\left(\tau \right)y_a\left(\tau \right)-(1-\kappa ){\beta }_{b}v\left(\tau \right)y_b\left(\tau \right)-v\left(\tau \right),\hfill \\ \hfill {\displaystyle \frac{d{i}_a\left(\tau \right)}{d\tau }}& =& (1-\kappa ){\beta }_{a}s\left(\tau \right)y_a\left(\tau \right)+(1-\kappa )(1-\epsilon ){\beta }_{a}v\left(\tau \right)y_a\left(\tau \right)+\gamma i_{ab}\left(\tau \right)-(1-\kappa ){\beta }_{ab}{i}_a\left(\tau \right)y_b\left(\tau \right)-(1+{\delta }_a)i_a\left(\tau \right),\hfill \\ \hfill {\displaystyle \frac{d{i}_b\left(\tau \right)}{d\tau }}& =& (1-\kappa ){\beta }_b(s\left(\tau \right)+v\left(\tau \right))y_b\left(\tau \right)-(1-\kappa ){\beta }_{ba}{i}_b\left(\tau \right)y_a\left(\tau \right)-(1+{\delta }_b+\gamma )i_b\left(\tau \right),\hfill \\ \hfill {\displaystyle \frac{d{i}_{ab}\left(\tau \right)}{d\tau }}& =& (1-\kappa ){\beta }_{ab}{i}_a\left(\tau \right)y_b\left(\tau \right)+(1-\kappa ){\beta }_{ba}{i}_b\left(\tau \right)y_a\left(\tau \right)-(1+{\delta }_{ab}+\gamma )i_{ab}\left(\tau \right),\hfill \\ \hfill {\displaystyle \frac{d{x}_a\left(\tau \right)}{d\tau }}& =& 1-(1-\kappa ){\alpha }_a{x}_a\left(\tau \right)(i_a\left(\tau \right)+i_{ab}\left(\tau \right))-({\xi }_a+{\sigma }_a)x_a\left(\tau \right),\hfill \\ \hfill {\displaystyle \frac{d{y}_a\left(\tau \right)}{d\tau }}& =& (1-\kappa ){\alpha }_a{x}_a\left(\tau \right)(i_a\left(\tau \right)+i_{ab}\left(\tau \right))-({\xi }_a+{\sigma }_a)y_a\left(\tau \right),\hfill \\ \hfill {\displaystyle \frac{d{x}_b\left(\tau \right)}{d\tau }}& =& 1-(1-\kappa ){\alpha }_b{x}_b\left(\tau \right)(i_b\left(\tau \right)+i_{ab}\left(\tau \right))-({\xi }_b+{\sigma }_b)x_b\left(\tau \right),\hfill \\ \hfill {\displaystyle \frac{d{y}_b\left(\tau \right)}{d\tau }}& =& (1-\kappa ){\alpha }_b{x}_b\left(\tau \right)(i_b\left(\tau \right)+i_{ab}\left(\tau \right))-({\xi }_b+{\sigma }_b)y_b\left(\tau \right).\hfill \end{array}$$

(2)

The following assumptions are introduced for simplicity where appropriate: $a_1=(1-\kappa ){\beta }_a,a_2=(1-\kappa )(1-\epsilon ){\beta }_a,a_3=1+{\delta }_a,a_4=(1-\kappa ){\alpha }_a,a_5={\xi }_a+{\sigma }_a,b_1=(1-\kappa ){\beta }_b,b_3=1+\gamma +{\delta }_b,b_4=(1-\kappa ){\alpha }_b,b_5={\xi }_b+{\sigma }_b,c_1=1+\varphi ,c_2=(1-\kappa ){\beta }_{ab},c_3=(1-\kappa ){\beta }_{ba},c_4=1+\gamma +{\delta }_{ab}$.

3.2. Yellow Fever Sub-Model

The yellow fever sub-model is a special case of the co-infection model (2) where yellow fever is the only disease affecting the population. Qualitative analysis of this special case is crucial for understanding the transmission dynamics and control interventions of yellow fever and is presented in this section. The yellow fever sub-model is determined by setting $i_a=i_{ab}=x_b=y_b=0$ in the model (2), to obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{ds\left(\tau \right)}{d\tau }}& =& 1-a_{1}s\left(\tau \right)y_a\left(\tau \right)-c_{1}s\left(\tau \right),\hfill \\ \hfill {\displaystyle \frac{dv\left(\tau \right)}{d\tau }}& =& \varphi s\left(\tau \right)-a_{2}v\left(\tau \right)y_a\left(\tau \right)-v\left(\tau \right),\hfill \\ \hfill {\displaystyle \frac{d{i}_a\left(\tau \right)}{d\tau }}& =& a_{1}s\left(\tau \right)y_a\left(\tau \right)+a_{2}v\left(\tau \right)y_a\left(\tau \right)-a_3{i}_a\left(\tau \right),\hfill \\ \hfill {\displaystyle \frac{d{x}_a\left(\tau \right)}{d\tau }}& =& 1-a_4{x}_a\left(\tau \right)i_a\left(\tau \right)-a_5{x}_a\left(\tau \right),\hfill \\ \hfill {\displaystyle \frac{d{y}_a\left(\tau \right)}{d\tau }}& =& a_4{x}_a\left(\tau \right)i_a\left(\tau \right)-a_5{y}_a\left(\tau \right).\hfill \end{array}$$

(3)

The transmission dynamics and the impact of control interventions of yellow fever transmissions, described by the analysis of sub-model (3), are presented here.

3.2.1. Equilibrium Points of the Yellow Fever Sub-Model (3)

Most mathematical epidemiological models have two important equilibrium points, namely: the disease-free equilibrium (DFE) and endemic equilibrium (EE) [5,6]. The DFE is the equilibrium point of the model when there is no disease in the system, whereas the EE is the equilibrium point of the model when there is disease in the system. The analytical representation of the DFE of the yellow fever sub-model (3) is given by

$$\begin{array}{c}\hfill (s^0,v^0,i_a^0,x_a^0,y_a^0)=\left({\displaystyle \frac{1}{c_1}},{\displaystyle \frac{\varphi }{c_1}},0,{\displaystyle \frac{1}{a_5}},0\right).\end{array}$$

(4)

The transmission dynamics of the yellow fever sub-model (3) about the EE will be explored numerically. This is due to the complexity of the equations involved in the analytical representation of the EE of the yellow fever sub-model (3).

