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Modelling effective antiretroviral therapy that inhibits HIVproduction in the liver.Hasifa Nampala1,∗, Livingstone S. Luboobi1, Joseph Y.T. Mugisha1, Celestino Obua3, MatyldaJab lo´ nska2, Matti Heili¨ 1 Department of Mathematics, Makerere University, Kampala, Uganda2 Department of Mathematics and Physics, Lappeenranta University of Technology,Lappeenranta, Finland3 Department of Pharmacology and Therapeutics, Makerere University, Kampala, Uganda∗ E-mail: Corresponding hnampala@yahoo.co.uk CD4+ and hepatocytes cells, both found in the liver, support all stages that lead to HIV production.

Among people infected with HIV, liver disease has become the second leading cause of morbidity andmortality.

Considering HIV infection and replication in hepatocytes as well as CD4+ cells, a mathematical model was developed and analysed to investigate the ability of different combinational therapy to inhibitviral production in liver cells. Therapy efficacy in form of a dose-response function was incorporated.

Analysis of the model suggested that it is possible to have the effective reproductive number Re belowunity provided the therapy efficacy is more than 90%. Within some range of parameter values, Re canalso be reduced below unity at an efficacy of 50%.

Simulation results showed that combinational therapy of DDI, 3TC, ATV and NFV as the most effective while AZT, d4T, ATV and NFV as the least effective, in terms of inhibiting viral production.

The findings showed that this model can possibly be used to recognize which of the current treatmentprotocols perform best in controlling HIV replication in the liver.

Infectious diseases are the second leading cause of death among humans worldwide, and the number onecause of death in developing countries [?]. Among all other previous pandemic like cholera and influenza,the Human Immunodeficiency Virus (HIV) has for three decades socially and economically affected theworld and has claimed over 25 million lives [?].

Among people infected with HIV, liver disease has become the second most cause of morbidity and mortality [?]. Various research have revealed that liver disease can occur solely due to HIV infection[?, ?, ?, ?].

During HIV infection, the virus uses envelope glycoprotein 120 (gp120) to access entry into the host cell by binding on the CD4 receptor or a coreceptor on the host cell. The most coreceptors for HIV areC-X-C chemokine receptor type 4 (CXCR4) and C-X-C chemokine receptor type 5 (CXCR5) [?, ?, ?, ?].

Recent studies have revealed that HIV infection can occur in other cells other than CD4+ cells providedthose cells posses either of the coreceptors [?].

Human hepatocytes possess CXCR4 making them susceptible to HIV invasion and hence causing hepatocytes apoptosis by viral signaling through CXCR4 [?]. A study by Kong et al [?] found thatalthough there has been a number of contradictions regarding HIV replication in hepatocytes [?, ?], thecells support the first and last stages of HIV production. Virions produced by hepatocytes can infectother cells. However, Kong et al [?], revealed that replication in hepatocytes is low compared to viral replication in CD4+ cells. In addition to hepatocytes, HIV productively infects other hepatic cells andmacrophages, especially, kupffer cells [?, ?].

Since the introduction of antiretroviral therapy (ART) scientists have aimed at getting the drug that can limit HIV replication and hence reduce the viral load in the body. To date, no drug with 100%efficacy and ability to eradicate the virus from within HIV infected bodies has been found.

Mathematical models have been used to study within-host dynamics of HIV. Gumel et al [?] used a heaviside function to investigate the effects of intermittent IL-2 plus ART on the dynamics of HIV, afterusing therapy for 200 days. The findings showed that in spite of the combined effect of the theoreticalmaximum, ant-HIV cytotoxic T-lymphocytes (CTLs) action and 100% efficacies of therapy coupled withIL-2 therapy, the virus continues to persist.

Rong et al [?] included a combination of reverse transcriptase inhibitors (RTIs) and protease inhibitors (PIs) in a mathematical model of HIV infection with two stains of HIV. They assessed the progressionrate of exposed CD4+ cells (eclipse phase) back to uninfected stage and viral production on the evolutionof drug-resistant virus. They further investigated the evolution of drug resistant strains in the presenceof antiretroviral treatment and the range of drug efficacies under which drug-resistant strain will be ableto invade and out-compete the wild-type strain. Results showed that when the drug efficacy is not highenough to exert sufficient selective pressure (RTI efficacy of 0.5 and PI efficacy of 0.3), the resistant strainwill be unable to invade the established sensitive strain.

