A Multi-Scale Model for the Spread of HIV in a Population Considering the Immune Status of People

Round 1

Reviewer 1 Report

Vásquez-Quintero et al. propose a multiscale model to study the propagation dynamics of the HIV in a group of young people between 15 and 24 years of age through sexual contact. The model takes into account the immune status of each individual. It also considers the adherence and nonadherence of antiretroviral therapy (ART) and therapeutic failure.

The paper talks a lot about various mathematical modelling but as a reader I fail to get the central hypothesis and the results of the study. The authors should clearly mention what the study adds to the available literature and give valid references wherever applicable.

The tile of the article should also be re-written. I leave it to the authors find a better tittle which gives a better glimpse of what the article is about.  For example, something like, “A multiscale mathematical model to study HIV propagation in a population group considering infection in the immune system of each infected individual…” or “Studying HIV propagation in a population group considering infection in the immune system of each infected individual...”

Moreover, there are numerous errors in English and grammar, few examples are mentioned below:

Line 21 …not only does ‘it’ increase…

Line 25 Once the virus enters the person’s ‘body’, it is…

Line 40 ‘In’ 2019, ‘out’ of the 38 million people

Line 47 This is the case of [13] that studies: This sentence does not make sense

Line 67-68 …active cytotoxic cells of eliminate infected CD4 T cells.

The Introduction section must be improved, for example:

Line 25-26 Once the virus enters the person’s organism, it is recognized by the body’s defense systems and starts the infection process. This line should be rewritten. The impression should be how the virus evades the body’s defense system and starts the infection process.

Line 41 Unfortunately, not everyone manages to have access to ART because they do not know their serological status; more so, many factors cause people, in spite of using ART, to abandon it, among the known factors, there are the high costs of medications, side effects, and non-adherence to therapy [12].

Comment: Is the author talking about a Global scenario or a geo-specific one. This should be clearly told. If it’s a global one, it should be broken down and discussed country/area-wise, based on its prevalence in these countries. For example, sub-Saharan Africa with more than two-thirds of HIV patients is the hardest hit region globally followed by Asia and the Pacific.

Finally, the conclusion section should also be re-framed making it easier for also the readers who come from a non-mathematical background. The main and relevant findings should be discussed point-wise.

Author Response

Reviewer 1

The authors thank the Reviewer 1 for the suggestions, which allowed us to make a deep review of the way in which our findings are presented and to accordingly restructure the manuscript, in order to make it more accessible to the audience of the journal.

Comment: The paper talks a lot about various mathematical modelling but as a reader I fail to get the central hypothesis and the results of the study. The authors should clearly mention what the study adds to the available literature and give valid references wherever applicable.

Answer: A text has been included in the abstract that describes the central hypothesis and some results, it also comments on what the authors consider contributions to the literature. Additionally, the last paragraph of the conclusions has been rewritten to consider the reviewer's comment.

Comment: The tile of the article should also be re-written. I leave it to the authors find a better tittle which gives a better glimpse of what the article is about. For example, something like, “A multiscale mathematical model to study HIV propagation in a population group considering infection in the immune system of each infected individual...” or “Studying HIV propagation in a population group considering infection in the immune system of each infected individual...”

 Answer: We propose to modify the title of the article to read as follows: "A multi-scale model for the spread of HIV in a population considering the immune status of people"

Comment: Line 41 Unfortunately, not everyone manages to have access to ART because they do not know their serological status; more so, many factors cause people, in spite of using ART, to abandon it, among the known factors, there are the high costs of medications, side effects, and non-adherence to therapy [12].

 Is the author talking about a Global scenario or a geo-specific one. This should be clearly told. If it’s a global one, it should be broken down and discussed country/area- wise, based on its prevalence in these countries. For example, sub-Saharan Africa with more than two-thirds of HIV patients is the hardest hit region globally followed by Asia and the Pacific.

Answer: The paragraph has been modified to include global and updated data on the number of infected people, with and without treatment. Particularly, it comments on the importance of considering access to ART and therapeutic failure in the development of mathematical models. Recent citations from UNAIDS are presented.

Comment: Finally, the conclusion section should also be re-framed making it easier for also the readers who come from a non- mathematical background. The main and relevant findings should be discussed point-wise.

Answer: The conclusions section has been revised and rewritten to use less technical language that allows a better understanding by non-mathematical professionals.

Reviewer 2 Report

In this work, the authors proposed a dynamic model to study the spread of the Human Immunodeficiency Virus (HIV) within a group of people. They considered the use of antiretroviral therapy (ART) and therapeutic failure. The spread was simulated over  a scale-free complex network.

In my opinion, the mathematical setting and the epidemic simulations are well postulated. However, the authors should justify more in details the choice of a scale-free topology and say why they did not consider for example the small-world topology which is more suitable for the characterisation of human social interactions.

Author Response

Reviewer 2

The authors thank the number 2 reviewer for his time and the valuable contributions he makes. Without a doubt he has allowed us to improve the model formulation.

Comment: In this work, the authors proposed a dynamic model to study the spread of the Human Immunodeficiency Virus (HIV) within a group of people. They considered the use of antiretroviral therapy (ART) and therapeutic failure. The spread was simulated over a scale-free complex network.

In my opinion, the mathematical setting and the epidemic simulations are well postulated. However, the authors should justify more in details the choice of a scale-free topology and say why they did not consider for example the small-world topology which is more suitable for the characterisation of human social interactions.

Answer: The network description has been supplemented with information that clarifies the reviewer's comment and new references have been included to support what was said. In summary, the following is explained: It should be noted that it is considered a complex network free of a Cluster type scale because it is the one that best adjusts to the phenomena that occur in nature and in everyday life, also due to its high heterogeneity, which means that there are nodes with very high few connections, moderately connected nodes and extremely connected nodes; It is not considered a small world model since it is a homogeneous network in which all nodes have approximately the same number of links and what is sought is to achieve a more realistic situation, therefore it must be considered that in our populations there will be people with a greater number of sexual partners than others.

