85 4.8 Profiles representing the scenario where only mnp and mpp are handled throughout the pop. Our results showed that of the eight strategies only six were able to reduce the rate of HIV transmission among serodiscordant married couples.
Background
Mathematical Epidemiology
Mathematical epidemiology is the field that uses mathematical models to capture the mechanism that affects the spread of diseases and seek better ways to control the spread of diseases [16]. In the second half of the twentieth century, further improvements were made to mathematical models to incorporate the entry and persistence of human pathogens.
Economic Epidemiology
Problem statement
In this study we want to formulate a special mathematical model of HIV and include economic aspects in the treatment rate. We want to investigate the best intervention strategy to prevent an increase in HIV transmission rates for HIV-serodiscordant couples.
Aim and Objectives
Aim
The problem is that there has not been a balance of focus in these two areas combined to appreciate the benefits and improvements that can be achieved from mathematical models of economic epidemiology.
Objectives
Significance
Plan of the study
Summary
The purpose of the study was to present and analyze an epidemic problem that can be controlled by vaccination as well as treatment. The aim of the study was to investigate the optimal control of a deterministic model of tuberculosis and interventions.
Economic Epidemiology Review
The health gap increased further and the marginal benefit and marginal cost curves reached a unique endemic steady state exposure. It was also indicated that total welfare was always higher along the transition path to the low exposure endemic steady state.
Preliminary Concepts
Deterministic models
Models of discrete systems are often an approximation to the models of continuous system, and simplifying assumptions may be necessary.
Tools to solve mathematical models
- Basic epidemiology terminology [1, 2]
- Invariance principle
- Equilibrium [3, 4, 5]
- The basic reproduction number
- The next generation matrix method
We also assume that the discrete system (2.4) can be linearized about the disease-free equilibrium and obtain the linearized system. By differentiation with respect to conditions Y0 and evaluated at the disease-free equilibrium we obtain the matrices F and T.
Summary
Some of the mathematical models and tools discussed in this chapter will be applied to the problem studied in the next chapter. In the submodels, we assume that the proportions are based on the smallest number of infected individuals receiving treatment in each compartment, based on the smallest number of infected individuals in the same compartment.
Serodiscordant couples formation from marriage of single individuals
Force of infection for S N
We have seven individuals in the serodiscordant pair formation model from married single individuals involved in HIV transmission dynamics at any given time, one from each of the compartments, SN, SP, TSP, and two from each of, MN P and TMN P .Each HIV-negative individual in the population has the opportunity to become infected with HIV-positive individuals from four sources of infection, SP, TSP, MN P, and TMN P. There are four infected individuals, one of the HIV-positive individuals of one of the serodiscordant pair's ward, and each of of treated compartments, TSP and TMN P. Therefore, the chances of an HIV-negative individual contracting HIV from each source of infection, SP, TSP, TMN P and MN P are 1. 4 for each source of infection. Therefore, the transmission rate from SP compared to that from MN P, TSP and TMN P , βη where η >1.
Feasible region
The solutions of the model are considered to be biologically and mathematically feasible in the region Ω, therefore it suffices to study the dynamics of the model in Ω. The system of continuous differential equations can be transformed into the following system of discrete difference equations:.
Existence of equilibria and Invasion reproduction number
The invasion reproduction number is determined in the same way as the basic reproduction number, but using the endemic equilibrium point E1 [40]. We will use the next generation matrix by Van den Driessche and Allen [41] to determine the incidence reproduction number of the discrete system. We require that the condition, β−4ΛN >0, always holds for the incidence reproduction number to be greater than zero and to be valid.
The Table 1.1 shows the real and imaginary roots for each possible change of signs that we can have in the characteristic equation (3.41).
Formation of serodiscordant couples through infection of HIV concordant negative
Forces of infection
The HIV-negative individual in the serodiscordant couple is at risk of being infected by individuals from the compartments, MP P, TMN P, TMP P. We assume that, γ, indicates greater chances that seropositive partner in the sero-discordant couple should infect the seronegative partner compared to the other infectious individuals. Therefore, the strength of infection for the HIV-negative individual in the serodiscordant couples, MN P, is denoted by, ωN P, given by.
We assume that the seronegative individuals of the treated serodiscordant couples compartment, TMN P, are also exposed to a risk of acquiring HIV from the infectious individuals in the population.
Feasible region
Therefore, it is sufficient to study the dynamics of the model in the region Γ, since all solutions of the model are both epidemiologically and mathematically feasible. The system of federal differential equations can be represented in the following discrete form.