Arnaout et al [?] used a basic within host model as by Perelson and Nelson [?] to analyse HIV dynamics in vivo. He incorporated treatment as drug effectiveness parameter between 0 and 1 to assessthe dynamics of infection below and above the threshold efficacy. Results from model analysis showedthat if effectiveness is below a certain threshold (1%), viral load may bounce back after a transientreduction. They further deduced that if effectiveness is below but sufficiently near the threshold, viralload may still be reduced to quite low level.

Liver disease in HIV infected people who are not co-infected with viral hepatitis, has been linked to use of ART [?] because of the toxic nature of all classes of ART. However, in recent studies it has beenfound that HIV infection and replication in the liver cells can cause liver disease in HIV mono-infectedprior to initiation of ART [?, ?, ?]. Despite their unwanted effects, antiretroviral drugs have improvedthe long term outlook of HIV infected patients. A number of wit-in host mathematical models of HIVdynamics have focused on viral progression in CD4+ cells [?, ?, ?]. Since there is evidence that HIVinfect other cells, we therefore study the progression of HIV in hepatocytes when antiretroviral therapyis administered.

This study therefore intends to use a mathematical model coupled with numerical simulations to study the ability of individual drugs as well as recommended therapy combinations, to inhibit viral replicationin liver cells. Unlike most studies, [?,?,?,?], this study considers drug efficacy as a dose-response functionas recommended by Perelson and Deeks [?].

Model development Despite HIV's high affinity for CD4+ cells as compared to hepatocytes [?], and based on the existenceof CD4+ cells in the liver [?], the study assumes that when HIV infects the liver, the virus eitherinfects CD4+ cells or hepatocytes. Both CD4+ cells [?] and hepatocytes [?, ?] support all stages of viralproduction. Like many researchers who have modelled HIV dynamics in vivo [?, ?, ?, ?, ?], the studyconsiders CTLs killing of infected cells.

In the model formulation, we define the eight variables as follows: uninfected CD4+ (Tc), exposed CD4+ cells (Ec), infectious CD4+ cells (Ic), uninfected hepatocytes (Th), latently infected and not acti-vated hepatocytes (If ) [?], productively infected hepatocytes (Ia), HIV-specific cytotoxic T lymphocytes(L) and viral load (V ).

Model parameters are as follows: CD4+ cells and hepatocytes are produced from within the body at rates λ1 and λ2, and die naturally at rates b1 and b3 respectively. At infection, the virus infects targethepatocytes with probability q at rate β2 and target CD4+ with probability 1 − q at rate β1. WhenHIV enters a resting CD4+ cell, the RNA may not be completely reverse transcribed into DNA and theun-integrated virus cell may decay before reverse transcription [?]. This results in a proportion of exposedcells reverting to the uninfected state at a rate α. If reverse transcription takes place, the cell becomesinfectious at a rate π. This implies that if reverse transcription takes place in a period 1/α, where 1/α <1/π then the exposed cell will revert to uninfected state, otherwise it will proceed to the infectious state.

Infected CD4+ die at rate b2 where b2 > b1 and are cleared by HIV-specific CTLs at a rate k1.

When a hepatocyte is exposed to the virus, there is a probability p that it becomes productively infected (viral replication will take place after successful reverse transcription) and probability (1 − p)that the cell will be latently infected, such that there is no viral production until cell activation (the extentof stimulation of cellular processes initiated as a response to external stimuli reaching the cell). Latentlyinfected hepatocytes are activated to become productively infected at rate µ. Decay rates for productivehepatocytes and latently infected hepatocytes are b4 and b3, respectively, where b4 > b3 [?]. Productivelyinfected hepatocytes are killed by HIV-specific CTLs at rate k1 and, until activated, latently infectedhepatocytes will not trigger the action of CTLs. This study assumes that latently infected hepatocyteswill either get activated to become infectious or die. There is no possibility of them becoming uninfectedagain [?].

With or without any pathogen in the body CTLs proliferate naturally at rate x and in the presence of HIV infection, they proliferate at rate k2 proportional to the number of infectious cells; they are clearedat rate b5. HIV is produced by infectious CD4+ and productively infected hepatocytes at average rates s1and s2 per cell, respectively. In addition to CD4+ and hepatocytes, HIV productively infects other cellsand macrophages, like kupffer cells in the liver, [?, ?, ?]. These cells produce virions at rate m. Virionsdie naturally at rate b6.

There are three types of antiretroviral drugs currently used as therapy for HIV. Non-nucleoside re- verse transcriptase inhibitors (NNRTI), nucleoside reverse transcriptase inhibitors (NRTI) and proteaseinhibitors (PI). NNRTIs prevent the enzyme (reverse transcriptase) from converting RNA of HIV toDNA, thus the HIV will not multiply. NRTI latches onto the new strand of DNA that reverse transcrip-tase is trying to build and PIs prevent final assembly and completion of new HIV viruses within the cell,resulting in the infected cells producing noninfectious virus. However, this study considers only infectiousvirus.