Reviewer 3 Report

The paper present well justified and generally well told story showing the anaalysis of spread of virus-related disease in the population including some micro-biological parameters. The population is modeled by some form of network and the development of the disease - by the system of ODE. The model is interesting and I generally support their publication in Processes. I would like to mention however some objections, which could in my opinion improve the paper.

1. Almost every plot in Figures 2-6 is characterized by oscillations. What is the source of these oscillations? 

2. Once more Figs 2-6. It seems that, after some time, the sum M+m^{*} stabilizes (for every case). I think that for people who are less familiar with biology of the process some explanation of this fact could be useful.

3. Once more Figs 2-6. For every figure, plots for V and T^{*} are colinear. Is it the expected effect?

4. This is very important question. Authors recall R_0 value as related to cells. But the definition relates it to individuals and the population understood as the community of people (see e.g. doi:10.3201/eid2501.171901). Also the use of adjective "basic" can be disputable and often leads to misunderstandings. I think that better substantiation of this part would be valuable. Especially due to the fact that R_0 value is always mentioned in captions of figs.2-6.

5. Authors state that they have the Holm-Kim modification of Barabasi-Albert type of network. I consider the descrition of this network unclear. Let us take a look at line 97 "\gamma is the partners’ distribution". How should we understand this sentence? \gamma is a value, which describes the distribution but not the distribution itself.

6. Following the former comment, I would like to pose some problems. What is the reality of the model? What is the maximum nauber of k for network with N=1000? I can imagine that such hubs can be adequate to social networks, but for networks with sexual contacts? Especially, that we read (line 455) "49% of the population is HIV carrier"

Finally, Authors should check their text where some spanish words srill exist (Cuadro, Figura).

Author Response

Reviewer 3

The authors thank the reviewer number 3 for his valuable comments, which have allowed us to clarify to potential readers some relevant aspects in the interpretation of results, as well as the approach of the network itself.

Comment: Almost every plot in Figures 2-6 is characterized by oscillations. What is the source of these oscillations?

Answer: The response to this comment has been considered in the text interpreting the figures and is given in the following terms. The oscillations that can be observed in each of the graphs can be explained from the mathematical point of view: when the local stability of the non-trivial equilibrium point is studied (in the presence of infection), the existence of complex eigenvalues ​​is expected. If it were the objective of the present work, it is possible to study and demonstrate the existence of interesting bifurcations when these values ​​are made pure imaginary. From a biological point of view, it can be considered that when the infection begins, the body loses healthy CD4 T cells that become infected, so that infectious viral particles also increase, then the immune system tries to recover by activating the cellular response (that is why the oscillations in the graphs of M and M * which are the inactive and activated CD8 T cells, respectively) and again increase their levels of healthy CD4 T cells and decrease the infected CD4 T cells and infectious viral particles, it is like this how a constant fight between the immune system and the virus is triggered until after a certain time, which in the simulations is noticeable after approximately 200 days, the values ​​of all cells and viral particles are stabilized; the same occurs in the simulations carried out below.

Comment: Once more Figs 2-6. It seems that, after some time, the sum M+m^{*} stabilizes (for every case). I think that for people who are less familiar with biology of the process some explanation of this fact could be useful.

Answer: we consider that there are different biological reasons why viral cells and particles stabilize after a certain time, one reason is that each of the populations involved in the model will finally reach the analytically determined equilibrium values; that is, regardless of the nature of the system’s transient, populations cannot grow indefinitely, and particularly T and M cannot be completely vanished, even in the absence of infection. This makes biological sense to the system and is analytically proven in paper.

Comment: Authors state that they have the Holm-Kim modification of Barabasi-Albert type of network. I consider the description of this network unclear. Let us take a look at line 97 "\gamma is the partners’ distribution". How should we understand this sentence? \gamma is a value, which describes the distribution but not the distribution itself.

Answer: The paragraph has been supplemented with information that clarifies the reviewer's comment and new references have been included to support what was said. In summary, the following is explained: The model considers a constant and finite population $N$, hence, it does not consider new young people entering the network, nor their deaths. A cluster scale-free network is generated (according to the Barabási–Albert model), being scale-free means that there is no preferred scale or preferred number of edges for each vertex with respect to the rest, in which the probability P(k,\gamma) of a vertex in the network interacting with other vertices decays through a power law P(k,\gamma)\ ~k^(-\gamma), where  k  is the average number of sexual partners and  \gamma is related to the distribution of the partners, it is a parameter whose value depends on the specific type of network; more precisely,  \gamma  modulates the probability  P (k, gamma), since for values greater than  \gamma,  P (k, gamma)  decreases faster.

Comment: Following the former comment, I would like to pose some problems. What is the reality of the model? What is the maximum number of k for network with N=1000? I can imagine that such hubs can be adequate to social networks, but for networks with sexual contacts? Especially, that we read (line 455) "49% of the population is HIV carrier".

Answer: The goal of this paper is not to illustrate a real scenario, applicable to a specific community, but to illustrate a methodology that we consider useful for this type of study - the coupling of the population complex network with the model in ODEs for the immunological state -; In this sense, the simulated scenarios are only illustrative and for this purpose the values of the parameters are selected based on the literature and some of them are ad hoc, as clarified in the document. It should be noted that the highest value used for k, the average number of sexual partners, is 10: since a population of 1,000 young people between 15 and 24 years old is considered, in a simulation time of 200 weeks (approximately 4 years), therefore it is not unreasonable to think that a person from a slightly liberal population could have an average of 10 sexual partners in a 4-year period.