3.2.2. The Basic Reproduction Number of the Yellow Fever Sub-Model (3)

The basic reproduction number of an epidemiological model represents the average number of infectious cases generated by one case in a population where all individuals are susceptible to the infection [5,6]. The value of the basic reproduction number indicates whether an outbreak will be eradicated or persist [5,6]. The basic reproduction number of the yellow fever sub-model (3), denoted by ${\mathcal{R}}_0^a$, is computed using the next generation matrix method [5]. The next generation matrix of the yellow fever sub-model (3) is

$$\begin{array}{c}\hfill F{V}^{-1}=\left(\begin{array}{cc}0& {\displaystyle \frac{a_1{s}^0+a_2{v}^0}{a_5}}\\ {\displaystyle \frac{a_4{x}_a^0}{a_5}}& 0\end{array}\right),\end{array}$$

(5)

where

$$\begin{array}{c}\hfill F=\left(\begin{array}{cc}0& a_1{s}^0+a_2{v}^0\\ a_4{x}_a^0& 0\end{array}\right)\mathrm{and}V=\left(\begin{array}{cc}{a}_3& 0\\ 0& a_5\end{array}\right).\end{array}$$

The dominant eigenvalue of the next generation matrix $F{V}^{-1}$ is the ${\mathcal{R}}_0^a$ and is determined as

$$\begin{array}{c}\hfill {\mathcal{R}}_0^a=\sqrt{{\displaystyle \frac{(a_1{s}^0+a_2{v}^0)}{a_5}}·{\displaystyle \frac{a_4{x}_a^0}{a_3}}}.\end{array}$$

(6)

The ${\mathcal{R}}_0^a$ can be rewritten explicitly as

$$\begin{array}{c}\hfill {\left({\mathcal{R}}_0^a\right)}^2={\displaystyle \frac{((1-\kappa ){\beta }_a{s}^0+(1-\kappa )(1-\epsilon ){\beta }_a{v}^0)}{({\xi }_a+{\sigma }_a)}}·{\displaystyle \frac{(1-\kappa ){\alpha }_a{x}_a^0}{(1+{\delta }_a)}}.\end{array}$$

(7)

3.2.3. Stability Analysis of the Yellow Fever Sub-Model (3)

The stability analysis of a model about its equilibrium point describes the dynamics of the model about the equilibrium point. The stability analysis of the yellow fever sub-model (3) about the DFE (4) is summarised in the following theorems.

Theorem 1.

Yellow fever sub-model (3) is locally asymptotically stable about the DFE (4) provided ${\mathcal{R}}_0^a<1$.

Proof of Theorem 1.

To prove Theorem 1, it suffices to show that all the eigenvalues of the Jacobian of the yellow fever sub-model (3) about the DFE point (4) have a negative real part. The Jacobian of the yellow fever sub-model (3) about the DFE (4) is

$$J_0=\left(\begin{array}{ccccc}-c_1& 0& 0& 0& -a_1{s}^0\\ \varphi & -1& 0& 0& -a_2{v}^0\\ 0& 0& -a_3& 0& a_1{s}^0+a_2{v}^0\\ 0& 0& -a_4{x}_a^0& -a_5& 0\\ 0& 0& a_4{x}_a^0& 0& -a_5\end{array}\right).$$

(8)

The eigenvalues of the Jacobian are

$$\begin{array}{ccc}\hfill {\lambda }_1& =& -1,\hfill \\ \hfill {\lambda }_2& =& -c_1,\hfill \\ \hfill {\lambda }_3& =& -a_5,\hfill \\ \hfill {\lambda }_4& =& {\displaystyle \frac{-(a_3+a_5)-\sqrt{{(a_3+a_5)}^2+4a_3{a}_5({\left({\mathcal{R}}_0^a\right)}^2-1)}}{2}},\hfill \\ \hfill {\lambda }_5& =& {\displaystyle \frac{-(a_3+a_5)+\sqrt{{(a_3+a_5)}^2+4a_3{a}_5({\left({\mathcal{R}}_0^a\right)}^2-1)}}{2}}.\hfill \end{array}$$

Obviously, ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4$ are negative. However, ${\lambda }_5<0$ if ${\mathcal{R}}_0^a<1$. Thus, we conclude that the sub-model (3) is locally asymptotically stable about the DFE (4) provided ${\mathcal{R}}_0^a<1$. □

The biological implication of Theorem 1 is that yellow fever can be eradicated if the initial number of infected humans and mosquito vectors is sufficiently small (within the neighbourhood of the DFE) and ${\mathcal{R}}_0^a<1$. However, if the initial infected population is large, the transmission dynamics of the yellow fever sub-model about the DFE require the investigation of the global stability of the yellow fever sub-model (3) about the DFE (4). The global stability analysis of the yellow fever sub-model (3) is presented in Theorem 2.

Theorem 2.

The yellow fever sub-model (3) is globally asymptotically stable about the DFE (4) provided ${\mathcal{R}}_0^a<1$.

The proof of Theorem 2 will be established using a stability result in [37], which is stated in Lemma 1.

Lemma 1

([37]). Consider a model system written in the form

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d{Z}_1}{dt}}& =& F(Z_1,Z_2),\hfill \\ \hfill {\displaystyle \frac{d{Z}_2}{dt}}& =& G(Z_1,Z_2),G(Z_1,0)=0,\hfill \end{array}$$

(9)

where $Z_1\in R^m$ and $Z_2\in R^n$. $Z_0=(Z_1^{*},0)$ denotes the disease-free equilibrium of the system. Assume that

H1.

For ${\displaystyle \frac{d{Z}_1}{dt}}=F(Z_1,0)$, $Z_1^{*}$ is globally asymptotically stable;

H2.

$G(Z_1,Z_2)=A{Z}_2-\widehat{G}(Z_1,Z_2)$, $\widehat{G}(Z_1,Z_2)\ge 0$ for $(Z_1,Z_2)\in \mathsf{\Omega },$ where the Jacobian $A={\displaystyle \frac{\partial G}{\partial Z_2}}(Z_1,0)$ is an M–matrix (the off-diagonal elements of A are non-negative) and Ω is the region where the model makes biological sense.