Mathematical models have been used to try and address issues in HIV infection during antiretroviral therapy [?, ?, ?, ?]. Therapy efficacy has been modeled as a number between 0 and 1. There are howevera number of underlying dynamics especially the pharmacokinetics of the medication that influence thedrug efficacy. In recent research by Perelson and Deeks [?], it was asserted that "non-nucleoside reversetranscriptase inhibitors and protease inhibitors exhibit cooperative dose-response curves; a finding thathas implications for the treatment of HIV as well as other viral infections. The notion that a drug's doseand effect are related is a basic tenet of pharmacology and is generally summarized by an experimentallyderived dose-response curve. Determining the dose that gives 50% of the maximum response is one way toquantify the potency of a drug". This assertion followed an earlier research by Shen et al [?] who used aHill equation to describe the effectiveness of HIV medication. Perelson and Deeks [?] hence recommendedthat it would be of great contribution if the efficacy of ART would be modeled as a dose-response function.

[?] validated the importance of the dose-response curves in terms of predicting which medication works perfectly in inhibiting viral replication. They found that PIs have higher gradientsimplying better efficacies and they concluded that, this would explain why some times PIs alone couldbe effective in treating HIV.

However, in a typical dose-response relationship, the response to a dose depends on a number of factors including the dose administered, the frequency of dosing as well as the pharmacokinetics of theparticular drug. In this research, we have assumed that dose and rate of dosing will lead to a "steady state" dose response, that is, after administering a particular dose repeatedly, the drug concentrationwill reach a steady state. We have therefore assumed a steady effective therapeutic exposure to the drugbecause with infections such as HIV, "Steady-State" pharmacokinetics as opposed to initial or loadingdoses is more reliable for the effects of the treatment.

The study further assumes sufficient exposure to the drug (no under dosing or poor exposure due to use of poor or substandard drugs) thus ruling out the possibility of partial suppression which leads toselection pressure. The model also considers early stages of treatment where the infection is presumedto be sensitive to the drugs thus assuming drug resistance as being negligible.

Taking φ1 as the therapeutic response of reverse transcriptase inhibitors and φ2 as the therapeutic response of protease inhibitors, where 0 ≤ φ1, φ2 ≤ 1, the therapeutic response function is defined as aHill equation (1) to describe the effectiveness of the drug [?] where φ = φ1orφ2, d is variable drug dose concentration, IC50 is variable drug concentration that leads to50% of the maximal viral inhibition and m is variable gradient of the dose-response curve correspondingto individual drugs. The response in this case is the drug efficacy or ability to inhibit viral replication [?].

The gradients of the dose-response curves of HIV drugs are given by Shen [?].

This study assumes that reverse transcription in CD4+ cells does not occur immediately at infection [?]. So reverse transcriptase inhibitors (RTIs) reduce the rate of transfer of cells from exposed to infectiousclass (π). In hepatocytes, it is the infection rate that gets reduced by RTIs, because it is assumed thatat infection, reverse transcription takes place and then the cell will become latent or productive. It is forthis reason that a cell cannot become uninfected again like the case of CD4+ cells.

In hepatocytes however, it is assumed that in the latent class, reverse transcription has already taken place though the final stage has not yet been attained. Hence, if protease inhibitors are 100% effective,no latently infected cell will become productive, otherwise, some will become infectious depending on theefficacy of the drug (PIs). The study therefore assumes that PIs will reduce the rate of activation fromlatent to infectious (µ). It is further assumed that the effect of medication is translated generally intominimal viral load. Thus viral production from macrophages is also inhibited by both RTIs and PIs. Thecombined response of PIs and RTIs in macrophages is therefore (1 − φ1)(1 − φ2) [?].

From the assumptions and description above we have the following system of ordinary differential equations (2)-(9).

