Phenomenon Of Backward Bifurcation Research Articles

Dynamic of a two-strain COVID-19 model with vaccination

COVID-19 is a respiratory illness caused by an ribonucleic acid (RNA) virus prone to mutations. In December 2020, variants with different characteristics that could affect transmissibility emerged around the world. To address this new dynamic of the disease, we formulate and analyze a mathematical model of a two-strain COVID-19 transmission dynamics with strain 1 vaccination. The model is theoretically analyzed and sufficient conditions for the stability of its equilibria are derived. In addition to the disease-free and endemic equilibria, the model also has single-strain 1 and strain 2 endemic equilibria. Using the center manifold theory, it is shown that the model does not exhibit the phenomenon of backward bifurcation, and global stability of the model equilibria are proved using various approaches. Simulations to support the model theoretical results are provided. We calculate the basic reproductive number R1 and R2 for both strains independently. Results indicate that – both strains will persist when R1>1 and R2>1 – Stain 2 could establish itself as the dominant strain if R1<1 and R2>1, or when R2>R1>1. However, because of de novo herd immunity due to strain 1 vaccine efficacy and provided the initial stain 2 transmission threshold parameter R2 is controlled to remain below unity, strain 2 will not establish itself/persist in the community.

Post pandemic fatigue: what are effective strategies?

Recurrent updates in non-pharmaceutical interventions (NPIs) aim to control successive waves of the coronavirus disease 2019 (COVID-19) but are often met with low adherence by the public. This study evaluated the effectiveness of gathering restrictions and quarantine policies based on a modified Susceptible-Exposed-Infectious-Hospitalized-Recovered (SEIHR) model by incorporating cross-boundary travellers with or without quarantine to study the transmission dynamics of COVID-19 with data spanning a nine-month period during 2020 in Hong Kong. The asymptotic stability of equilibria reveals that the model exhibits the phenomenon of backward bifurcation, which in this study is a co-existence between a stable disease-free equilibrium (DFE) and an endemic equilibrium (EE). Even if the basic reproduction number ({mathscr {R}}_{0}) is less than unity, this disease cannot be eliminated. The effect of each parameter on the overall dynamics was assessed using Partial Rank Correlation Coefficients (PRCCs). Transmission rates (i.e., beta _1 and beta _2), effective contact ratio a_6 between symptomatic individuals and quarantined people, and transfer rate theta _3 related to infection during quarantine were identified to be the most sensitive parameters. The effective contact ratios between the infectors and susceptible individuals in late July were found to be over twice as high as that in March of 2020, reflecting pandemic fatigue and the potential existence of infection during quarantine.

A size-structured model describing flocculation of unicellular algae

Flocculation is a chemically-based microorganisms separation technology, which is widely used in the field of environmental engineering. The size of algae cell has a significant effect on photosynthesis and flocculation. In view of this, a size-structured model describing flocculation of unicellular algae is constructed for the first time in this paper. In Section 2, the model is introduced and then transformed into a dimensionless form. In Section 3, the well-posedness of the solutions of the model is demonstrated through the use of the theory of integral semigroups. In Section 4, the existence of the steady states of the model is analysed. It is proved that, when the threshold parameter R0>1, there is at least one positive steady state; when the threshold parameter V0<1, there is no positive steady state. Furthermore, the sufficient conditions are obtained for local and global stability of the boundary steady state. In Sections 5 and 6, some numerical examples are given, which show the existence of the interesting phenomenon of backward bifurcation when R0<1<V0.

MODELING THE ROLE OF VACCINATION, ENVIRONMENTAL SANITATION, AND SATURATED TREATMENT ON THE SPREAD OF TYPHOID FEVER

A deterministic nonlinear mathematical model is developed for typhoid transmission dynamics in human hosts, coupled with multiple transmission routes. The model aims to examine the role of control interventions such as vaccination, environmental sanitation, and saturated treatment on the prevalence of typhoid. First, the qualitative analysis of the model with constant control interventions is performed. The model exhibits a backward bifurcation phenomenon. Sensitivity analysis is also conducted to identify impactful parameters for effective control of the disease. Then, the model is extended to a corresponding optimal control problem to investigate the optimum intervention strategies by assessing their effects on typhoid prevalence and economic load. The characterization of the optimal controls is determined using Pontryagin’s Maximum Principle, and the optimality system is developed. Numerical results suggest that, in the absence of treatment, the combination of vaccination and environmental sanitation controls plays an important role in reducing the typhoid burden and economic load. Moreover, the comprehensive use of the three control interventions is more effective than using any single or two combined control interventions. It reduces the number of infective humans and environmental bacteria as well as the cost burden associated with applied controls and opportunity loss. Thus, the comprehensive effect of the three control interventions is found to be more economical during typhoid outbreaks.