Then the disease-free equilibrium $Z_0$ is globally asymptotically stable provided that the basic reproduction number is less than one [37].

Proof of Lemma 1.

To apply Lemma 1, it suffices to show that the disease-free part of model (3) is globally stable (condition (H1)) and the disease part of model (3) satisfies condition H2 of the Lemma. From the yellow fever sub-model Equation (3), let $Z_1=(s,v,x_a),Z_2=(i_a,y_a)$. So, the disease-free part of model (3) is given by

$${\displaystyle \frac{d{Z}_1}{d\tau }}=F(Z_1,0)=\left(\begin{array}{c}{\displaystyle \frac{ds\left(\tau \right)}{d\tau }}\\ {\displaystyle \frac{dv\left(\tau \right)}{d\tau }}\\ {\displaystyle \frac{d{x}_a\left(\tau \right)}{d\tau }}\end{array}\right)=\left(\begin{array}{c}1-c_{1}s\left(\tau \right)\\ \varphi s\left(\tau \right)-v\left(\tau \right)\\ 1-a_5{x}_a\left(\tau \right)\end{array}\right),$$

(10)

while the disease part of model (3) is

$$\begin{array}{c}\hfill {\displaystyle \frac{d{Z}_2}{d\tau }}=G(Z_1,Z_2)=\left(\begin{array}{c}{a}_{1}s\left(\tau \right)y_a\left(\tau \right)+a_{2}v\left(\tau \right)y_a\left(\tau \right)-a_3{i}_a\left(\tau \right)\\ a_4{x}_a\left(\tau \right)i_a\left(\tau \right)-a_5{x}_a\left(\tau \right)\end{array}\right).\end{array}$$

(11)

The disease-free part (10) is made up of a set of linear ordinary differential equations, and its exact solution can be obtained as $s\left(\tau \right)={\displaystyle \frac{1}{c_1}}+A_1{e}^{-\tau },v\left(\tau \right)={\displaystyle \frac{\varphi }{c_1}}+\left({\displaystyle \frac{A_1\varphi }{1-c_1}}+A_2\right)e^{-\tau },x_a\left(\tau \right)={\displaystyle \frac{1}{a_5}}+A_3{e}^{-a_5\tau }$, where $A_1,A_2,A_3$ are constants of integration. As $\tau ⟶\infty$, $s\left(\tau \right)⟶s^0={\displaystyle \frac{1}{c_1}},v\left(\tau \right)⟶v^0={\displaystyle \frac{\varphi }{c_1}}$ and $x_a\left(\tau \right)⟶x_a^0={\displaystyle \frac{1}{a_5}}$, confirming that the disease-free part of model (3) is globally asymptotically stable based on Lemma 1 (i.e., based on the model and its assumptions).

For the disease part, the Jacobian of (11) about the disease-free equilibrium is determined as

$$\begin{array}{c}\hfill A=\left(\begin{array}{cc}-a_3& a_1{s}^0+a_2{v}^0\\ a_4{x}_a^0& a_5\end{array}\right),\end{array}$$

(12)

where A is an M–matrix with all its off-diagonal elements non-negative. From Equations (11) and (12), the term $\widehat{G}(Z_1,Z_2)$ is determined as

$$\begin{array}{c}\hfill \widehat{G}(Z_1,Z_2)=\left(\begin{array}{c}{a}_1{y}_a(s^0-s)+a_2{y}_a(v^0-v)\\ a_4{i}_4(x_a^0-x_a)\end{array}\right).\end{array}$$

(13)

Clearly, $\widehat{G}(Z_1,Z_2)\ge 0$, since $s^0\ge s$, $v^0\ge v$, and $x_a^0\ge x_a$, confirming that condition $\left(H_2\right)$ holds; thus, the proof is complete. □

The epidemiological implication of this result is that the yellow fever disease outbreak could be eradicated irrespective of the size of the infected humans and mosquito vectors at the initial stage of the outbreak, provided the control interventions are effective enough to keep ${\mathcal{R}}_0^a$ below unity.

3.3. Malaria Sub-Model

The malaria sub-model is another special case of the co-infection model (2) where malaria is the only disease affecting the population. Qualitative analysis of this special case is crucial for understanding the transmission dynamics and control interventions of malaria and is presented in this section. The malaria sub-model is obtained by setting $i_a=i_{ab}=x_a=y_a=0$ in the model (2), to obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{ds\left(\tau \right)}{d\tau }}& =& 1-b_{1}s\left(\tau \right)y_b\left(\tau \right)-c_{1}s\left(\tau \right)+\gamma i_b\left(\tau \right),\hfill \\ \hfill {\displaystyle \frac{dv\left(\tau \right)}{d\tau }}& =& \varphi s\left(\tau \right)-b_{1}v\left(\tau \right)y_b\left(\tau \right)-v\left(\tau \right),\hfill \\ \hfill {\displaystyle \frac{d{i}_b\left(\tau \right)}{d\tau }}& =& b_1(s\left(\tau \right)+v\left(\tau \right))y_b\left(\tau \right)-b_3{i}_b\left(\tau \right),\hfill \\ \hfill {\displaystyle \frac{d{x}_b\left(\tau \right)}{d\tau }}& =& 1-b_4{x}_b\left(\tau \right)i_b\left(\tau \right)-b_5{x}_b\left(\tau \right),\hfill \\ \hfill {\displaystyle \frac{d{y}_b\left(\tau \right)}{d\tau }}& =& b_4{x}_b\left(\tau \right)i_b\left(\tau \right)-b_5{y}_b\left(\tau \right).\hfill \end{array}$$

(14)

The analysis of sub-model (14) that will reveal the qualitative dynamics and the impact of control interventions of malaria disease outbreak is presented in this section.

3.3.1. Equilibrium Points of the Malaria Sub-Model (14)

The analytical representation of the DFE of the malaria sub-model (14) is given by

$$\begin{array}{c}\hfill (s^0,v^0,i_b^0,x_b^0,y_b^0)=\left({\displaystyle \frac{1}{c_1}},{\displaystyle \frac{\varphi }{c_1}},0,{\displaystyle \frac{1}{b_5}},0\right).\end{array}$$

(15)

Due to the complexity of equations involved in the computations and establishment of the existence of the EE of the malaria sub-model (14), the determination of the EE and the transmission dynamics of the malaria sub-model (14) about the EE will be explored numerically.