λ1 − (1 − q)β1TcV − b1Tc + αEc (1 − q)β1TcV − b1E − αEc − (1 − φ1)πEc (1 − φ1)πEc − b1Ic − b2Ic − k1IcL λ2 − (1 − φ1)qβ2ThV − b3Th (1 − φ1)(1 − p)qβ2ThV − b3If − (1 − φ2)µIf (1 − φ1)pqβ2ThV − b4Ia − k1IaL + (1 − φ2)µIf x + k2(Ic + Ia)L − b5L (1 − φ2)s1Ic + (1 − φ2)s2Ia + (1 − φ1)(1 − φ2)m − b6V The system of equations (2)-(9) settles to a a disease-free equilibrium point A0(Tc, E0, Ic, Th, Ia, If , L, V ) =(λ1/b1, 0, 0, λ2/b3, 0, 0, x/b5, 0). The effective reproduction number for the system (2)-(9), calculated usingthe next generation method as in [?] is (1 − φ1)(1 − φ2)b5s1πβ1(1 − q) b1b6(b1 + α + (1 − φ1)π)(k1x + b5(b1 + b2)) (1 − φ1)(1 − φ2)b5s1β1(1 − q) (b1 + α + (1 − φ1)π) b1b6(k1x + b5(b1 + b2)) b5(1 − φ1)(1 − φ2)s2qβ2λ2[p(b3 + (1 − φ2)µ) + (1 − p)(1 − φ2)µ] b3b6(b4b5 + k1x)(b3 + (1 − φ2)µ) b5(1 − φ1)(1 − φ2)2s2(1 − p)qµβ2λ2 b5(1 − φ1)(1 − φ2)s2pqβ2λ2 b3b6(b3 + (1 − φ2)µ)(b4b5 + k1x) Rf and Ra is the number of secondary infections from latently and productively infected hepatocytes respectively. Rc1 and Rc2 is the number of secondary infections produced by cells in the eclipse phase(exposed) and virus producing CD4+ cells respectively. Rc and Rh is the number of secondary infectionsproduced by one virus in CD4+ and hepatocyte, respectively. R0 is the total number of secondaryinfections in the liver. The total number of secondary infections is directly proportional to the clearancerate of CTLs and inversely proportional to the clearance rate of virions. Secondary infections in eithertype of cells largely depend on the drug efficacy. It can be seen that if the drug is 100% effective, thenthere is no secondary infections in either cell type.

Generally, the number of secondary infections (R0) is dependent on antigen-independent CTLs prolif- eration rate (x) and independent of antigen-dependant proliferation rate (k2). This indicates that if theCTLs are boosted prior to infection, then the body can handle infection better than when they proliferatein the presence of infection.

We then study the behaviour of the effective reproduction number for specific model parameter values as presented in Table 1. When every exposed hepatocyte becomes latently infected for some time beforeit is activated to produce virions, we have p = 0. Then with the activation rate µ = 0.019 and 50%effectiveness of both protease inhibitors and reverse transcriptase inhibitors and all other parameters asshown in Table 1, numerical simulations show that the effective reproductive number of hepatocytes canbe reduced below unity, as shown in Figure 1. However, if the probability p = 1, that is, every exposedcell becomes infectious at infection, then the number of secondary infections are greater than unity. Itcan also be seen that to keep Rh below unity, p should be less than 0.4093. Thus, it can be consideredimportant to increase drug efficacy as well as reducing activation rate of latently infected hepatocytesin order to reduce the basic reproductive number below unity. It is also shown in Figure 1 that withonly 30% of hepatocytes becoming productive at infection, the threshold activation rate below which thehepatocytes effective reproductive number is below unity is 0.0096.

Analysing the combined dependence of the effective reproductive number on p and µ at the same time, it can be seen from the right panel of Figure 2 that, there are multiple parameter value combinations forp and µ at which the effective reproductive number is unity. That is, with the probability of a hepatocytebecoming productive at infection less than 0.6, corresponding activation rate µ lower than 0.019, therapy efficacy of 50% and all other parameters as stated in Table 1, HIV infection in hepatocytes can possiblybe managed.

Considering CD4+ cells, if the rate of transfer of exposed to infectious stage is π = 0.23 and the therapy is 50% effective, then the effective reproductive number is below unity when the probability pis above 0.9266, as presented in Figure 3. In other words it could be possible to manage HIV infectionin CD4+ cells given the parameter values given in Table 1 and if almost all HIV infect hepatocytescells. However, it has been stated that HIV has higher affinity for CD4+ cells than hepatocytes [?]. Wetherefore assume a probability of 0.8 that HIV infects a CD4+ as shown in the right panel of Figure3. The basic reproductive number is seen to be below unity given the rate of transfer from exposed toinfectious CD4+ cells is below 0.019.

The range of values of q and π that will give the effective reproductive number below unity are as shown in Figure 4. This suggests a possibility to manage HIV in CD4+ cells given the range of parametervalues shown in Figure 4 and Table 1 with therapy efficacy of 50%.

In all the previous simulations, the therapy efficacy has been fixed at 50% for both drug classes.