Dynamics of novel COVID-19 in the presence of Co-morbidity.

A novel coronavirus (COVID-19) has emerged as a global serious public health issue from December 2019. People having a weak immune system are more susceptible to coronavirus infection. It is a double challenge for people of any age with certain underlying medical conditions including cardiovascular disease, diabetes, high blood pressure and cancer etc. Co-morbidity increases the probability of COVID-19 complication. In this paper a deterministic compartmental model is formulated to understand the transmission dynamics of COVID-19. Rigorous mathematical analysis of the model shows that it exhibits backward bifurcation phenomenon when the basic reproduction number is less than unity. For the case of no re-infection it is shown that having the reproduction number less than one is necessary and sufficient for the effective control of COVID-19, that is, the disease free equilibrium is globally asymptotically stable when the reproduction threshold is less than unity. Furthermore, in the absence of reinfection, a unique endemic equilibrium of the model exists which is globally asymptotically stable whenever the reproduction number is greater than unity. Numerical simulations of the model, using data relevant to COVID-19 transmission dynamics, show that the use of efficacious face masks publicly could lead to the elimination of COVID-19 up to a satisfactory level. The study also shows that in the presence of co-morbidity, the disease increases significantly.

EBOLA VIRUS DISEASE DYNAMICS WITH SOME PREVENTIVE MEASURES: A CASE STUDY OF THE 2018–2020 KIVU OUTBREAK

To fight against Ebola virus disease, several measures have been adopted. Among them, isolation, safe burial and vaccination occupy a prominent place. In this paper, we present a model which takes into account these three control strategies as well as the indirect transmission through a polluted environment. The asymptotic behavior of our model is achieved. Namely, we determine a threshold value [Formula: see text] of the control reproduction number [Formula: see text], below which the disease is eliminated in the long run. Whenever the value of [Formula: see text] ranges from [Formula: see text] and 1, we prove the existence of a backward bifurcation phenomenon, which corresponds to the case, where a locally asymptotically stable positive equilibrium co-exists with the disease-free equilibrium, which is also locally asymptotically stable. The existence of this bifurcation complicates the control of Ebola, since the requirement of [Formula: see text] below one, although necessary, is no longer sufficient for the elimination of Ebola, more efforts need to be deployed. When the value of [Formula: see text] is greater than one, we prove the existence of a unique endemic equilibrium, locally asymptotically stable. That is the disease may persist and become endemic. Numerically, we fit our model to the reported data for the 2018–2020 Kivu Ebola outbreak which occurred in Democratic Republic of Congo. Through the sensitivity analysis of the control reproduction number, we prove that the transmission rates of infected alive who are outside hospital are the most influential parameters. Numerically, we explore the usefulness of isolation, safe burial combined with vaccination and investigate the importance to combine the latter control strategies to the educational campaigns or/and case finding.

Analysis of deterministic models for dengue disease transmission dynamics with vaccination perspective in Johor, Malaysia.

Dengue is a mosquito-borne disease which has continued to be a public health issue in Malaysia. This paper investigates the impact of singular use of vaccination and its combined effort with treatment and adulticide controls on the population dynamics of dengue in Johor, Malaysia. In a first step, a compartmental model capturing vaccination compartment with mass random vaccination distribution process is appropriately formulated. The model with or without imperfect vaccination exhibits backward bifurcation phenomenon. Using the available data and facts from the 2012 dengue outbreak in Johor, basic reproduction number for the outbreak is estimated. Sensitivity analysis is performed to investigate how the model parameters influence dengue disease transmission and spread in a population. In a second step, a new deterministic model incorporating vaccination as a control parameter of distinct constant rates with the efforts of treatment and adulticide controls is developed. Numerical simulations are carried out to evaluate the impact of the three control measures by implementing several control strategies. It is observed that the transmission of dengue can be curtailed using any of the control strategies analysed in this work. Efficiency analysis further reveals that a strategy that combines vaccination, treatment and adulticide controls is most efficient for dengue prevention and control in Johor, Malaysia.