3.3.2. The Basic Reproduction Number of the Malaria Sub-Model (14)

The basic reproduction number for the malaria sub-model (14) denoted by ${\mathcal{R}}_0^b$ is determined using the next generation matrix method [5]. The next generation matrix of the malaria sub-model (14) is computed as

$$\begin{array}{c}\hfill F{V}^{-1}=\left(\begin{array}{cc}0& {\displaystyle \frac{b_1(s^0+v^0)}{b_5}}\\ {\displaystyle \frac{b_4{x}_b^0}{a_5}}& 0\end{array}\right),\end{array}$$

(16)

where

$$\begin{array}{c}\hfill F=\left(\begin{array}{cc}0& b_1(s^0+v^0)\\ b_4{x}_b^0& 0\end{array}\right)\mathrm{and}V=\left(\begin{array}{cc}{b}_3& 0\\ 0& b_5\end{array}\right).\end{array}$$

The ${\mathcal{R}}_0^b$ associated with the malaria model is therefore determined as the dominant eigenvalue of the next generation matrix $F{V}^{-1}$ and is given by

$$\begin{array}{c}\hfill {\mathcal{R}}_0^b=\sqrt{{\displaystyle \frac{(b_1(s^0+v^0)}{b_5}}·{\displaystyle \frac{b_4{x}_b^0}{b_3}}},\end{array}$$

(17)

which can be rewritten explicitly as

$$\begin{array}{c}\hfill {\left({\mathcal{R}}_0^b\right)}^2={\displaystyle \frac{(1-\kappa ){\beta }_b(s^0+v^0)}{({\xi }_b+{\sigma }_b)}}·{\displaystyle \frac{(1-\kappa ){\alpha }_b{x}_b^0}{(1+{\delta }_b+\gamma )}}.\end{array}$$

(18)

3.3.3. Stability Analysis of the Malaria Sub-Model (14)

The results of the stability analysis of the malaria sub-model (14) are summarised in the subsequent theorems.

Theorem 3.

The malaria sub-model (14) is locally asymptotically stable about the DFE (15) provided ${\mathcal{R}}_0^b<1$.

The proof of Theorem 3 can be established using a similar approach to that used in the proof of Theorem 1. The implication of Theorem 3 is that malaria disease can be eradicated if the initial infection is small (within the neighbourhood of the DFE) and provided the control interventions are effective enough to keep the ${\mathcal{R}}_0^b$ below unity. However, if the initial infection is large, the actual transmission dynamics of the malaria sub-model about the DFE requires the investigation of the global stability analysis of the malaria sub-model (14) about the DFE (15). The results of the global stability analysis of the malaria sub-model (14) are presented in Theorem 4.

Theorem 4.

The malaria sub-model (14) is globally asymptotically stable about the DFE (15) provided ${\mathcal{R}}_0^b<1$.

The proof of Theorem 4 can be established using a similar approach to that used in the proof of Theorem 2. The implication of this result is that malaria can be eradicated regardless of the initial infections, provided the control interventions are effective to keep ${\mathcal{R}}_0^b$ below unity.

3.4. Analysis of the Yellow Fever and Malaria Co-Infection Model (2)

The dynamical system analysis of the yellow fever and malaria co-infection model (2) is presented in this section. This analysis is crucial for understanding the transmission dynamics and the impact of the control interventions on yellow fever and malaria co-infection.

3.4.1. Equilibrium Points of the Yellow Fever and Malaria Co-Infection Model (2)

The yellow fever and malaria co-infection model (2) has equilibrium points. The analytical representation of one of the equilibrium points (DFE) of the yellow fever and malaria co-infection model (2) is given by

$$\begin{array}{c}\hfill (s^0,v^0,i_a^0,i_b^0,i_{ab}^0,x_a^0,y_a^0,x_b^0,y_b^0)=\left({\displaystyle \frac{1}{c_1}},{\displaystyle \frac{\varphi }{c_1}},0,0,0,{\displaystyle \frac{1}{a_5}},0,{\displaystyle \frac{1}{b_5}},0\right).\end{array}$$

(19)

The equations involved in the determination of the EE of the yellow fever and malaria co-infection model (2) is complex to represent analytically, so it will be explored numerically.

3.4.2. Basic Reproduction Number of the Yellow Fever and Malaria Co-Infection Model (2)

The basic reproduction number of the yellow fever and malaria co-infection model (2) denoted by ${\mathcal{R}}_0$ is computed using the next generation matrix method [5]. The next generation matrix of the yellow fever and malaria co-infection model (2) is

$$\begin{array}{c}\hfill F{V}^{-1}=\left(\begin{array}{ccccc}0& 0& 0& {\displaystyle \frac{a_1{s}^0+a_2{v}^0}{a_5}}& 0\\ 0& 0& 0& 0& {\displaystyle \frac{b_1(s^0+v^0)}{b_5}}\\ 0& 0& 0& 0& 0\\ {\displaystyle \frac{a_4{x}_a^0}{a_3}}& 0& {\displaystyle \frac{a_4{x}_b^0}{c_4}}+{\displaystyle \frac{a_4\gamma x_a^0}{a_3{c}_4}}& 0& 0\\ 0& {\displaystyle \frac{b_4{x}_b^0}{b_3}}& {\displaystyle \frac{b_4{x}_b^0}{b_3}}& 0& 0\end{array}\right),\end{array}$$

(20)

where

$$\begin{array}{c}\hfill F=\left(\begin{array}{ccccc}0& 0& 0& a_1{s}^0+a_2{v}^0& 0\\ 0& 0& 0& 0& b_1(s^0+v^0)\\ 0& 0& 0& 0& 0\\ a_4{x}_a^0& 0& a_4{x}_a^0& 0& 0\\ 0& b_4{x}_b^0& b_4{x}_b^0& 0& 0\end{array}\right)\mathrm{and}V=\left(\begin{array}{ccccc}{a}_3& 0& -\gamma & 0& 0\\ 0& b_3& 0& 0& 0\\ 0& 0& c_4& 0& 0\\ 0& 0& 0& a_5& 0\\ 0& 0& 0& 0& b_5\end{array}\right).\end{array}$$