However, medically it is not the case that all classes of ART are 50% effective. We therefore investigatedrug efficacy that leads the effective reproductive number (R0) of liver cells less than unity. Figure 5shows that, given µ = 0.0096, p = 0.4093, q = 0.9266 and π = 0.019, it is possible to have R0 < 1provided the therapy efficacies are greater than 90%.

In Figure 6 we study the dependence of the effective reproduction number R0 on the infection rates β1 and β2. Apparently, the infection might not proceed to endemic state if drug efficacies are fixed at50% given the infection rates β1 < 0.0015 and β2 < 0.00015 for CD4+ cells are hepatocytes respectively.

This raises a big challenge given that research has revealed infection rates as high as 0.005 [?].

Numerical simulations In this section we present numerical simulations of the model equations (2)-(9) proposed in this work.

The dynamics of infection are first considered when there is no therapy. This is shown in Figure 7. Theviral load (V ) grows steeply in the first days, leading to increased number of latently infected CD4+ (Ec),productively infected CD4+ (Ic), latently infected hepatocytes (If ) and productively infected hepatocytes(Ia). This results in a clear drop in the numbers of uninfected cells, both CD4+ (Tc) and hepatocytes(Th). That significant decrease takes place within the first day of infection and it can be seen that mostof those previously uninfected cells start to contribute to all the classes of infected cells. Following theprogression of the infection, there is a significant response of HIV-specific CTLs to infection at a ratek2 as shown in equation (8). That helps the liver to reduce the viral population but cannot eliminateit completely. As we have seen in the R0 analysis in Figure 5, the number of secondary infections willalways be greater than unity when the drug efficacies are φ1 = φ2 = 0. The graphs show that the infectionis apparently destructive to the liver without any medical intervention. Numerical simulations show adiscrepancy between equilibrium population ratios of hepatocytes and CD4+ cells prior to therapy andtheir physiological concentrations in the liver. This could be due to high concentration of hepatocytes ascompared to CD4+ cells yet HIV production is highest the latter.

Analysing the model with therapeutic effect of the drugs (equations (2)-(9)), the study considers medication as listed in Tables 2-4. The sampled drugs under study are representatives of all classes ofART currently used as medication for HIV, namely, NRTIs, NNRTIs and PIs. Tables 2-4 present allthe parameters which were used in calculating of drug efficacy as shown in equation (1). All doses areexpressed as concentrations in moles per liter.

Antiretroviral treatments are always used in combinations of three or four drugs from specific classes.

However, we first simulate the infection dynamics when each drug is administered individually with itsusual dose. The goal is to verify how every individual drug is able to inhibit viral production and hencereduce the viral load. Figure 8 depicts the dynamics. The first immediate observation is that the infection level reduces when either drug is used. However, the severity of the influence varies significantly fromone medicine to another. Apparently, a drug that reduces the number of infected CD4+ most effectivelydoes not perform equally well in heapatocytes. The most distinct aspects of infection dynamics are thetime delay before the infection peaks and the maximum level reached by the infection.

In particular, the study considers the detailed Figures 9, 10 and 11 for Ic, Ia and V respectively for individual drugs. If drug efficacy is measured by reduction in the number of productively infectedhepatocytes or the reduction in viral population, then atazanavir (ATV) is clearly the best performingdrug. However, it actually reduces the number of productively infected CD4+ the least of all. On theother hand, stavudine (d4T) provides the least improvement in productive hepatocytes and viral load.

It is only the CD4+ that benefit most from the use of d4T.

Of all medications considered in this study, ATV is clearly the treatment which is capable of delaying and dampening the peak of infection. Considering 2007 World Health Organisation recommendationsof using ART, that is, two NRTI and one NNRTI drug (2NRTI+1NNRTI) or two NRTI and two PIdrugs (2NRTI+2PI). The study presents simulation results when the aforementioned combinations areconsidered. Out of the drugs used in the study, as shown in Tables 2-4, we obtain six different pairs ofNRTI drugs combined with a single NNRTI drug or with the two PI drugs. Figure 12 shows the infectiondynamics for the 2NRTI+1NNRTI combinations.

Combinations have higher efficacy than each single drug on its own. The number of uninfected cells remain at higher levels when combinations are used as compared to single drug. Consequently, the numberof infected cells and viral populations are reduced more with combinations than with single drugs. Inmost of the cell types it is visible that the best combination in all aspects is DDI+3TC+EFV, whereasthe worst one is AZT+d4T+EFV.