Assessing the potential impact of immunity waning on the dynamics of COVID-19 in South Africa: an endemic model of COVID-19.

We developed an endemic model of COVID-19 to assess the impact of vaccination and immunity waning on the dynamics of the disease. Our model exhibits the phenomenon of backward bifurcation and bi-stability, where a stable disease-free equilibrium coexists with a stable endemic equilibrium. The epidemiological implication of this is that the control reproduction number being less than unity is no longer sufficient to guarantee disease eradication. We showed that this phenomenon could be eliminated by either increasing the vaccine efficacy or by reducing the disease transmission rate (adhering to non-pharmaceutical interventions). Furthermore, we numerically investigated the impacts of vaccination and waning of both vaccine-induced immunity and post-recovery immunity on the disease dynamics. Our simulation results show that the waning of vaccine-induced immunity has more effect on the disease dynamics relative to post-recovery immunity waning and suggests that more emphasis should be on reducing the waning of vaccine-induced immunity to eradicate COVID-19.

Stability analysis and backward bifurcation on an SEIQR epidemic model with nonlinear innate immunity

&lt;abstract&gt;&lt;p&gt;Infectious diseases have a great impact on the economy and society. Dynamic models of infectious diseases are an effective tool for revealing the laws of disease transmission. Quarantine and nonlinear innate immunity are the crucial factors in the control of infectious diseases. Currently, there no mathematical models that comprehensively study the effect of both innate immunity and quarantine. In this paper, we propose and analyze an SEIQR epidemic model with nonlinear innate immunity. The boundedness and positivity of the solutions are discussed. Employing the next-generation matrix, we compute the expression of the basic reproduction number. Under certain conditions, the phenomenon of backward bifurcation may occur. That is to say, the stable disease-free equilibrium point and the stable endemic equilibrium point coexist when the basic reproduction ratio is less than one. And the basic reproduction number is no longer the threshold value to determine whether the disease breaks out. We investigate the globally asymptotical stability of the disease-free equilibrium point for the system by constructing Lyapunov function. Also, we research the global stability of the endemic equilibrium by using geometric approach. Numerical simulations are carried out to reveal the theoretical results and find some complex dynamics (for example, the existence of Hopf bifurcation) of the system. Both theoretical and numerical results indicate that the nonlinear innate immunity may cause backward bifurcation and Hopf bifurcation, which makes more difficult to eliminate the disease.&lt;/p&gt;&lt;/abstract&gt;

The co-dynamics of malaria and tuberculosis with optimal control strategies

Malaria and Tuberculosis are both the severe and causing death diseases in the world. The occurrence of TB and malaria as a coinfection is also an alarming threat to the human. Therefore, we consider a mathematical model of the dynamics of malaria and tuberculosis coinfection and explore its theoretical results. We formulate the model and obtain their basic properties. We show that at the disease free case each model is locally asymptotically stable, when the basic reproduction number less than unity. Further, we analyze the phenomenon of backward bifurcation for coinfection model. For the sub models, we present the local stability for the disease free case whenever the basic reproduction number less than 1. Further, an optimal control problem is presented to investigate the dynamics of malaria and tuberculosis coinfection. The numerical results with different scenarios are presented. The mathematical model with and without control problemare solved numerically using the Runge-Kutta backward and forward scheme of order four.

Analysis of a Tuberculosis Infection Model considering the Influence of Saturated Recovery (Treatment)

Tuberculosis (TB) is a serious global health threat that is caused by the bacterium Mycobacterium tuberculosis, is extremely infectious, and has a high mortality rate. In this paper, we developed a model of TB infection to predict the impact of saturated recovery. The existence of equilibrium and its stability has been investigated based on the effective reproduction number R C . The model displays interesting dynamics, including backward bifurcation and Hopf bifurcation, which further results in the stable disease-free and stable endemic equilibria to be coexisting. Bifurcation analysis demonstrates that the saturation parameter is accountable for the phenomenon of backward bifurcation. We derive a condition that guarantees that the model is globally asymptotically stable using the Lyapunov function approach to global stability. The numerical simulation also reveals that the extent of saturation of TB infection is the mechanism that is fuelling TB disease in the population.