The dominant eigenvalue of the next generation matrix $F{V}^{-1}$ is the ${\mathcal{R}}_0$ and is obtained as

$$\begin{array}{c}\hfill {\mathcal{R}}_0=max\{{\mathcal{R}}_0^a,{\mathcal{R}}_0^b\},\end{array}$$

(21)

where

$$\begin{array}{c}\hfill {\mathcal{R}}_0^a=\sqrt{{\displaystyle \frac{(a_1{s}^0+a_2{v}^0)}{a_5}}·{\displaystyle \frac{a_4{x}_a^0}{a_3}}},{\mathcal{R}}_0^b=\sqrt{{\displaystyle \frac{(b_1(s^0+v^0)}{b_5}}·{\displaystyle \frac{b_4{x}_b^0}{b_3}}}.\end{array}$$

(22)

This shows that ${\mathcal{R}}_0$ of the full co-infection model exceeds that of the individual yellow fever and malaria. Epidemiologically, an increase in the basic reproduction number implies an increase in the severity of the infection. This supports the findings that co-infections by yellow fever and malaria might lead to greater severity of infections [1].

3.4.3. Stability Analysis of the Yellow Fever and Malaria Co-Infection Model (2)

The result of stability analysis of the yellow fever and malaria co-infection model (2) is summarised in the theorem below.

Theorem 5.

For ${\mathcal{R}}_0<1$, the yellow fever and malaria co-infection model (2) is not globally asymptotically stable about the DFE (19).

The proof of Theorem 5 will be established using Lemma 1.

Proof of Theorem 5.

To prove Theorem 5, it is sufficient to show that the model fails to satisfy one of the conditions of the global stability conditions of Lemma 1 [37]. From the yellow fever and malaria co-infection model (2), let $Z_1=(s,v,x_a,x_b),Z_2=(i_a,i_b,i_{ab},y_a,y_b)$. So, the disease-free part of the yellow fever and malaria co-infection model (2) is given by

$${\displaystyle \frac{d{Z}_1}{d\tau }}=F(Z_1,0)=\left(\begin{array}{c}1-c_{1}s\left(\tau \right)\\ \varphi s\left(\tau \right)-v\left(\tau \right)\\ 1-a_5{x}_a\left(\tau \right)\\ 1-b_5{x}_b\left(\tau \right)\end{array}\right),$$

(23)

whereas the disease part of the yellow fever and malaria co-infection model (2) is

$$\begin{array}{c}\hfill {\displaystyle \frac{d{Z}_2}{d\tau }}=G(Z_1,Z_2)=\left(\begin{array}{c}{a}_{1}s\left(\tau \right)y_a\left(\tau \right)+a_{2}v\left(\tau \right)y_a\left(\tau \right)+\gamma i_{ab}\left(\tau \right)-c_2{i}_a\left(\tau \right)y_b\left(\tau \right)-a_3{i}_a\left(\tau \right)\\ b_1(s\left(\tau \right)+v\left(\tau \right))y_b\left(\tau \right)-c_3{i}_b\left(\tau \right)y_a\left(\tau \right)-b_3{i}_b\left(\tau \right)\\ c_2{i}_a\left(\tau \right)y_b\left(\tau \right)+c_3{i}_b\left(\tau \right)y_a\left(\tau \right)-c_4{i}_{ab}\left(\tau \right)\\ a_4{x}_a\left(\tau \right)(i_a\left(\tau \right)+i_{ab}\left(\tau \right))-a_5{y}_a\left(\tau \right)\\ b_4{x}_b\left(\tau \right)(i_b\left(\tau \right)+i_{ab}\left(\tau \right))-b_5{y}_b\left(\tau \right)\end{array}\right).\end{array}$$

(24)

The Jacobian of the disease part (24) about the disease-free equilibrium is

$$\begin{array}{c}\hfill A=\left(\begin{array}{ccccc}-a_3& 0& \gamma & a_1{s}^0+a_2{v}^0& 0\\ 0& -b_3& 0& 0& b_1(s^0+v^0)\\ 0& 0& -c_4& 0& 0\\ a_4{x}_a^0& 0& a_4{x}_a^0& -a_5& 0\\ 0& b_4{x}_b^0& b_4{x}_b^0& 0& -b_5\end{array}\right),\end{array}$$

(25)

where A is an M–matrix with all its off-diagonal elements non-negative. From Equations (24) and (25), the expression $\widehat{G}(Z_1,Z_2)$ is calculated as

$$\begin{array}{c}\hfill \widehat{G}(Z_1,Z_2)=\left(\begin{array}{c}{a}_1{y}_a(s^0-s)+a_2{y}_a(v^0-v)+c_2{i}_a{y}_a\\ b_1{y}_b(s^0-s)+b_1{y}_b(v^0-v)+c_3{i}_b{y}_b\\ -c_2{i}_a{y}_a-c_3{i}_b{y}_b\\ a_4{i}_a(x_a^0-x_a)+a_4{i}_{b}ab(x_a^0-x_a)\\ b_4{i}_b(x_b^0-x_b)+b_4{i}_{b}ab(x_b^0-x_b)\end{array}\right).\end{array}$$

(26)

Clearly, $\widehat{G}(Z_1,Z_2)$ is not positive, showing that condition $\left(H_2\right)$ does not hold and hence completes the proof. □

The implication of these results is that it will be difficult to eradicate yellow fever and malaria co-infection simultaneously by only lowering ${\mathcal{R}}_0$ below unity. This could explain why yellow fever and malaria co-infections are more severe and difficult to eradicate [1].

4. Numerical Simulations

In this section, numerical simulations are considered to explore the transmission dynamics and impact of control interventions on the yellow fever and malaria co-infection model (2). Insights gained from this study can inform broader strategies aimed at combating yellow fever and malaria outbreaks in endemic areas. The values of the parameters used for the numerical simulations are given in

Table 3. Parameter values used in the simulation.