We now consider the drug combinations of 2NRTI+2PI. As presented in Figure 13, these options are even more efficient in infection reduction. This is consistent with the previous simulations of individualdrugs that revealed how ATV is the best among the considered drugs, in viral reduction. When combinedwith another PI drug and two more drugs from NRTI class, ATV proves strongest of all treatmentsstudied in this paper. Simulation results show that DDI+3TC+ATV+NFV is the best combination andAZT+d4T+ATV+NFV is the worst.

We finally analyse the dynamics of the variables when the best medication in the previous simulations is used (ATV). This is considered in a situation while more (p = 0.8) and less (p = 0.3) hepatocytes becomeproductive at infection as shown in Figure 14. When more hepatocytes become productive at infection ascompared to latency (p = 0.8), viral load will peak in the first 10 days. Consequently, uninfected CD4+cells and hepatocytes will become fewer due to viral production from the many productive hepatocytes.

However, in the long run the infected CD4+ cells will become more when fewer hepatocytes are productiveat infection, as compared to latency (p = 0.3). This could be explained by earlier findings in this workthat the use of ATV leads to most infectious CD4+ cells. On the other hand, even when we consider thebest combination in the previous simulations and use it, Figure 15 shows that there is not much differencewhen less infected cells become latently infected during combination therapy. The significant differenceis only in Ia and If . This shows that during HIV infection, whether a hepatocyte becomes productive atinfection or latent, it does not create much difference in infectious CD4+ cell. This could be so becauseit is the CD4+ cells that produce the biggest number of virions in the liver. This could be consistentwith one dilemma perturbing researchers in the field of HIV medication, 'the virus's ability to harbourin cells for a long time'.

The aim of this study was to understand HIV infection dynamics in the liver with administration ofantiretroviral therapy. The work was based on a number of biological facts as well as feasible assump-tions. HIV dynamics in the liver were analysed based on three main cell types; CD4+, hepatocytes and HIV-specific T-lymphocytes. Nevertheless, an aggregated effect of infection in macrophages was also con-sidered. In CD4+ cells, reverse transcription was considered not to occur immediately at infection. Thesame consideration was taken for hepatocytes, though unlike CD4+ cells that can return to uninfectedstate , hepatocytes could only die or proceed to infectious state after being exposed to the virus.

The model proposed in this study was formulated as a system of ordinary differential equations with eight variables. Drug efficacy was considered as a dose-response function with parameters obtained exper-imentally in pharmaceutical studies [?, ?]. HIV therapy included the three classes of enzyme inhibitors,namely, NRTIs, NNRTIs and PIs.

Analysis of the model's basic reproduction number revealed that the key parameters to control the infection are: p – the probability that at infection, a hepatocyte becomes productively infected, q – theprobability that HIV infects hepatocyte and not CD4+, µ – the rate at which latently infected hepatocytesbecome productive, and π – the rate at which exposed CD4+ become infectious. In particular, consideringall the other parameters as shown in Table 1 and fixing each drug efficacy at 50% it was revealed thatin order to possibly keep the number of secondary infections below unity, the crucial parameters need tosatisfy the conditions, p < 0.4093, q > 0.9266, µ < 0.0096 and π < 0.019. However, most of the knownvalues of those parameters are significantly outside the these limits [?, ?]. The effective reproductionnumber was found to be below unity only when the infection rates for CD4+ cells and hepatocytes wererespectively β1 < 0.0015 and β2 < 0.00015, whereas these values are sometimes suggested as high as0.005 [?]. Finally, with all the parameters fixed at their theoretical values it was seen that strict controlof the infection may be possible when the drug efficacy of either therapy exceeds 90%.

The model was used to simulate infection progression scenarios in the cases of no medical treatment, single drug administration and full ART combinations. Results showed that all enzyme inhibitors in HIVinfection significantly reduced viral load. The ability to inhibit was significantly higher when combinationtherapy was used as opposed to single drug.

Simulations results further suggested that atazanavir (ATV) was possibly the best single drug and Stavudine (4dT) the worst in terms of viral load reduction. This is consistent with the claims that proteaseinhibitors are more effective than reverse transcriptase inhibitors in terms of viral load reduction. Amongthe considered full ART combinations, with effectiveness measured in terms of reducing the viral loadas well as the number of infectious hepatocytes, then DDI+3TC+ATV+NFV proved to be possibly thebest option and AZT+d4T+ATV+NFV the worst. This was however not consistent with with reducingthe number of infectious CD4+ cells. Simulation results also suggested that it is not of any advantagehaving more hepatocytes becoming latent than infectious at the point of infection because in the longrun the implications is not significantly different.