A mathematical model for the coinfection of Buruli ulcer and Cholera

We propose to study the modeling and analysis of the coinfection of Buruli and cholera. We developed the model based on the literature for the coinfection and its optimal control. First, we analyze the sub-models at their steady states and present its mathematical results. The global stability for the sub-models are investigated for the special case. We show that the sub-models are locally as well as globally asymptotically stable whenever R0 is less or greater than one. Further, the co-infection model is analyzed by computing its R0 and it is proven that the coinfection model is locally asymptotically stable. We study the bifurcation analysis for the coinfection model and determine the conditions for the possible existence of backward bifurcation phenomenon. Moreover, we use five different control variables and obtain the control problem. The details mathematical results involve in the optimality system are shown. We use the Pontryagin’s Maximum Principle to determine the best strategy in controlling both the diseases. Lastly, we perform the numerical experiments using different set of controls for the possible eliminations of infection. We observe from our numerical results that the preventions and treatments are the best controls for the infection minimization.

Mathematical modeling and projections of a vector-borne disease with optimal control strategies: A case study of the Chikungunya in Chad

In this work, we extend existing models of vector-borne diseases by including density-dependent rates and some existing control mechanisms to decrease the disease burden in the human population. We begin by analyzing the vector model dynamics and by determining the offspring reproductive number denoted by N as well as the trivial and nontrivial equilibria. Using theory of cooperative systems and the general theory of Lyapunov, we prove that, although there is a possibility that the trivial equilibrium coexists with a positive equilibrium, it remains globally asymptotically stable whenever N ≤ 1 . The fact that the non-trivial equilibrium is globally asymptotically stable permits us to reduce the study of the full model to the study of a reduced model whenever N > 1 . Thus, we analyze the reduced model by computing the basic reproduction number R 0 , equilibrium points as well as asymptotic stability of each equilibrium point. We also explore the nature of the bifurcation for the disease-free equilibrium from R 0 = 1 . By the application of the centre manifold theory, we prove that the backward bifurcation phenomenon can occur in our model, which means that the necessary condition R 0 < 1 is not sufficient to guarantee the final extinction of the disease in human populations. To calibrate our model, we estimate model parameters on clinical data from the last Chikungunya epidemic which occurred in Chad, using the non-linear least-square method. We find out that R 0 = 1.8519 , which means that we are in an endemic state since R 0 > 1 . To determine model parameters that are responsible for disease spread in the human community, we perform sensitivity analysis (SA) using a global method. It follows that the density-dependent death rate of mosquitoes and the average number of mosquito bites are key parameters in the disease dynamics. Following this, we thus formulate an optimal control model by including in the autonomous model, four time-dependent control functions to fight the disease spread. Pontryagin’s maximum principle is used to characterize our optimal controls. Numerical simulations, using parameter values of Chikungunya transmission dynamics, and efficiency analysis, are conducted to determine the better control strategy which guaranteed the final extinction of the disease in human populations.

Projections and fractional dynamics of the typhoid fever: A case study of Mbandjock in the Centre Region of Cameroon

In this work, we formulate a mathematical model with a non-integer order derivative to investigate typhoid fever transmission dynamics. To combat the spread of this disease in the human community, control measures like vaccination are included in the proposed model. We calculate the epidemiological threshold called the control reproduction number, Rc, and perform the asymptotic stability of the typhoid-free equilibrium point. We prove that the typhoid-free equilibrium for both integer and non-integer models is locally and globally asymptotically stable whenever Rc is less than one. We also prove that both models admit only one endemic equilibrium point which is globally asymptotically stable whenever Rc>1 and no endemic equilibrium point otherwise. This means that the backward bifurcation phenomenon does not occur. In absence of vaccination, Rc is equal to the basic reproduction number R0. We found out that Rc<R0 which means that vaccination can permit to decrease of the spread of typhoid fever in the human community. Using fixed point theory, we perform existence and uniqueness analysis of solutions of the fractional model. We use the Adams-Bashforth-Moulton method to construct a numerical scheme of the fractional model. We prove the stability of the proposed numerical scheme. To calibrate our model, we estimate model parameters on clinical data of Mbandjock District Hospital in the Centre Region of Cameroon, using the Non-linear Least-Square method. This permits us to find Rc=1.3722, which means that we are in an endemic state (since Rc>1), and then to predict new cases of typhoid fever per month at Mbandjock in the next new year. To determine model parameters that are responsible for disease spread in the human community, we perform sensitivity analysis (SA). This analysis shows that the vaccination rate, the human-bacteria contact rate, as well as the recovery rate, are the most important parameters in the disease spread. To validate our analytical results, and to see the impact of some control measures in the spread of typhoid fever in the human community, as well as the impact of the fractional-order on typhoid transmission dynamics, we perform several numerical simulations.