Parameters	Values	References
${\beta }_a$	0.000004–0.000075	[15,38,39]
${\beta }_b$	0.0044	[12,26,40]
${\beta }_{ab}$	0.5 ${\beta }_a$	Assumed
${\beta }_{ba}$	0.5 ${\beta }_b$	Assumed
$\mu$	${\displaystyle \frac{1}{65}}$	[15,41]
${\delta }_a$	0.0003–0.0005	[15,38]
${\delta }_b$	0.0002	[12]
${\delta }_{ab}$	${\delta }_a+{\delta }_b$	Assumed
$\varphi$	0–1	[15,17,42]
$\epsilon$	0.0–0.99	[43,44]
$\gamma$	0.00019	[9,31,45]
${\alpha }_a$	0.0000015–0.00001	[15,38,39]
${\alpha }_b$	0.0044	[12,26,40]
${\xi }_a$	${\displaystyle \frac{1}{30}}$	[15,41]
${\xi }_b$	${\displaystyle \frac{1}{15}}$	[9,12,26]
${\sigma }_a$	0–1	[15,46,47]
${\sigma }_b$	0–1	[15,46,47]
$\kappa$	0–0.99	Assumed

.

4.1. Numerical Simulation of the Yellow Fever Sub-Model (3)

The numerical simulations for the yellow fever sub-model (3) are presented in this section. This is crucial for understanding the transmission dynamics of yellow fever infections in the regions where only yellow fever is endemic.

Figure 2 is a graphical illustration of the transmission dynamics of the yellow fever sub-model (3) when ${\mathcal{R}}_0^a<1$. The figure shows that effective vaccination, such that ${\mathcal{R}}_0^a<1$, may lead to eradication of humans and mosquitoes infected with the yellow fever virus for the scenario where only yellow fever is affecting the community. Therefore, effective vaccination that will keep the ${\mathcal{R}}_0^a$ below unity is strongly recommended for the possible eradication of yellow fever.

To confirm whether yellow fever eradication when ${\mathcal{R}}_0^a<1$ is dependent on the initial number of infections, the numerical simulation when ${\mathcal{R}}_0^a<1$ was conducted for various initial conditions as shown in Figure 3. The figure reveals that for various initial conditions when ${\mathcal{R}}_0^a<1$, the trajectories of the yellow fever sub-model (3) converge to the disease-free equilibrium. This result validates Theorem 2. Based on these results and model assumptions, yellow fever can be eradicated irrespective of the initial number of infections, provided that the control interventions are effective enough to keep the ${\mathcal{R}}_0^a$ below unity.

Figure 4 is a graphical illustration of the transmission dynamics of the yellow fever sub-model (3) when ${\mathcal{R}}_0^a>1$. The figure shows that when yellow fever vaccination coverage is not adequate enough, such that ${\mathcal{R}}_0^a>1$, yellow fever-infected humans and mosquitoes may increase, leading to the disease becoming endemic. This type of scenario can be found in rural areas with little or no yellow fever vaccination coverage due to limited access to basic amenities. Therefore, effective vaccination, especially in the rural areas where yellow fever is endemic, is strongly recommended.

To confirm whether the endemicity of yellow fever when ${\mathcal{R}}_0^a>1$ is dependent on the initial number of infections, the numerical simulation when ${\mathcal{R}}_0^a>1$ was conducted for various initial conditions as shown in Figure 5. The figure reveals that for various initial conditions when ${\mathcal{R}}_0^a>1$, the trajectories of the yellow fever sub-model (3) converge to an endemic equilibrium state of the model. This suggests the existence of an endemic equilibrium for the yellow fever sub-model (3) and its local stability. Based on these results and model assumptions, yellow fever infections are likely to remain endemic irrespective of the initial number of infections, provided that vaccination coverage is not enough to keep the ${\mathcal{R}}_0^a$ below unity.

4.2. Numerical Simulation of the Malaria Sub-Model (14)

The numerical simulations for the malaria sub-model (14) are presented in this section. This is essential for understanding the transmission dynamics of malaria infections in regions where malaria is endemic.

Figure 6 is a numerical illustration of the transmission dynamics of the malaria sub-model (14) when ${\mathcal{R}}_0^b<1$. The figure shows that effective control interventions, such that ${\mathcal{R}}_0^b<1$, may lead to eradication of humans and mosquitoes infected with the malaria parasite for the scenario where only malaria is endemic in the population. Therefore, effective control interventions that will keep the ${\mathcal{R}}_0^b$ below unity are strongly recommended to ensure malaria eradication.

To determine if malaria eradication for ${\mathcal{R}}_0^b<1$ depends on initial infection levels, we conduct numerical simulations for various initial conditions, as displayed in Figure 7. The result shows that, regardless of these initial conditions, the malaria sub-model (14) consistently converges to the disease-free equilibrium when ${\mathcal{R}}_0^b<1$. This outcome supports Theorem 4. Based on this result and model assumptions, malaria can be eradicated regardless of the initial number of infections, provided that control interventions are effective enough to keep ${\mathcal{R}}_0^b$ below unity.

Figure 8 is a graphical illustration of the transmission dynamics of the malaria sub-model (3) when ${\mathcal{R}}_0^b>1$. The figure shows that when control interventions for malaria are insufficient, such that ${\mathcal{R}}_0^b>1$, both infected humans and mosquitoes may increase and persist in the population. This scenario is predominant in rural areas with inadequate control interventions, consequently leading to malaria becoming endemic. Therefore, effective control interventions, especially in areas where malaria is endemic, are strongly recommended.

To confirm whether the endemicity of malaria when ${\mathcal{R}}_0^b>1$ is dependent on the initial number of infections, numerical simulation when ${\mathcal{R}}_0^b>1$ was conducted for various initial conditions as shown in Figure 9. The figure reveals that for various initial conditions, when ${\mathcal{R}}_0^b>1$, the trajectories of the malaria sub-model (14) converge to a possible endemic equilibrium state of the sub-model (14). This supports the existence and stability of the malaria sub-model (14) about an endemic equilibrium when ${\mathcal{R}}_0^b>1$. Based on these results and model assumptions, malaria disease may remain endemic irrespective of the initial number of infections, provided that the control interventions are not effective enough to keep ${\mathcal{R}}_0^b$ below unity.