Existence of equilibria and the basic reproduction number
The discrete system has four infectious compartments, therefore we have to make use of the next generation matrix as in [41]. The eigenvalues of the matrix F(I −T)−1 are given by;. 3.73) We investigate the existence of endemic equilibrium points using the fixed point theory. We express the forces of infection in terms of MN N, MN P, MP P, TMN P and TMP P and determine the equilibrium points of the model by finding the fixed points of the functions.
We also consider the first derivative of the functionϕ(ω3 N N∗ ,ω∗T)(ωNP) with respect toωNP and is given as.
Stability of equilibria
It is not clear whether the first and second derivatives are less than zero, or greater than zero, or equal to zero due to the complexity of the derivatives and different signs of parameters. We set the infection forces equal to zero and the corresponding Jacobian matrix of the equations is given by. Every possibility we have in Table 3.4 shows that at least one positive real root exists.
Sensitivity analysis of parameters
Sensitivity with respect to the transmission rate ,β,
Sensitivity with respect to the treatment rate, ,
This implies that given the condition that the HIV-induced death rate is less than the natural death rate, the treatment rate is highly and positively effective at R0. This implies that given the condition that the HIV-induced death rate times a positive constant, bn, is greater than the natural death rate, the treatment rate will not be effective in reducing the number of new infections. The rate of change of the invasion reproduction number with respect to the treatment rate is found as.
Equation (3.90) indicates that the change in invasion reproduction number versus treatment rate indicates that ∂Rinv.
Single’s and married couples HIV transmission and serodiscordant couples formation
Forces of infection
The HIV-negative individuals in the serodiscordant couples also have chances of being infected by the three sources of infection. We assume that seronegative partners of the treated HIV-serodiscordant are also exposed to the risk of being infected by the HIV-positive individuals from the sources of the infection wards. The amplification factor, σ, corresponds to the individual HIV-positive individual in the infectivity of HIV-negative concordant pairs.
The amplification factor, ψ, corresponds to HIV-positive concordant couples in the strength of infection for HIV serodiscordant married couples, indicating the greater chance that infectious individuals from the compartment, MP P, have to infect the seronegative partner in the serodiscordant married couple. compared to everyone. divisions other than, MN P,.
Feasible region
Existence of Equilibria
- Existence of equilibria
- Incorporating economic aspects
- Influence of prevalence and price on the demand for treatment
- The value function
- Price dependent deterministic demand functions
We also assume that everyone who is infected and receives treatment reduces the rate of HIV transmission in the population. Therefore, the market price of treatment also rises to a certain level, and then remains unchanged for any further increase in prevalence. If the prevalence of HIV decreases, the demand for treatment in the population also decreases.
As a result, the treatment market price drops to a certain level and then remains constant for any further decrease in incidence.
Summary
The demand for treatment depends on prevalence and market price, as stated in subsection (3.62), and both prevalence and market price are time dependent. Next, assume that the treatment rate as a function of demand increases significantly during the first few years of implementation and begins to increase at a decreasing rate and then remains constant over time. In Chapter 3, we analytically designed and solved submodels to understand the dynamics of the formation of serodiscordant married couples and HIV transmission in the subpopulation of single individuals and married couples.
In this chapter, simulations are based on the model where the treatment rate is a price- and prevalence-dependent demand function rather than a constant as in the sub-models in chapter 3.
Intervention strategies
Best intervention strategies
This strategy reduces the HIV transmission rate among the serodiscordant married couples and the HIV-concordant positive married couples compartments. The second effective strategy is the treatment of the HIV-positive single individuals, sp, and the serodiscordant couples, mnp. This strategy reduces the HIV transmission rate among the serodiscordant married couples and the HIV-positive single individuals.
This strategy reduces HIV transmission among the serodiscordant married couples and the HIV-concordant married couples.
Influence of α and β in the population
This study was motivated by concerns about high HIV transmission among serodiscordant married couples in the sub-Saharan region. We then combined the dynamics of the two submodels to formulate a complex model of single individuals and married couples. We found that six of the eight strategies were able to reduce transmission rates among the serodiscordant married couples.
Therefore, we recommend that one of the strategies, which is the treatment of serodiscordant married couples, be implemented in the sub-Saharan region.
Parameters and description
Descartes rule of signs on the characteristic equation (3.41)
Slope and concavity of the function ϕ (ω
Descartes rule of signs on the characteristic equation 3.86
State variables initial conditions
Parameter values
Intervention strategic scenarios