We therefore conclude that mathematical modelling creates hints that can be used as basis for under- standing the dynamics of HIV in the liver during antiretroviral therapy. The model used in this study,under the assumptions mentioned, suggested a possible tool for proposing the best and worst antretroviralcombinations for reduction of viral load in the liver. We therefore recommend that this approach couldbe improved and used to optimize antiretroviral combinations.

The authors would like to acknowledge the Sida/SAREC bilateral research cooperation programme ofMakerere University for funding this research. We thank the Center for International Mobility (CIMO)Finland for funding research visits at Lappeenranta University of Technology.

Figure 1. Basic reproduction number Rh of hepatocytes. The number is calculated withvarying parameter p and fixed µ = 0.006 (left panel) and varied µ with fixed p = 0.3 (right panel), withall the other parameters as given in Table 1 and the drug efficacies assumed as φ1 = 0.5 and φ2 = 0.5.

Figure 2. Basic reproduction number Rh of hepatocytes (left panel) and its correspondinglevel lines (right panel). The number is calculated with varying parameters p and µ, and all theother parameters as given in Table 1 and the drug efficacies assumed as φ1 = 0.5 and φ2 = 0.5.

Figure 3. Basic reproduction number Rc of CD4+ cells. The number is calculated with variedparameter q and fixed π = 0.23 (left panel) and varied π with fixed q = 0.2 (right panel), with all theother parameters given in Table 1 and drug efficacies assumed as φ1 = 0.5 and φ2 = 0.5.

Figure 4. Basic reproduction number Rc of CD4+ cells (left panel) and its correspondinglevel lines (right panel). The number is calculated with varying parameters q and π, with all theother parameters as given in Table 1 and drug efficacies assumed as φ1 = 0.5 and φ2 = 0.5.

Figure 5. Basic reproduction number R0 (left panel) and its corresponding level lines(right panel). The number is calculated with varying drug efficacies φ1 and φ2, with values of p, µ, q,π and m as optimized with respect to Rh = 1 and Rc = 1, and with all the other parameters as given inTable 1.

Figure 6. Basic reproduction number R0 (left panel) and its corresponding level lines(right panel). The number is calculated with varying drug infection rates β1 and β2, with values of p,µ, q and π as optimized with respect to Rh = 1 and Rc = 1, and with all the other parameters as givenin Table 1.

Figure 7. Dynamics of HIV mono-infection in the liver with no medical treatment. Verticalaxes represent the variables and horizontal axes are time in days. Parameter values are as indicated inTable 1.

Figure 8. Dynamics of HIV mono-infection in the liver on single drug therapy. Vertical axesrepresent the variables and horizontal axes are time in days. Parameter values are as indicated in Table1.

Figure 9. Productively infected CD4+ in the liver in HIV mono-infection on single drugtherapy. Vertical axis representa the Ic variable and the horizontal axis is time in days. Parametervalues are as indicated in Table 1.

Figure 10. Productively infected hepatocytes in the liver in HIV mono-infection on singledrug therapy. Vertical axis represents the Ia variable and the horizontal axis is time in days.

Parameter values are as indicated in Table 1.

Figure 11. Viral load in the liver in HIV mono-infection on single drug therapy. Verticalaxis represents the V variable and the horizontal axis is time in days. Parameter values are as indicatedin Table 1.

Figure 12. HIV mono-infection dynamics in the liver on 2NRTI+1NNRTI combinationtherapy. Vertical axes represent the variables and the horizontal axes are time in days. Parametervalues are as indicated in Table 1.

Figure 13. HIV mono-infection dynamics in the liver on 2NRTI+2PI combinationtherapy. Vertical axes represent the variables and the horizontal axes are time in days. Parametervalues are as indicated in Table 1.

Figure 14. Infection dynamics with the best single drug therapy. The simulation is done withvarying probability that a hepatocyte gets infectious at infection. Vertical axes represent the variablesand the horizontal axes are time in days. Parameter values are as indicated in Table 1.

Figure 15. Infection dynamics with the best combination therapy. The simulation is donewith varying probability that a hepatocyte gets infectious at infection. Vertical axes represent thevariables and the horizontal axes are time in days. Parameter values are as indicated in Table 1.