4.3. Numerical Simulation of the Yellow Fever and Malaria Co-Infection Model (2)

The numerical simulations for the yellow fever and malaria co-infection model (2) are presented in this section. This is essential for understanding the transmission dynamics and the impact of control interventions of the yellow fever and malaria co-infection model.

Figure 10 is a numerical illustration of the transmission dynamics of the yellow fever and malaria co-infection model (2) when ${\mathcal{R}}_0<1$. The figure shows that effective control interventions (targeting both yellow fever and malaria) that can keep ${\mathcal{R}}_0$ below unity may lead to the eradication of yellow fever and malaria. Targeting both yellow fever and malaria to ensure total disease eradication is crucial since the diseases coexist and ${\mathcal{R}}_0=max\{{\mathcal{R}}_0^a,{\mathcal{R}}_0^b\}$. Based on these results and model assumptions, multiple control interventions (targeting both infections simultaneously) that will keep ${\mathcal{R}}_0$ below unity are strongly recommended, to ensure total disease eradication in this scenario where there is yellow fever and malaria co-infection.

A situation where the rate of spread of yellow fever is high, resulting in ${\mathcal{R}}_0^a>1$ while the rate of spread of malaria is low, resulting in ${\mathcal{R}}_0^b<1$ for a yellow fever and malaria co-infection model (2) is presented in Figure 11. Intuitive expectation is that malaria will die out while yellow fever persists. However, the figure reveals that when ${\mathcal{R}}_0^a>1$ and ${\mathcal{R}}_0^b<1$ in a co-infectious setting, humans and mosquitoes infected with malaria increase and persist, humans co-infected with yellow fever and malaria also increase and persist, while humans and mosquitoes infected with yellow fever decrease but do not get eradicated. This could explain the severity of co-infections being more difficult to control than a single infection. So, trying to eradicate only malaria in a community where yellow fever and malaria are endemic will be difficult due to the severity of co-infections and possibly because yellow fever, which is endemic, is a viral infection and has no cure at the moment. Thus, controlling both infections simultaneously is strongly recommended for effective disease eradication.

Another situation where the rate of spread of yellow fever is low, resulting in ${\mathcal{R}}_0^a<1$ while the rate of spread of malaria is high, resulting in ${\mathcal{R}}_0^b>1$ for a yellow fever and malaria co-infection model (2) is presented in Figure 12. The figure reveals that when ${\mathcal{R}}_0^a<1$ and ${\mathcal{R}}_0^b>1$ in a co-infectious disease setting, humans and mosquitoes infected with malaria decrease and get eradicated, humans co-infected with yellow fever and malaria also decrease and get eradicated, while humans and mosquitoes infected with yellow fever increase and persist. These results agree with intuitive expectations. A possible explanation for this could be that malaria is treatable. So, it is possible to eradicate yellow fever while malaria persists. However, it is preferable that both infections be controlled simultaneously using appropriate control interventions for possible disease eradication.

An endemic situation when ${\mathcal{R}}_0>1$ (i.e., ${\mathcal{R}}_0^a>1$ and ${\mathcal{R}}_0^b>1$) and both infections (yellow fever and malaria) are endemic in the system is presented in Figure 13. The figure reveals that both infected humans and mosquitoes persist when ${\mathcal{R}}_0>1$ (i.e., ${\mathcal{R}}_0^a>1$ and ${\mathcal{R}}_0^b>1$). However, malaria infections for both humans and mosquitoes dominated. This agrees with the statistical report globally, as the number of humans infected with malaria is greater than the number of humans infected with yellow fever [1,2]. Therefore, reducing the number of infections using appropriate control interventions for both diseases is strongly recommended for effective disease eradication.

A graphical illustration of the effects of the yellow fever vaccination rate $\varphi$ on the transmission dynamics of the yellow fever and malaria co-infection model (2) is presented in Figure 14. The figure reveals that increasing yellow fever vaccination coverage may increase vaccinated humans and decrease yellow fever-infected humans to almost zero. For mosquitoes, increasing the yellow fever vaccination rate may increase susceptible mosquitoes that can host the yellow fever virus, but decrease infected mosquitoes that can transmit yellow fever infection. Thus, effective yellow fever vaccination coverage is strongly recommended for the eradication of the yellow fever virus.

A graphical illustration of the effects of using mosquito nets on the transmission dynamics of the yellow fever and malaria co-infection model (2) is presented in Figure 15 and Figure 16. Since the use of insecticide-treated mosquito nets has a direct impact on disease dynamics for both humans and mosquitoes, the human dynamics are presented in Figure 15, whereas the mosquito dynamics are presented in Figure 16. Figure 15 revealed that increasing the use of insecticide-treated mosquito nets appears to have a significant impact in decreasing both yellow fever and malaria infections. Specifically, effective use of insecticide-treated mosquito nets may lead to the eradication of yellow fever and malaria infections.

Furthermore, Figure 16 revealed that increasing the use of insecticide-treated mosquito nets appears to have a significant impact in decreasing the infected mosquito population that can transmit either yellow fever or malaria infections. Effective use of insecticide-treated mosquito nets may lead to the eradication of infected mosquitoes that can transmit either yellow fever or malaria infections. Thus, the effective use of insecticide-treated mosquito nets is strongly recommended for the possible eradication of yellow fever and malaria.

Figure 17 shows the effects of treating malaria-infected humans on the transmission dynamics of the yellow fever and malaria co-infection model (2). Treatment appears to significantly affect humans infected with malaria but not those co-infected with yellow fever and malaria. Increasing malaria treatment rates may decrease the number of humans infected with malaria. Therefore, effective malaria treatment is strongly recommended.

A graphical illustration of the effects of using appropriate mosquito insecticide on the transmission dynamics of the yellow fever and malaria co-infection model (2) is presented in Figure 18 and Figure 19. Since the use of mosquito insecticide has a direct impact on the disease transmission dynamics for both humans and mosquitoes, the human dynamics are presented in Figure 18 while the mosquito dynamics are captured in Figure 19. Figure 18 shows that increasing the use of appropriate mosquito insecticides appears to have a significant impact in decreasing both yellow fever and malaria infections but enhances the susceptibility of humans. Specifically, effective use of appropriate mosquito insecticides may lead to the eradication of yellow fever and malaria infections, as well as co-infections.