Mathematical modeling of the impact of temperature variations and immigration on malaria prevalence in Nigeria

The study examines the population-level impact of temperature variability and immigration on malaria prevalence in Nigeria, using a novel deterministic model. The model incorporates disease transmission by immigrants into the community. In the absence of immigration, the model is shown to exhibit the phenomenon of backward bifurcation. The disease-free equilibrium of the autonomous version of the model was found to be locally asymptotically stable in the absence of infective immigrants. However, the model exhibits an endemic equilibrium point when the immigration parameter is greater than zero. The endemic equilibrium point is seen to be globally asymptotically stable in the absence of disease-induced mortality. Uncertainty and sensitivity analysis of the model, using parameter values and ranges relevant to malaria transmission dynamics in Nigeria, shows that the top three parameters that drive malaria prevalence (with respect to [Formula: see text]) are the mosquito natural death rate ([Formula: see text]), mosquito biting rate ([Formula: see text]) and the transmission rates between humans and mosquitoes ([Formula: see text]). Numerical simulations of the model show that in Nigeria, malaria burden increases with increasing mean monthly temperature in the range of 22–28[Formula: see text]. Thus, this study suggests that control strategies for malaria should be intensified during this period. It is further shown that the proportion of infective immigrants has marginal effect on the transmission dynamics of the disease. Therefore, the simulations suggest that a reduction in the fraction of infective immigrants, either exposed or infectious, would significantly reduce the malaria incidence in a population.

Mathematical modelling of the transmission dynamics of hepatitis B virus in the presence of imperfect vaccination

Hepatitis B infection remains a global problem since the 1990s and the reasons for which disease is still in existence remain poorly understood. However, understanding the important role played by vaccination in the transmission dynamics of Hepatitis B virus is critical to its control and management. In this paper, an epidemiological model is proposed to model the spread of the Hepatitis B virus disease in the presence of imperfect vaccination. The basic reproduction number, R 0 and the equilibria of the model are determined and the stabilities of the equilibria determined. It is shown that the disease-free equilibrium point is both locally and globally asymptotically stable when R 0 1. The model is also shown to exhibit a backward bifurcation phenomenon. Numerical simulations are carried out and it is observed that increasing both the vaccination and treatment rates reduces the populations of both the acutely infected and chronic carriers which eventually leads to the containment of the disease. We conclude that the combination of both vaccination and treatment with the use of a vaccine with a high efficacy is essential in the control of Hepatitis B virus disease.

Malaria and COVID-19 co-dynamics: A mathematical model and optimal control

Malaria, one of the longest-known vector-borne diseases, poses a major health problem in tropical and subtropical regions of the world. Its complexity is currently being exacerbated by the emerging COVID-19 pandemic and the threats of its second wave and looming third wave. We formulate and analyze a mathematical model incorporating some epidemiological features of the co-dynamics of both malaria and COVID-19. Sufficient conditions for the stability of the malaria only and COVID-19 only sub-models’ equilibria are derived. The COVID-19 only sub-model has globally asymptotically stable equilibria while under certain condition, the malaria-only could undergo the phenomenon of backward bifurcation whenever the sub-model reproduction number is less than unity. The equilibria of the dual malaria-COVID19 model are locally asymptotically stable as global stability is precluded owing to the possible occurrence of backward bifurcation. Optimal control of the full model to mitigate the spread of both diseases and their co-infection are derived. Pontryagin’s Maximum Principle is applied to establish the existence of the optimal control problem and to derive the necessary conditions for optimal control of the diseases. Though this is not a case study, simulation results to support theoretical analysis of the optimal control suggests that concurrently applying malaria and COVID-19 protective measures could help mitigate their spread compared to applying each preventive control measure singly as the world continues to deal with this unprecedented and unparalleled COVID-19 pandemic.

Analysis of COVID-19 and comorbidity co-infection model with optimal control.