Table 1. Parameters for the basic model of HIV in the liver.

rate of creation of CD4+ from within the body natural death rate of uninfected CD4+ probability that HIV infects hepatocytes probability that at infection, hepatocyte becomes productively infected rate at which latently infected hepatocytes become rate of transmission of HIV in CD4+ antigen-independent CTLs proliferation rate antigen-dependent proliferation rate of CTLs rate at which exposed CD4+ become uninfected rate at which exposed CD4+ become infectious death rate of infected CD4+ due to infection rate at which CTLs kill infected CD4+ and hepato- rate of creation of hepatocytes from within the body natural death rate of hepatocytes rate of transmission of HIV in hepatocytes death rate of hepatocytes due to infection rate of clearance of CTLS by all means average rate of production of virions by an infected average rate of production of virions by an infected death rate of HIV rate of production of virions from macroghages The time in this study is considered in days and, therefore, all the rates are per day.

Table 2. Example NRTI medications used in antiretroviral therapy, and their parameters.

Table 3. Example NNRTI medication used in antiretroviral therapy, and its parameters.

Table 4. Example PI medications used in antiretroviral therapy, and their parameters.

Source: http://personal.lut.fi/wiki/lib/exe/fetch.php/en/jablonska/plos_model2_revision1.pdf

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Origin and of Teleosts Honoring Gloria Arratia Joseph S. Nelson, Hans-Peter Schultze & Mark V. H. Wilson (editors) More advanced teleosts stem-based Verlag Dr. Friedrich Pfeil • München Acknowledgments . Gloria Arratia's contribution to our understanding of lower teleostean phylogeny and classifi cation – Joseph S. Nelson .

### thebsa.org.uk

I am sorry to hear you have been troubled by symptoms of imbalance and dizziness. Your query appears to contain two elements: 1) dizziness and imbalance related to the time of year, and 2) complementary therapies or supplements for dizziness and imbalance. I will cover these in two sections below. I admit I do not routinely provide much advice in these areas, and don't think other clinicians allied to a more 'medical' model commonly would either. I think this is because there is not enough evidence to back up any recommendations, although I would hope anyone would be open minded to the possibility of stronger evidence becoming available. I have looked at the scientific literature for you, in case I have been missing something. Both of these areas appear to be poorly understood and controversial. The scientific research proves confusing and contradictory, but I will attempt to summarise what I have discovered. Some of the difficulty in carrying out research in these areas relates to different definitions of dizziness, imbalance and vertigo. Often people use the term dizziness to cover feeling light-headed, woozy, giddy, floaty, or unsteady. Vertigo is a term more often used by clinicians and scientists with a stricter definition of an illusion of movement of one's self or the environment, often spinning or rotational sensations. Individuals often mean very different things when they say they are dizzy, and research studies have often defined and categorised dizziness differently when you compare them. Patients get allocated into treatment groups in the studies in quite varying ways. Dizziness and vertigo can be related to many different underlying conditions and it is likely that these conditions need to be considered separately in research studies, but this is often not the case. Finding effective treatments for specific conditions can be compromised when patients are grouped in research studies into a general ‘dizzy patient' category. Vertigo is often the subject of research studies rather than dizziness as vertigo is often considered a defining feature of vestibular disorders (conditions related specifically to the vestibular or balance system, including the vestibular or balance organs in the inner ear), which it is sometimes possible to more clearly define. Any understanding of vertigo might not relate to symptoms of dizziness or imbalance though. I have put my overall summary of these areas first so that you might obtain some quick guidance. More specific and detailed information is then provided if you wish to read further, but these sections are denser to read. In summary There are relatively small numbers of studies into both the seasonality of dizziness and balance disorders, and complementary medicine and therapy in dizziness and imbalance. The studies tend to present conflicting results. Those that have been carried out often use unsatisfactory research methods, use small numbers of subjects, and are liable to publications bias (for example, studies published by the manufacturers of supplements). There is a distinct lack of the ‘gold standard' randomised, double-blind, controlled studies. If you are experiencing seasonal symptoms it is difficult to advise how to manage this as it is difficult to assert any control over the seasons, the weather, or barometric changes. If you are able to determine any triggers that are controllable or avoidable then that would be recommended. Hain (2015) advises to treat any allergy or migraine triggers appropriately, and to try any appropriate treatment prior to the anticipated onset of symptoms. If someone has migraine then we would expect there might be triggers that cause symptoms. Being able to identify triggers might help someone determine that they suffer from migraine. It is important to note that an individual can have migraine or vestibular migraine without any headache, which can mean this diagnosis can be missed. VEDA (2015) advice to patients is to avoid anxiety, as we know this can potentially make any symptoms of dizziness worse, and to educate themselves. I have attached a leaflet from the Meniere's Society that outlines some possible causes of dizziness and imbalance which might help you to decide whether any of the conditions fit with your symptoms. I have also attached a leaflet on