For the mosquitoes’ transmission dynamics, Figure 19 shows that increasing the use of mosquito insecticide appears to have some impact in decreasing both of the mosquito populations that transmit yellow fever and malaria. Therefore, effective use of mosquito insecticide is strongly recommended.

5. Discussion

Yellow fever and malaria co-infections are real public health concerns in Africa, especially in Nigeria, where mosquitoes carrying either pathogen (Aedes for yellow fever, Anopheles for malaria) coexist. This study considered a mathematical model to investigate the transmission dynamics and control interventions of yellow fever and malaria co-infections. A mathematical model that considers the critical factors that influence the transmission dynamics and control interventions of yellow fever and malaria co-infections was formulated. Dynamical system analysis supported with numerical simulations was applied to explore the intrinsic dynamics of the model and the impact of control interventions systematically.

A special case of the model, which represents a situation where there is only yellow fever infection, was considered. The essential epidemiological features of this special case, such as the disease-free equilibrium, ${\mathcal{R}}_0^a$, were determined. The analysis of this special case revealed that it is possible to eradicate yellow fever by making sure that ${\mathcal{R}}_0^a$ is kept below unity. This was achieved analytically by showing that the model is globally asymptotically stable when ${\mathcal{R}}_0^a$ is less than one. These analytical results were supported numerically by showing that, irrespective of the initial infections, the model converges to the disease-free equilibrium. However, when ${\mathcal{R}}_0^a$ becomes greater than one, it was shown numerically that yellow fever disease is likely to become endemic irrespective of the initial number of infections. Based on these results and model assumptions, effective implementation of control interventions (vaccination and the use of mosquito nets) that reduce ${\mathcal{R}}_0^a$ below one, such that the disease can be eradicated, is strongly recommended.

Another special case of the model, which represents a situation where there is only malaria infection, was also considered. The essential epidemiological features of this special case, such as the disease-free equilibrium, ${\mathcal{R}}_0^b$, were computed. The analysis of this special case revealed that it is possible to eradicate malaria by making sure that the associated ${\mathcal{R}}_0^b$ is kept below unity. This was also achieved analytically by showing that the model is globally asymptotically stable when ${\mathcal{R}}_0^b$ is less than one. These analytical results were supported numerically by showing that, irrespective of the initial infections, the model converges to the disease-free equilibrium. However, when ${\mathcal{R}}_0^b$ becomes greater than one, it was shown numerically that malaria disease might become endemic irrespective of the initial number of infections. Based on these results and model assumptions, effective control interventions such as treatment and use of mosquito nets that reduce ${\mathcal{R}}_0^b$ below one, such that the disease is eradicated, are strongly recommended.

Next, we considered the general case, which captures the transmission dynamics and control interventions of yellow fever and malaria co-infections. The essential epidemiological features of the general case, such as the disease-free equilibrium, ${\mathcal{R}}_0$, were computed. The analysis of this general case revealed that it will be difficult to eradicate both yellow fever and malaria by decreasing the ${\mathcal{R}}_0$ below unity. This was established analytically by showing that the model is not globally asymptotically stable about the disease-free equilibrium when ${\mathcal{R}}_0$ is less than one. These results support the severity of co-infections [1]. Based on these results, we recommend additional effort beyond reducing ${\mathcal{R}}_0$ below unity to ensure disease eradication in a co-infectious setting. It was also discovered that when ${\mathcal{R}}_0^a>1$ and ${\mathcal{R}}_0^b<1$ in a co-infectious setting, humans and mosquitoes infected with malaria increase and persist, humans co-infected with yellow fever and malaria also increase and persist, while humans and mosquitoes infected with yellow fever decrease but do not get eradicated. This implies that trying to eradicate malaria only in a community where yellow fever and malaria are endemic will be difficult due to the severity of co-infections and possibly because yellow fever, which is endemic, is a viral infection and has no cure at the moment. The analysis of an endemic situation when ${\mathcal{R}}_0>1$ (i.e., ${\mathcal{R}}_0^a>1$ and ${\mathcal{R}}_0^b>1$) revealed that both humans and mosquitoes infected with either or both diseases persist, with malaria infections dominating. This result supports statistical reports globally, as the number of humans infected with malaria is greater than the number of humans infected with yellow fever [1,2]. Therefore, reducing the number of infections by applying appropriate control interventions for both diseases is strongly recommended for disease eradication.

The impact of various control interventions on the transmission dynamics of yellow fever and malaria co-infections was investigated numerically. The result of the analysis revealed that increasing yellow fever vaccination coverage might decrease yellow fever-infected humans to almost zero and also decrease infected mosquitoes that can transmit the yellow fever virus. Thus, effective yellow fever vaccination coverage is strongly recommended for yellow fever eradication. Increasing the use of insecticide-treated mosquito nets appears to have a significant impact in decreasing both yellow fever and malaria infections. Specifically, effective use of insecticide-treated mosquito nets may lead to the eradication of yellow fever and malaria infections, as well as co-infections. Treatment of malaria-infected humans appears to have a significant impact on humans infected with malaria but not on humans co-infected with yellow fever and malaria. Specifically, increasing the rate of treatment of malaria-infected humans appears to decrease the number of humans infected with malaria. Thus, effective treatment of malaria infection is strongly recommended. Increasing the use of appropriate mosquito insecticides appears to have a significant impact in decreasing both yellow fever and malaria infections but increases the susceptibility of humans. Specifically, effective use of appropriate mosquito insecticides may lead to the eradication of yellow fever and malaria infections, as well as co-infections.

In general, this study used a mathematical model to highlight the transmission dynamics and impact of control interventions, which is crucial for the proper management of yellow fever and malaria co-infections. The results of this study will aid in better management of yellow fever and malaria co-infections in endemic regions.

Even though this study has made a significant contribution using a mathematical model to understand the transmission dynamics and control interventions for yellow fever and malaria co-infections, it still has some limitations. One of the limitations is that the results obtained in this study are based on secondary data and have not been validated with primary (real) data due to limited access to real data on yellow fever and malaria co-infections. Another limitation is that the model used in this study is deterministic, which gives less-accurate predictions compared to a stochastic model that captures the variability in variables and parameters. These will be considered in future research for more accurate disease outbreak predictions that will lead to improved management of yellow fever and malaria co-infections in endemic areas.