In this work, we develop and analyze a mathematical model for the dynamics of COVID‐19 with re‐infection in order to assess the impact of prior comorbidity (specifically, diabetes mellitus) on COVID‐19 complications. The model is simulated using data relevant to the dynamics of the diseases in Lagos, Nigeria, making predictions for the attainment of peak periods in the presence or absence of comorbidity. The model is shown to undergo the phenomenon of backward bifurcation caused by the parameter accounting for increased susceptibility to COVID‐19 infection by comorbid susceptibles as well as the rate of reinfection by those who have recovered from a previous COVID‐19 infection. Simulations of the cumulative number of active cases (including those with comorbidity), at different reinfection rates, show infection peaks reducing with decreasing reinfection of those who have recovered from a previous COVID‐19 infection. In addition, optimal control and cost‐effectiveness analysis of the model reveal that the strategy that prevents COVID‐19 infection by comorbid susceptibles is the most cost‐effective of all the control strategies for the prevention of COVID‐19.

Transmission dynamics of SARS-CoV-2: A modeling analysis with high-and-moderate risk populations

Nigeria is second to South Africa with the highest reported cases of COVID-19 in sub-Saharan Africa. In this paper, we employ an SEIR-based compartmental model to study and analyze the transmission dynamics of SARS-CoV-2 outbreaks in Nigeria. The model incorporates different group of populations (that is, high- and- moderate risk populations) and is use to investigate the influence on each population on the overall transmission dynamics.The model, which is fitted well to the data, is qualitatively analyzed to evaluate the impacts of different schemes for controlstrategies. Mathematical analysis reveals that the model has two equilibria; i.e., disease-free equilibrium (DFE) which is local asymptotic stability (LAS) if the basic reproduction number (R0) is less than 1; and unstable for R0>1, and an endemic equilibrium (EE) which is globally asymptotic stability (LAS) whenever R0>1. Furthermore, we find that the model undergoes a phenomenon of backward bifurcation (BB, a coexistence of stable DFE and stable EE even if the R0<1). We employ Partial Rank Correlation coefficients (PRCCs) for sensitivity analyses to evaluate the model’s parameters. Our results highlight that proper surveillance, especially movement of individuals from high risk to moderate risk population, testing, as well as imposition of other NPIs measures are vital strategies for mitigating the COVID-19 epidemic in Nigeria. Besides, in the absence of an exact solution for the proposed model, we solve the model with the well-known ODE45 numerical solver and the effective numerical schemes such as Euler (EM), Runge–Kutta of order 2 (RK-2), and Runge–Kutta of order 4 (RK-4) in order to establish approximate solutions and to show the physical features of the model. It has been shown that these numerical schemes are very effective and efficient to establish superb approximate solutions for differential equations.

A co-infection model for HPV and syphilis with optimal control and cost-effectiveness analysis

A co-infection model for human papillomavirus (HPV) and syphilis with cost-effectiveness optimal control analysis is developed and presented. The full co-infection model is shown to undergo the phenomenon of backward bifurcation when a certain condition is satisfied. The global asymptotic stability of the disease-free equilibrium of the full model is shown not to exist when the associated reproduction number is less than unity. The existence of endemic equilibrium of the syphilis-only sub-model is shown to exist and the global asymptotic stability of the disease-free and endemic equilibria of the syphilis-only sub-model was established, for a special case. Sensitivity analysis is also carried out on the parameters of the model. Using the syphilis associated reproduction number, [Formula: see text], as the response function, it is observed that the five-ranked parameters that drive the dynamics of the co-infection model are the demographic parameter [Formula: see text], the effective contact rate for syphilis transmission, [Formula: see text], the progression rate to late stage of syphilis [Formula: see text], and syphilis treatment rates: [Formula: see text] and [Formula: see text] for co-infected individuals in compartments [Formula: see text] and [Formula: see text], respectively. Moreover, when the HPV associated reproduction number, [Formula: see text], is used as the response function, the five most dominant parameters that drive the dynamics of the model are the demographic parameter [Formula: see text], the effective contact rate for HPV transmission, [Formula: see text], the fraction of HPV infected who develop persistent HPV [Formula: see text], the fraction of individuals vaccinated against incident HPV infection [Formula: see text] and the HPV vaccine efficacy [Formula: see text]. Numerical simulations of the optimal control model showed that the optimal control strategy which implements syphilis treatment controls for singly infected individuals is the most cost-effective of all the control strategies in reducing the burden of HPV and syphilis co-infections.